Stochastic Resonance

Stochastic resonance

Luca Gammaitoni Dipartimento di Fisica, Universita` di Perugia, and Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, VIRGO-Project, I-06100 Perugia, Italy

Peter Ha¨nggi Institut fu¨r Physik, Universita¨t Augsburg, Lehrstuhl fu¨r Theoretische Physik I, D-86135 Augsburg, Germany

Peter Jung Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701

Fabio Marchesoni Department of Physics, University of Illinois, Urbana, Illinois 61801 and Istituto Nazionale di Fisica della Materia, Universita` di Camerino, I-62032 Camerino, Italy

Over the last two decades, stochastic resonance has continuously attracted considerable attention. The term is given to a phenomenon that is manifest in nonlinear systems whereby generally feeble input information (such as a weak signal) can be be amplified and optimized by the assistance of noise. The effect requires three basic ingredients: (i) an energetic activation barrier or, more generally, a form of threshold; (ii) a weak coherent input (such as a periodic signal); (iii) a source of noise that is inherent in the system, or that adds to the coherent input. Given these features, the response of the system undergoes resonance-like behavior as a function of the noise level; hence the name stochastic resonance. The underlying mechanism is fairly simple and robust. As a consequence, stochastic resonance has been observed in a large variety of systems, including bistable ring lasers, semiconductor devices, chemical reactions, and mechanoreceptor cells in the tail fan of a crayfish. In this paper, the authors report, interpret, and extend much of the current understanding of the theory and physics of stochastic resonance. They introduce the readers to the basic features of stochastic resonance and its recent history. Definitions of the characteristic quantities that are important to quantify stochastic resonance, together with the most important tools necessary to actually compute those quantities, are presented. The essence of classical stochastic resonance theory is presented, and important applications of stochastic resonance in nonlinear optics, solid state devices, and neurophysiology are described and put into context with stochastic resonance theory. More elaborate and recent developments of stochastic resonance theory are discussed, ranging from fundamental quantum properties—being important at low temperatures—over spatiotemporal aspects in spatially distributed systems, to realizations in chaotic maps. In conclusion the authors summarize the achievements and attempt to indicate the most promising areas for future research in theory and experiment. [S0034-6861(98)00101-9]

CONTENTS D. Weak-noise limit of stochastic resonance—power spectra 244 V. Applications 246 I. Introduction 224 A. Optical systems 246 II. Characterization of Stochastic Resonance 226 1. Bistable ring laser 246 A. A generic model 226 2. Lasers with saturable absorbers 248 1. The periodic response 226 2. Signal-to-noise ratio 228 3. Model for absorptive optical bistability 248 B. Residence-time distribution 229 4. Thermally induced optical bistability in 1. Level crossings 229 semiconductors 249 2. Input-output synchronization 230 5. Optical trap 250 C. Tools 231 B. Electronic and magnetic systems 251 1. Digital simulations 231 1. Analog electronic simulators 251 2. Analog simulations 231 2. Electron paramagnetic resonance 253 3. Experiments 231 3. Superconducting quantum interference III. Two-State Model 232 devices 253 IV. Continuous Bistable Systems 234 C. Neuronal systems 254 A. Fokker-Planck description 234 1. Neurophysiological background 254 1. Floquet approach 234 2. Stochastic resonance, interspike interval B. Linear-response theory 236 histograms, and neural response to periodic 1. Intrawell versus interwell motion 238 stimuli 255 2. Role of asymmetry 239 3. Neuron firing and Poissonian spike trains 257 3. Phase lag 240 4. Integrate-and-fire models 259 C. Residence-time distributions 240 5. Neuron firing and threshold crossing 260

VI. Stochastic Resonance—Carried On 261 A. Quantum stochastic resonance 261 1. Quantum corrections to stochastic resonance 262 2. Quantum stochastic resonance in the deep cold 263 B. Stochastic resonance in spatially extended systems 267 1. Global synchronization of a bistable string 267 2. Spatiotemporal stochastic resonance in excitable media 268 C. Stochastic resonance, chaos, and crisis 270 D. Effects of noise color 272 VII. Sundry Topics 274 A. Devices 274 1. Stochastic resonance and the dithering effect 274 B. Stochastic resonance in coupled systems 274 1. Two coupled bistable systems 274 2. Collective response in globally coupled bistable systems 275 3. Globally coupled neuron models 275 C. Miscellaneous topics on stochastic resonance 275 1. Multiplicative stochastic resonance 275 2. Resonant crossing 276 3. Aperiodic stochastic resonance 277 D. Stochastic resonance—related topics 277 1. Noise-induced resonances 277 2. Periodically rocked molecular motors 278 3. Escape rates in periodically driven systems 279 FIG. 1. Stochastic resonance in a symmetric double well. (a) VIII. Conclusions and Outlook 279 Sketch of the double-well potential V(x)ϭ(1/4)bx4 Acknowledgments 281 2 Ϫ(1/2)ax . The minima are located at Ϯxm , where Appendix: Perturbation Theory 281 1/2 xmϭ(a/b) . These are separated by a potential barrier with References 283 the height given by ⌬Vϭa2/(4b). The barrier top is located at xbϭ0. In the presence of periodic driving, the double-well potential V(x,t)ϭV(x)ϪA0x cos(⍀t) is tilted back and forth, thereby raising and lowering successively the potential barriers I. INTRODUCTION of the right and the left well, respectively, in an antisymmetric manner. This cyclic variation is shown in our cartoon (b). A Users of modern communication devices are annoyed suitable dose of noise (i.e., when the period of the driving by any source of background hiss. Under certain circum- approximately equals twice the noise-induced escape time) will stances, however, an extra dose of noise can in fact help make the ‘‘sad face’’ happy by allowing synchronized hopping rather than hinder the performance of some devices. to the globally stable state (strictly speaking, this holds true There is now even a name for the phenomenon: stochas- only in the statistical average). tic resonance. It is presently creating a buzz in fields such as physics, chemistry, biomedical sciences, and engineer- the potential barrier separating the two minima. The ing. noise strength DϭkBT is related to the temperature T. The mechanism of stochastic resonance is simple to If we apply a weak periodic forcing to the particle, the explain. Consider a heavily damped particle of mass m double-well potential is tilted asymmetrically up and and viscous friction ␥, moving in a symmetric double- down, periodically raising and lowering the potential well potential V(x) [see Fig. 1(a)]. The particle is sub- barrier, as shown in Fig. 1(b). Although the periodic ject to fluctuational forces that are, for example, induced forcing is too weak to let the particle roll periodically by coupling to a heat bath. Such a model is archetypal from one potential well into the other one, noise- ¨ for investigations in reaction-rate theory (Hanggi, induced hopping between the potential wells can be- Talkner, and Borkovec, 1990). The fluctuational forces come synchronized with the weak periodic forcing. This cause transitions between the neighboring potential statistical synchronization takes place when the average wells with a rate given by the famous Kramers rate waiting time TK(D)ϭ1/rK between two noise-induced (Kramers, 1940), i.e., interwell transitions is comparable with half the period

␻0␻b ⌬V T⍀ of the periodic forcing. This yields the time-scale r ϭ exp Ϫ . (1.1) K 2␲␥ ͩ D ͪ matching condition for stochastic resonance, i.e., 2 2TK͑D͒ϭT⍀ . (1.2) with ␻0ϭVЉ(xm)/m being the squared angular frequency of the potential in the potential minima at Ϯxm , In short, stochastic resonance in a symmetric double- 2 and ␻bϭ͉VЉ(xb)/m͉ the squared angular frequency at well potential manifests itself by a synchronization of the top of the barrier, located at xb; ⌬V is the height of activated hopping events between the potential minima

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 225 with the weak periodic forcing (Gammaitoni, Over time, the notion of stochastic resonance has Marchesoni, et al., 1989). For a given period of the forc- been widened to include a number of different mecha- ing T⍀ , the time-scale matching condition can be ful- nisms. The unifying feature of all these systems is the filled by tuning the noise level Dmax to the value deter- increased sensitivity to small perturbations at an optimal mined by Eq. (1.2). noise level. Under this widened notion of stochastic The concept of stochastic resonance was originally put resonance, the first non-bistable systems discussed were forward in the seminal papers by Benzi and collabora- excitable systems (Longtin, 1993). In contrast to bistable tors (Benzi et al., 1981, 1982, 1983) wherein they address systems, excitable systems have only one stable state the problem of the periodically recurrent ice ages. This (the rest state), but possess a threshold to an excited very suggestion that stochastic resonance might rule the state which is not stable and decays after a relatively periodicity of the primary cycle of recurrent ice ages was long time (in comparison to the relaxation rate of small raised independently by C. Nicolis and G. Nicolis (Nic- perturbations around the stable state) to the rest state. olis, 1981, 1982, 1993; Nicolis and Nicolis, 1981). A sta- Soon afterwards, threshold detectors (see Sec. V.C, tistical analysis of continental ice volume variations over which presents cartoon versions of excitable systems) the last 106 yr shows that the glaciation sequence has an were discovered as a class of simple systems exhibiting average periodicity of about 105 yr. This conclusion is stochastic resonance (Jung, 1994; Wiesenfeld et al. 1994; intriguing because the only comparable astronomical Gingl, Kiss, and Moss, 1995; Gammaitoni, 1995a; Jung, time scale in earth dynamics known so far is the modu- 1995). In the same spirit, stochastic-resonance-like fea- lation period of its orbital eccentricity caused by plan- tures in purely autonomous systems have been reported etary gravitational perturbations. The ensuing variations (Hu, Ditzinger, et al., 1993; Rappel and Strogatz, 1994). of the solar energy influx (or solar constant) on the earth The framework developed for excitable and threshold surface are exceedingly small, about 0.1%. The question dynamical systems has paved the way for stochastic climatologists (still) debate is whether a geodynamical resonance applications in neurophysiology: stochastic model can be devised, capable of enhancing the climate resonance has been demonstrated in mechanoreceptor sensitivity to such a small external periodic forcing. Sto- neurons located in the tail fan of crayfish (Douglass chastic resonance provides a simple, although not con- et al., 1993) and in hair cells of crickets (Levin and clusive answer to this question (Matteucci, 1989, 1991; Miller, 1996). Winograd et al., 1992; Imbrie et al., 1993). In the model In the course of an ever-increasing flourishing of sto- of Benzi et al. (1981, 1982, 1983), the global climate is chastic resonance, new applications with novel types of represented by a double-well potential, where one mini- stochastic resonance have been discovered, and there mum represents a small temperature corresponding to a seems to be no end in sight. Most recently, the notion of largely ice-covered earth. The small modulation of the stochastic resonance has been extended into the domain earth’s orbital eccentricity is represented by a weak pe- of microscopic and mesoscopic physics by addressing the riodic forcing. Short-term climate fluctuations, such as quantum analog of stochastic resonance (Lo¨ fstedt and the annual fluctuations in solar radiation, are modeled Coppersmith, 1994a, 1994b; Grifoni and Ha¨nggi, 1996a, by Gaussian white noise. If the noise is tuned according 1996b) and also into the world of spatially extended, to Eq. (1.2), synchronized hopping between the cold and pattern-forming systems (spatiotemporal stochastic warm climate could significantly enhance the response resonance) (Jung and Mayer-Kress, 1995; Lo¨ cher et al., of the earth’s climate to the weak perturbations caused 1996; Marchesoni et al., 1996; Wio, 1996; Castelpoggi by the earth’s orbital eccentricity, according to argu- and Wio, 1997; Vilar and Rubı´, 1997). Other important ments by Benzi et al. (1981, 1982). extensions of stochastic resonance include stochastic A first experimental verification of the stochastic reso- resonance phenomena in coupled systems, reviewed in nance phenomenon was obtained by Fauve and Heslot Sec. VII.B, and stochastic resonance in deterministic (1983), who studied the noise dependence of the spectral systems exhibiting chaos (see Sec. VI.C). line of an ac-driven Schmitt trigger. The field then re- Stochastic resonance is by now a well-established phe- mained somewhat dormant until the modern age of sto- nomenon. In the following sections, the authors have chastic resonance was ushered in by a key experiment in attempted to present a comprehensive review of the a bistable ring laser (McNamara, Wiesenfeld, and Roy, present status of stochastic resonance theory, applica- 1988). Soon after, prominent dynamical theories in the tions, and experimental evidences. After having intro- adiabatic limit (Gammaitoni, Marchesoni, Menichella- duced the reader into different quantitative measures of Saetta, and Santucci, 1989; McNamara and Wiesenfeld, stochastic resonance, we outline the theoretical tools. A 1989; Presilla, Marchesoni, and Gammaitoni, 1989; Hu series of topical applications that are rooted in the physi- et al., 1990) and in the full nonadiabatic regime (Jung cal and biomedical sciences are reviewed in some detail. and Ha¨nggi, 1989, 1990, 1991a) have been proposed. The authors trust that with the given selection of top- Moreover, descriptions in terms of the linear-response ics and theoretical techniques a reader will enjoy the approximation have frequently been introduced to char- tour through the multifaceted scope that underpins the acterize stochastic resonance (Dykman et al., 1990a, physics of stochastic resonance. Moreover, this compre- 1990b; Gammaitoni et al., 1990; Dykman, Haken, et al., hensive review will put the reader at the very forefront 1993; Jung and Ha¨nggi, 1991a; Hu, Haken, and Ning, of present and future stochastic resonance studies. The 1992). reader may also profit by consulting other, generally

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 226 Gammaitoni et al.: Stochastic resonance more confined reviews and historical surveys, which in Sec. IV.A, the potential parameters a and b can be several aspects complement our work and/or provide ad- eliminated such that Eq. (2.2a) assumes the dimension- ditional insights into topics covered herein. In this con- less form text we refer the reader to the accounts given by Moss 1 1 (1991, 1994), Jung (1993), Moss, Pierson, and O’Gorman V͑x͒ϭϪ x2ϩ x4. (2.2b) (1994), Moss and Wiesenfeld (1995a, 1995b), Wiesenfeld 2 4 and Moss (1995), Dykman, Luchinsky, et al. (1995), Bul- In Eq. (2.1) ␰(t) denotes a zero-mean, Gaussian white sara and Gammaitoni (1996), as well as to the compre- noise with autocorrelation function hensive proceedings of two recent conferences (Moss, Bulsara, and Shlesinger, 1993; Bulsara et al., 1995). ͗␰͑t͒␰͑0͒͘ϭ2D␦͑t͒ (2.3) and intensity D. The potential V(x) is bistable with minima located at Ϯxm , with xmϭ1. The height of the potential barrier between the minima is given by ⌬Vϭ II. CHARACTERIZATION OF STOCHASTIC RESONANCE 1 4 [see Fig. 1(a)]. Having elucidated the main physical ideas of stochas- In the absence of periodic forcing, x(t) fluctuates tic resonance in the preceding section, we next define around its local stable states with a statistical variance the observables that actually quantify the effect. These proportional to the noise intensity D. Noise-induced observables should be physically motivated, easily mea- hopping between the local equilibrium states with the surable, and/or be of technical relevance. In the seminal Kramers rate paper by Benzi et al. (1981), stochastic resonance was 1 ⌬V quantified by the intensity of a peak in the power spec- rKϭ exp Ϫ (2.4) trum. Observables based on the power spectrum are in- &␲ ͩ D ͪ deed very convenient in theory and experiment, since enforces the mean value ͗x(t)͘ to vanish. they have immediate intuitive meaning and are readily In the presence of periodic forcing, the reflection sym- measurable. In the neurophysiological applications of metry of the system is broken and the mean value ͗x(t)͘ stochastic resonance another measure has become fash- does not vanish. This can be intuitively understood as ionable, namely the interval distributions between acti- the consequence of the periodic biasing towards one or vated events such as those given by successive neuronal the other potential well. firing spikes or consecutive barrier crossings. Filtering all the information about x(t), except for We follow here the historical development of stochas- identifying in which potential well the particle resides at tic resonance and first discuss important quantifiers of time t (known as two-state filtering), one can achieve a stochastic resonance based on the power spectrum. binary reduction of the two-state model (McNamara and Along with the introduction of the quantifiers, we dem- Wiesenfeld, 1989). The starting point of the two-state onstrate their properties for two generic models of sto- model is the master equation for the probabilities nϮ of chastic resonance; the periodically driven bistable two- being in one of the two potential wells denoted by their state system and the double-well system. The detailed equilibrium positions Ϯxm , i.e., mathematical analysis of these models is the subject of Secs. III, IV, and the Appendix. Important results n˙ Ϯ͑t͒ϭϪWϯ͑t͒nϮϩWϮ͑t͒nϯ , (2.5) therein are used within this section to support a more with corresponding transition rates W (t). The periodic intuitive approach. In a second part, we discuss quanti- ϯ bias toward one or the other state is reflected in a peri- fiers that are based on the interval distribution; these odic dependence of the transition rates; see Eq. (3.3) latter measures emphasize the synchronization aspect of below. stochastic resonance. We finish the section with a list of other, alternative methods and tools that have been used to study stochastic resonance. In addition, we present a list of experimental demonstrations. 1. The periodic response A. A generic model For convenience, we choose the phase of the periodic driving ␸ϭ0, i.e., the input signal reads explicitly We consider the overdamped motion of a Brownian A(t)ϭA cos(⍀t). The mean value ͗x(t)͉x ,t ͘ is ob- particle in a bistable potential in the presence of noise 0 0 0 tained by averaging the inhomogeneous process x(t) and periodic forcing with initial conditions x0ϭx(t0) over the ensemble of x˙ ͑t͒ϭϪVЈ͑x͒ϩA0 cos͑⍀tϩ␸͒ϩ␰͑t͒, (2.1) the noise realizations. Asymptotically (t0 Ϫϱ), the memory of the initial conditions gets→ lost and where V(x) denotes the reflection-symmetric quartic x(t)͉x ,t becomes a periodic function of time, i.e., potential ͗ 0 0͘ ͗x(t)͘asϭ͗x(tϩT⍀)͘as with T⍀ϭ2␲/⍀. For small am- a b plitudes, the response of the system to the periodic input V͑x͒ϭϪ x2ϩ x4. (2.2a) 2 4 signal can be written as ¯ By means of an appropriate scale transformation, cf. ͗x͑t͒͘asϭ¯x cos͑⍀tϪ␾͒, (2.6)

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 227

turbed bistable system with A0ϭ0 switches spontane- ously between its stable states with rate rK . The input signal modulates the symmetric bistable system, making successively one stable state less stable than the other over half a period of the forcing. Tuning the noise intensity so that the random-switching frequency rK is made to agree closely with the forcing angular frequency ⍀, the system attains the maximum probability for an escape out of the less stable state into the more stable one, before a random back switching event takes place. When the noise intensity D is too small (DӶDSR), the switching events become very rare; thus the periodic component(s) of the interwell dynamics are hardly vis- FIG. 2. Amplitude ¯x (D) of the periodic component of the ible. Under such circumstances, the periodic component system response (2.6) vs the noise intensity D (in units of ⌬V) of the output signal x(t) is determined primarily by mo- for the following values of the input amplitude: tion around the potential minima—the intrawell motion. A0xm /⌬Vϭ0.4 (triangles), A0xm /⌬Vϭ0.2 (circles), and A similar loss of synchronization happens in the oppo- A0xm /⌬Vϭ0.1 (diamonds) in the quartic double-well poten- site case when DӷDSR : The system driven by the ran- 4 −1 tial (2.2a) with aϭ10 s , xmϭ10 (in units [x] used in the dom source flips too many times between its stable experiment), and ⍀=100 s−1. states within each half forcing period for the forced components of the interwell dynamics to be statistically relevant. with amplitude ¯x and a phase lag ␾¯. Approximate ex- In this spirit, the time-scale matching condition in Eq. pressions for the amplitude and phase shift read (1.2), which with TKϭ1/rK is recast as ⍀ϭ␲rK , pro- 2 vides a reasonable condition for the maximum of the A0͗x ͘0 2rK ¯x ͑D͒ϭ (2.7a) response amplitude ¯x . Although the time-scale match- D 2 2 ͱ4rKϩ⍀ ing argument yields a value for DSR that is reasonably close to the exact value it is important to note that it is and not exact (Fox and Lu, 1993). Within the two-state ⍀ model, the value DSR obeys the transcendental equation ␾¯͑D͒ϭarctan , (2.7b) ͩ 2r ͪ 2 2 K 4rK͑DSR͒ϭ⍀ ͑⌬V/DSRϪ1͒, (2.8) 2 where ͗x ͘0 is the D-dependent variance of the station- obtained from Eq. (2.7a). The time-scale matching con- ary unperturbed system (A0ϭ0). Equation (2.7) has dition obviously does not fulfill Eq. (2.8); thus underpin- been shown to hold in leading order of the modulation ning its approximate nature. amplitude A0xm /D for both discrete and continuous The phase lag ␾¯ exhibits a transition from ␾¯=␲/2 at one-dimensional systems (Nicolis, 1982; McNamara and ϩ ¯ Wiesenfeld, 1989; Presilla, Marchesoni, and Gammai- Dϭ0 to ␾ϰ⍀ in the vicinity of DSR . By taking the toni, 1989; Hu, Nicolis, and Nicolis, 1990). While post- second derivative of the function ␾¯ in Eq. (2.7b) and poning a more accurate discussion of the validity of the comparing with Eq. (2.8) one easily checks that DSR lies above equations for ¯x and ␾¯ to Sec. IV.B, we notice on the right-hand side of the point of inflection of ␾¯, ¯ here that Eq. (2.7a) allows within the two-state approxi- being ␾Љ(DSR)Ͼ0. 2 2 mation, i.e., ͗x ͘0ϭxm , a direct estimate for the noise It is important to note that the variation of the angu- intensity DSR that maximizes the output ¯x versus D for lar frequency ⍀ at a fixed value of the noise intensity D fixed driving strength and driving frequency. does not yield a resonance-like behavior of the response The first and most important feature of the amplitude amplitude. This behavior is immediately evident from ¯x is that it depends on the noise strength D, i.e., the Eq. (2.7a) and also from numerical studies (for those periodic response of the system can be manipulated by who don’t trust the theory). A more refined analysis changing the noise level. At a closer inspection of Eq. (Thorwart and Jung, 1997) shows that the decomposi- (2.7), we note that the amplitude ¯x first increases with tion of the susceptibility into its real and imaginary parts increasing noise level, reaches a maximum, and then de- restores a nonmonotonic frequency dependence—see creases again. This is the celebrated stochastic resonance also the work on dynamical hysteresis and stochastic effect. In Fig. 2, we show the result of a simulation of the resonance by Phillips and Schulten (1995), and Mahato double-well system [Eqs. (2.1)–(2.3)] for several weak and Shenoy (1994). amplitudes of the periodic forcing A0 . Upon decreasing Finally, we introduce an alternative interpretation of the driving frequency ⍀, the position of the peak moves the quantity ¯x (D) due to Jung and Ha¨nggi (1989, to smaller noise strength (see Fig. 6, below). 1991a): the integrated power p1 stored in the delta-like 2 Next we attempt to assign a physical meaning to the spikes of S(␻)atϮ⍀is p1ϭ␲¯x (D). Analogously, the 2 value of DSR . The answer was given originally by Benzi modulation signal carries a total power pAϭ␲A0. and co-workers (Benzi et al. 1981, 1982, 1983): an unper- Hence the spectral amplification reads

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 228 Gammaitoni et al.: Stochastic resonance

2n such as A0 , we can restrict ourselves to the first spectral spike, being consistent with the linear-response assumption implicit in Eq. (2.6). For small forcing amplitudes, SN(␻) does not deviate much from the power 0 spectral density SN(␻) of the unperturbed system. For a bistable system with relaxation rate 2rK , the hopping 0 contribution to SN(␻) reads 0 2 2 2 SN͑␻͒ϭ4rK͗x ͘0 /͑4rKϩ␻ ͒. (2.11) The spectral spike at ⍀ was verified experimentally (Debnath, Zhou, and Moss, 1989; Gammaitoni, Marchesoni, et al., 1989; Gammaitoni, Menichella- Saetta, Santucci, Marchesoni, and Presilla, 1989; Zhou and Moss, 1990) to be a delta function, thus signaling the presence of a periodic component with angular frequency ⍀ in the system response [Eq. (2.6)]. In fact, for A0xmӶ⌬V we are led to separate x(t) into a noisy background (which coincides, apart from a normalization constant, with the unperturbed output signal) and a periodic component with x(t) given by Eq. (2.6) FIG. 3. Characterization of stochastic resonance. (a) A typical ͗ ͘as ¨ power spectral density S(␯) vs frequency ␯ for the case of the (Jung and Hanggi, 1989). On adding the power spectral quartic double-well potential in Eq. (2.2a). The delta-like density of either component, we easily obtain spikes at (2nϩ1)␯⍀ , ⍀ϭ2␲␯⍀ , with nϭ0, 1, and 2, are dis- 2 S͑␻͒ϭ͑␲/2͒¯x ͑D͒ ͓␦͑␻Ϫ⍀͒ϩ␦͑␻ϩ⍀͔͒ϩSN͑␻͒, played as finite-size histogram bins. (b) Strength of the first (2.12) delta spike, Eq. (2.12), and the signal-to-noise ratio SNR, Eq. 0 2 (2.13), vs D (in units of ⌬V). The arrow denotes the D value with SN(␻)ϭS (␻)ϩ (A ) and ¯x (D) given in Eq. N O 0 corresponding to the power spectral density plotted in (a). The (2.7a). In Fig. 3(b) the strength of the delta-like spike of 4 −1 other parameters are Axm /⌬Vϭ0.1, aϭ10 s , and xmϭ10 S(␻) (more precisely ¯x ) is plotted as a function of D. (in units [x] used in the experiment). Stochastic resonance can be envisioned as a particular problem of signal extraction from background noise. It is quite natural that a number of authors tried to char- ␩ϵp /p ϭ ¯x D /A 2. (2.9) 1 A ͓ ͑ ͒ 0͔ acterize stochastic resonance within the formalism of In the linear-response regime of Eq. (2.7), ␩ is indepen- data analysis, most notably by introducing the notion of dent of the input amplitude. This spectral amplification signal-to-noise ratio (SNR) (McNamara et al., 1988; ␩ will frequently be invoked in Sec. IV, instead of Debnath et al., 1989; Gammaitoni, Marchesoni, et al., ¯x (D). 1989; Vemuri and Roy, 1989; Zhou and Moss, 1990; Gong et al., 1991, 1992). We adopt here the following definition of the signal-to-noise ratio 2. Signal-to-noise ratio ⍀ϩ⌬␻ Instead of taking the ensemble average of the system SNRϭ2 lim S͑␻͒d␻ SN͑⍀͒. (2.13) ͵⍀Ϫ⌬␻ Ͳ response, it sometimes can be more convenient to ex- ͫ⌬␻ 0 ͬ → tract the relevant phase-averaged power spectral density Hence on combining Eqs. (2.11) and (2.12), the SN ratio S(␻), defined here as (see Secs. III and IV.A) for a symmetric bistable system reads in leading order ϩϱ Ϫ 2 S͑␻͒ϭ e i␻␶͗͗x͑tϩ␶͒x͑t͒͘͘d␶, (2.10) SNRϭ␲͑A0xm /D͒ rK . (2.14) ͵Ϫϱ Note that the factor of 2 in the definition (2.13) was where the inner brackets denote the ensemble average introduced for convenience, in view of the power spec- over the realizations of the noise and outer brackets in- tral density symmetry S(␻)ϭS(Ϫ␻). The SN ratio dicate the average over the input initial phase ␸. In Fig. SNR for the power spectral density plotted in Fig. 3(a) 3(a) we display a typical example of S(␯)(␻=2␲␯) for versus frequency ␯ (␻ϭ2␲␯) is displayed in Fig. 3(b). the bistable system. Qualitatively, S(␻) may be de- ¯ The noise intensity DSR at which SNR assumes its maxi- scribed as the superposition of a background power mum does not coincide with the value DSR that maxi- spectral density SN(␻) and a structure of delta spikes mizes the response amplitude ¯x , or equivalently the centered at ␻ϭ(2nϩ1)⍀ with n=0,Ϯ1,Ϯ2... . The strength of the delta spike in the power spectrum given generation of only odd higher harmonics of the input by Eq. (2.12). As a matter of fact, if the prefactor of the frequency are typical fingerprints of periodically driven Kramers rate is independent of D, we find that the SN symmetric nonlinear systems (Jung and Ha¨nggi, 1989). ratio of Eq. (2.14) has a maximum at Since the strength (i.e., the integrated power) of such ¯ spectral spikes decays with n according to a power law DSRϭ⌬V/2. (2.15)

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 229

escape rate rK . However, any experimentalist who ever tried to reproduce stochastic resonance in a real system (including here the analog circuits) knows by experience that a synchronization phenomenon takes place any time the condition rKϳ⍀ is established by varying either D or ⍀. In Figs. 4(a) and 4(b) we depict the typical input-output synchronization effect in the bistable system Eqs. (2.1)–(2.3). In Fig. 4(a) the noise intensity is increased from low (rare random switching events) up to very large values, crossing the resonance values DSR of Eqs. (2.8). In the latter case the output signal x(t) becomes tightly locked to the periodic input. In Fig. 4(b), the noise intensity D is kept ﬁxed and the forcing frequency ⍀ is increased. At low values of ⍀, we notice an alternate asymmetry of the output signal towards either positive or negative values, depending on the sign of the input signal. However, many switches occur in both directions within any half forcing period. At large values of ⍀, the effect of the time modulation is averaged out and the symmetry of the output signal seems to be fully restored. Finally, at ⍀ϳrK the synchronization mechanism is established with clear resemblance to Fig. 4(a). In the following subsection we characterize stochastic resonance as a ‘‘resonant’’ synchronization phenomenon, resulting from the combined action of noise and periodic forcing in a bistable system. The tool employed to this purpose is the residence-time distribution. Intro- duced as a tool (Gammaitoni, Marchesoni, et al., 1989; Zhou and Moss, 1990; Zhou, et al., 1990; Lo¨ fstedt and Coppersmith, 1994b; Gammaitoni, Marchesoni, and Santucci, 1995), such a notion proved useful for applications in diverse areas of natural sciences (Bulsara et al., 1991; Longtin et al., 1991; Simon and Libchaber, 1992; Carroll and Pecora, 1993b; Gammaitoni, Marchesoni, et al. 1993; Mahato and Shenoy, 1994; Mannella et al., 1995; Shulgin et al., 1995).

1. Level crossings A deeper understanding of the mechanism of stochas- FIG. 4. Example of input/output synchronization in the sym- tic resonance in a bistable system can be gained by map- metric bistable system of Eqs. (2.1)–(2.2a). (a) Varying the ping the continuous stochastic process x(t) (the system noise intensity D with ⍀ held constant. The sampled signal output signal) into a stochastic point process ͕ti͖. The shown with dashes is the input A(t) (arbitrary units). The re- symmetric signal x(t) is converted into a point process maining trajectories are the corresponding system output (in by setting two crossing levels, for instance at xϮϭϮc units of xm) for increasing D values (from bottom to top). (b) with 0рcрxm . On sampling the signal x(t) with an ap- Effect of varying ⍀ with D held constant. The three output propriate time base, the times ti are determined as fol- samples x(t) (in units of x ) are displayed for increasing ⍀ m lows: data acquisition is triggered at time t0ϭ0 when values (from top to bottom). The parameters for (a) and (b) x(t) crosses, say, xϪ with negative time derivative are Ax /⌬Vϭ0.1, aϭ104 s−1, and x ϭ(a/b)1/2 ϭ 10, cf. in m m ͓x(0)ϭϪc, x˙(0)Ͻ0]; t1 is the subsequent time when Fig. 2. x(t) ﬁrst crosses xϩ with positive derivative [x(t1)ϭc, x˙ (t1)Ͼ0]; t2 is the time when x(t) switches back to B. Residence-time distribution negative values by recrossing xϪ with negative derivative, and so on. The quantities T(i)ϭtiϪtiϪ1 represent In Sec. II.A we interpreted the resonant-like depen- the residence times between two subsequent switching dence of the amplitude ¯x (D) of the periodic response events. For simplicity and to make contact with the on the noise intensity D by means of a synchronization theory of Sec. IV.C, we set cϭxm . The statistical prop- argument, originally formulated by Benzi and co- erties of the stochastic point process ͕ti͖ are the subject workers (Benzi et al., 1981). Moreover, we pointed out of intricate theorems of probability theory (Rice, 1944; that the response amplitude does not show this synchro- Papoulis, 1965; Blake and Lindsey, 1973). In particular, nization if the driving frequency ⍀ is tuned against the no systematic way is known to ﬁnd the distribution of

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 230 Gammaitoni et al.: Stochastic resonance threshold crossing times. An exception is the symmetric bistable system: here, the long intervals T of consecutive crossings obey Poissonian statistics with an exponential distribution (Papoulis, 1965)

N͑T͒ϭ͑1/TK͒exp͑ϪT/TK͒. (2.16) The distribution (2.16) is important for the forthcom- ing discussion, because it describes to a good approximation the ﬁrst-passage time distribution between the potential minima in unmodulated bistable systems [see also Ha¨nggi, Talkner, and Borkovec (1990), and references therein].

2. Input-output synchronization In the absence of periodic forcing, the residence time distribution has the exponential form of Eq. (2.16). In the presence of the periodic forcing (Fig. 5), one observes a series of peaks, centered at odd multiples of the 1 half driving period T⍀ϭ2␲/⍀, i.e., at Tnϭ(nϪ 2 )T⍀ , with nϭ1,2,... . The heights of these peaks decrease exponentially with their order n. These peaks are simply explained: the best time for the system to switch between the potential wells is when the relevant potential barrier assumes a minimum. This is the case when the potential V(x,t)ϭV(x)ϪA0x cos(⍀tϩ␸) is tilted most extremely to the right or the left (in whichever well the system is residing). If the system switches at this time into the other well it then takes half a period waiting time in the other well until the new relevant barrier assumes a minimum. Thus T⍀/2 is a preferred residence interval. If the system ‘‘misses’’ a ‘‘good opportunity’’ to jump, it has to wait another full period until the relevant potential barrier for a switch again assumes the minimum. The second peak in the residence-time distribution is therefore located at 3/2T⍀ . The location of the other peaks is evident. The peak heights decay exponentially because the probabilities of the system to jump over a minimal barrier are statistically independent. We now argue that the strength P1 of the first peak at T⍀/2 (the area under the peak) is a measure of the synchronization between the periodic forcing and the switching between the wells: If the mean residence time of the system in one potential well is much larger than the period of the driving, the system is not likely to jump the first time the relevant potential barrier assumes its minimum. The escape-time distribution exhibits in such a FIG. 5. Residence time distributions N(T) for the symmetric case a large number of peaks where P1 is small. If the bistable system of Eqs. (2.1)–(2.2a). (a) Increasing D (from mean residence-time of the system in one well is much below) with ⍀ held constant; inset: the strength P1 of the first shorter than the period of the driving, the system will peak of N(T)vsD(in units of ⌬V). The definition of P1 is as not ‘‘wait’’ with switching until the relevant potential in Eq. (4.67) with ␣ϭ1/4. (b) Increasing ⍀ (from below) with barrier assumes its maximum and the residence-time dis- D held constant; inset: P1 versus ␯⍀. Here, ⍀ is in units of rK. tribution has already decayed practically to zero before The numbers 1–4 in the insets correspond to the D values (a) the time T /2 is reached and the weight P is again and the ⍀ values (b) of the distributions on display. The pa- ⍀ 1 rameters for (a) and (b) are Ax /⌬Vϭ0.1, aϭ104 s−1, and small. Optimal synchronization, i.e., a maximum of P , m 1 x ϭ(a/b)1/2 ϭ 10, cf. Fig. 2. is reached when the mean residence time matches half m the period of the driving frequency, i.e., our old time- scale matching condition Eq. (1.2). This resonance con- have plotted the strength of the peak at T⍀/2 as a func- dition can be achieved by varying either ⍀ or D. This is tion of the noise strength D [Fig. 5(a)] and as a function demonstrated in Figs. 5(a) and 5(b). In the insets we of the driving frequency ⍀ [Fig. 5(b)]. In passing, we

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 231

anticipate that also the remaining peaks of N(T)atTn and small driving frequencies, due to the truncation pro- with nϾ1 exhibit stochastic resonance [see Sec. IV.C]. cedure, are the main limitations of this algorithm. We conclude with a comment on the multipeaked residence-time distributions: the existence of peaks in 2. Analog simulations the residence-time distribution N(T)atTn with nϾ1 should not mislead the reader to think that the power This type of simulation allows more flexibility than spectral density S(␻) exhibits subharmonics of the fun- digital simulation and for this reason has been preferred damental frequency ⍀ (i.e., delta spikes at frequencies by a number of researchers. Rather than quoting all of smaller than ⍀). Although it may happen that the sys- them individually, we mention here the prominent tem waits for an odd number of half forcing periods (i.e., groups, including those active in Perugia (Italy) (Gam- an integer number of extra ‘‘wait loops’’) before switch- maitoni, Marchesoni, et al., 1989, 1993, 1994, 1995; Gam- ing states, such occurrences are randomly spaced in time maitoni, Menichella-Saetta, et al., 1989), in St. Louis and, therefore, do not correspond to any definite spec- (USA) (Debnath et al., 1989; Zhou and Moss, 1990; tral component (Papoulis, 1965). Moss, 1991, 1994), in Lancaster (England) (Dykman, Mannella, et al., 1990b, 1992) and in Beijing (China) (Hu et al., 1991; Gong et al., 1991, 1992). Analog simulators C. Tools of stochastic processes are easy to design and assemble. Their results are not as accurate as digital simulations, The seminal paper by Benzi et al. (1981) provoked no but offer some advantages: (a) a large range of param- immediate reaction in the literature. Apart from a few eter space can be explored rather quickly; (b) high- early theoretical studies by Nicolis (1982), Eckmann and dimensional systems may be simulated more readily Thomas (1982), and Benzi et al. (1982, 1983, 1985), only than by computer, though systematic inaccuracies must one experimental paper (Fauve and Heslot, 1983) ad- be estimated and treated carefully. The block diagram of dressed the phenomenon of stochastic resonance. One the Perugia simulator is illustrated in Sec. V.B.1 for the reason may be the technical difficulty of treating nonsta- case of a damped quartic double-well oscillator sub- tionary Fokker-Planck equations with time-dependent jected to both noisy and periodic driving. In passing, we coefficients (Jung, 1993). Moreover, extensive numerical mention here that Figs. 1–5 were actually obtained by computations were not yet everyday practice. The ex- means of that simulation circuit. In order to give the perimental article by McNamara, Wiesenfeld, and Roy reader an idea of the reliability of analog simulation, we (1988) marked a renaissance of stochastic resonance, point out that all directly measured quantities are given which has flourished and developed ever since in differ- with a maximum error of about 5%. ent directions. Our present knowledge of this topic has been reached through a variety of investigation tools. In the following paragraphs, we outline the most popular 3. Experiments ones, with particular attention given to their advantages By now, stochastic resonance has been repeatedly ob- and limitations. served in a large variety of experiments. The first experiment was on an electronic circuit. Fauve and Heslot 1. Digital simulations (1983) used a Schmitt trigger to demonstrate the effect. The first evidence of stochastic resonance was pro- A first in situ physical experiment by McNamara, Wie- duced by simulating the Budyko-Sellers model of cli- senfeld, and Roy (1988) used a bistable ring laser to mate change (Benzi et al., 1982) on a Digital Instru- demonstrate stochastic resonance in the noise-induced ments mainframe computer (model PDP 1000), an switching between the two counter-propagating laser advanced computer at the time! Nowadays accurate modes. Stochastic resonance has also been demon- digital simulations of either continuous or discrete sto- strated optically in a semiconductor feedback laser (Ian- chastic processes can be carried out at home on unso- nelli et al., 1994), in a unidirectional photoreactive ring phisticated personal desktop computers. Regardless of resonator (Jost and Sahleh, 1996), and in optical hetero- the particular algorithm adopted in the diverse cases, dyning (Dykman, Golubev, et al., 1995). Relevance of digital simulations proved particularly useful in the stochastic resonance in electronic paramagnetic reso- study of stochastic resonance in numerous cases (Nicolis nance has been identified by Gammaitoni, Martinelli, et al., 1990; Dayan et al., 1992; Mahato and Shenoy, Pardi, and Santucci (1991, 1993). Simon and Libchaber 1994; Masoliver et al., 1995) and in chaotic (Carroll and (1992) observed SR in a beautifully designed optical Pecora, 1993a, 1993b; Hu, Haken, and Ning, 1993; Ippen trap, where a dielectric particle moves in the field of two et al., 1993; Anishchenko et al., 1994) or spatially ex- overlapping Gaussian laser-beams that try to pull the tended systems (Neiman and Schimansky-Geier, 1994, particle into their center. Spano and collaborators 1995; Jung and Mayer-Kress, 1995; Lindner et al., 1995). (1992) have observed stochastic resonance in a paramag- A decisive contribution to the understanding of the sto- netically driven bistable buckling ribbon. Magnetosto- chastic resonance phenomenon was produced in Augs- chastic resonance in ferrite-garnet films has been mea- burg (Germany) by Jung and Ha¨nggi (1989, 1990, 1991a, sured by Grigorenko et al. (1994) and in yttrium-iron 1991b, 1993), who encoded the matrix-continued frac- spheres by Reibold et al. (1997). I and Liu (1995) ob- tion algorithm (Risken, 1984; Risken and Vollmer, served stochastic resonance in weakly ionized magneto- 1989). Convergence problems at low noise intensities plasmas, and Claes and van den Broeck (1991) for dis-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 232 Gammaitoni et al.: Stochastic resonance persion of particles suspended in time-periodic flows. A Let us consider a symmetric unperturbed system that first demonstration of stochastic resonance in a semicon- switches between two discrete states Ϯxm with rate W0 ductor device, more precisely in the low-temperature out of either state. We define nϮ(t) to be the probabili- ionization breakdown in p-type germanium, has been re- ties that the system occupies either state Ϯ at time t, ported by Kittel et al. (1993). Stochastic resonance has that is x(t)ϭϮxm . In the presence of a periodic input been observed in superconducting quantum interference signal A(t)ϭA0 cos(⍀t), which biases the state Ϯ alter- devices (SQUID) by Hibbs et al. (1995) and Rouse et al. natively, the transition probability densities Wϯ(t) out (1995). Furthermore, Rouse et al. provided the first ex- of the states Ϯxm depend periodically on time. Hence perimental evidence of noise-induced resonances (see the relevant master equation for nϮ(t) reads Sec. VII.D.1) in their SQUID system. In recent experi- ˙ ments by Mantegna and Spagnolo (1994, 1995, 1996), nϮ͑t͒ϭϪWϯ͑t͒nϮϩWϮ͑t͒nϯ (3.1a) stochastic resonance was demonstrated in yet another or, making use of the normalization condition semiconductor device, a tunnel diode. Stochastic reso- nϩϩnϪϭ1, nance has also been observed in modulated bistable chemical reaction dynamics (Minimal-Bromate and n˙ Ϯ͑t͒ϭϪ͓WϮ͑t͒ϩWϯ͑t͔͒nϮϩWϮ͑t͒. (3.1b) ¨ Belousov-Zhabotinsky reactions) by Hohmann, Muller, The solution of the rate equation (3.1) is given by and Schneider (1996). Undoubtedly, the neurophysiological experiments on t n ͑t͒ϭg͑t͒ n ͑t ͒ϩ W ͑␶͒gϪ1͑␶͒d␶ , stochastic resonance constitute a cornerstone in the Ϯ Ϯ 0 ͵ Ϯ ͫ t0 ͬ field. They have triggered the interest of scientists from biology and biomedical engineering to medicine. Long- t g͑t͒ϭexp Ϫ ͓W ͑␶͒ϩW ͑␶͔͒d␶ , (3.2) tin, Bulsara, and Moss (1991) have demonstrated the ͵ ϩ Ϫ ͩ t0 ͪ surprising similarity between interspike interval histograms of periodically stimulated sensory neurons and with unspecified initial condition nϮ(t0). For the transi- residence-time distributions of periodically driven tion probability densities Wϯ(t), McNamara and Wie- bistable systems (see Secs. II.B and IV.C). Stochastic senfeld (1989) proposed to use periodically modulated resonance in a living system was first demonstrated by escape rates of the Arrhenius type

Douglass et al. (1993) (see also in Moss et al., 1994) in A0xm W ͑t͒ϭr exp ϯ cos͑⍀t͒ . (3.3) the mechanoreceptor cells located in the tail fan of cray- ϯ K ͫ D ͬ fish. A similar experiment using the sensory hair cells of a cricket was performed by Levin and Miller (1996). A On assuming, as in Sec. II.A, that the modulation ampli- cricket can detect an approaching predator by the coher- tude is small, i.e., A0xmӶD, we can use the following ent motion of the air although the coherence is buried expansions in the small parameter A0xm /D, under a huge random background. Fairly convincing ar- A0xm W ͑t͒ϭr 1ϯ cos͑⍀t͒ guments had been given by Levin and Miller that sto- ϯ Kͫ D chastic resonance is actually responsible for this capabil- 2 1 A0xm ity of the cricket. Since the functionality of neurons is ϩ cos2͑⍀t͒ϯ... , based on gating ion channels in the cell membrane, 2 ͩ D ͪ ͬ Bezrukov and Vodyanoy (1995) have studied the impact 1 A x 2 0 m 2 of stochastic resonance on ion-channel gating. Stochastic Wϩ͑t͒ϩWϪ͑t͒ϭ2rK 1ϩ cos ͑⍀t͒ϩ... . resonance has been studied in visual perceptions [Riani ͫ 2 ͩ D ͪ ͬ and Simonotto, 1994, 1995; Simonotto et al. (1997); Chi- (3.4) alvo and Apkarian (1993)] and in the synchronized re- In Sec. IV.C, we discuss in more detail the validity of the sponse of neuronal assemblies to a global low-frequency expression (3.3) for the rates. The most important con- field (Gluckman et al., 1996). dition is a small driving frequency (adiabatic assumption). The integrals in Eq. (3.2) can be performed analytically to first order in A0xm /D, 1 III. TWO-STATE MODEL nϩ͑t͉x0 ,t0͒ϭ1ϪnϪ͑t͉x0 ,t0͒ϭ 2 ͕exp͓Ϫ2rK͑tϪt0͔͒ ϫ͓2␦ Ϫ1Ϫ␬͑t ͔͒ϩ1ϩ␬͑t͖͒, (3.5) In this section we discuss the simplest model that x0 ,xm 0 epitomizes the class of symmetric bistable systems intro- where ␬(t)ϭ2r (A x /D)cos(⍀tϪ␾¯)/ 4r2 ϩ⍀2, and duced in Sec. II.A. Such a discrete model was proposed K 0 m ͱ K ¯ originally as a stochastic resonance study case by Mc- ␾ϭarctan͓⍀/(2rK)͔. The quantity nϩ(t͉x0 ,t0) in Eq. Namara and Wiesenfeld (1989), who also pointed out (3.5) should be read as the conditional probability that that under certain restrictions it renders an accurate rep- x(t) is in the state + at time t, given that its initial state is x ϵx(t ). Here the Kronecker delta ␦ is 1 when resentation of most continuous bistable systems. For this 0 0 x0 ,xm reason, we discuss the two-state model in some detail. the system is initially in the state +. Most of the results reported below are of general valid- From Eq. (3.5), any statistical quantity of the discrete ity and provide the reader with a preliminary analytical process x(t) can be computed to first order in A0xm /D, scheme on which to rely. namely:

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 233

(a) The time-dependent response ͗x(t)͉x0 ,t0͘ to (c) The power spectral density S(␻). The power spec- the periodic forcing. From the definition tral density commonly reported in the literature is the ͗x(t)͉x0 ,t0͘ϭ͐xP(x,t͉x0 ,t0)dx with P(x,t͉x0 ,t0) Fourier transform of Eq. (3.10) [see Eq. (2.10)] Ϫ ϩ ϩ ϵnϩ(t)␦(x xm) nϪ(t)␦(x xm), it follows immedi- 2 2 2 Ϫ 1 A0xm 4rK 4rKxm ately that in the asymptotic limit t0 ϱ, S͑␻͒ϭ 1Ϫ → 2 ͩ D ͪ 4r2 ϩ⍀2 4r2 ϩ␻2 ¯ ͫ K ͬ K lim ͗x͑t͉͒x0 ,t0͘ϵ͗x͑t͒͘asϭ¯x͑D͒cos͓⍀tϪ␾͑D͔͒, t Ϫϱ 2 2 2 0→ ␲ A0xm 4rKxm (3.6) ϩ ͓␦͑␻Ϫ⍀͒ϩ␦͑␻ϩ⍀͔͒, 2 ͩ D ͪ 4r2 ϩ⍀2 K with (3.11) 2 A0xm 2rK ¯x ͑D͒ϭ (3.7a) which has the same form as the expression for S(␻) D 2 2 ͱ4rKϩ⍀ derived in Eq. (2.12). As a matter of fact, SN(␻) is the and product of the Lorentzian curve obtained with no input signal A0ϭ0 and a factor that depends on the forcing ⍀ amplitude A , but is smaller than unity. The total out- ␾¯͑D͒ϭarctan . (3.7b) 0 ͩ 2r ͪ put power, signal plus noise, for the two-state model K discussed here, is 2␲x2 , independent of the input-signal Equation (3.7) coincides with Eq. (2.7) for x2 ϭx2 . m ͗ ͘0 m amplitude A and frequency ⍀. Hence the effect of the (b) The autocorrelation function x(tϩ )x(t) x ,t . 0 ͗ ␶ ͉ 0 0͘ input signal is to transfer power from the broadband The general definition noise background into the delta spike(s) of the power ͗x͑tϩ␶͒x͑t͉͒x0 ,t0͘ spectral density. Finally, the SN ratio follows as 2 A0xm 4 ϭ xyP͑x,tϩ␶͉y,t͒ SNRϭ␲ rKϩ ͑A0͒. (3.12) ͵͵ ͩ D ͪ O To leading order in A x /D, Eq. (3.12) coincides with ϫP͑y,t͉x0 ,t0͒dxdy (3.8) 0 m Eq. (2.14). greatly simplifies in the stationary limit t Ϫϱ, 0→ The residence-time distribution N(T) for the two- lim ͗x͑tϩ␶͒x͑t͉͒x0 ,t0͘ state model was calculated by Zhou, Moss, and Jung t Ϫϱ (1990), and by Lo¨ fstedt and Coppersmith (1994a, 1994b) 0→ 2 2 within a two-state model, yielding in leading order of ϵ͗x͑tϩ␶͒x͑t͒͘asϭxmexp͑Ϫ2rK͉␶͉͓͒1Ϫ␬͑t͒ ͔ A0xm /D [cf. Sec. IV.D], 2 2 ϩxm␬͑tϩ␶͒␬͑t͒. (3.9) N͑T͒ϭ ͓1Ϫ͑1/2͒͑A x /D͒ cos͑⍀T͔͒r N0 0 m K In Eq. (3.9) we can easily separate an exponentially de- ϫexp͑ϪrKT͒, (3.13) caying branch due to randomness and a periodically os- Ϫ1 2 2 cillating tail driven by the periodic input signal. Note with 0 ϭ1Ϫ(1/2)(A0xm /D) /͓1ϩ(⍀/rK) ͔. Note N that even in the stationary limit t Ϫϱ, the output- that N(T) exhibits the peak structure of Fig. 5, with 0→ signal autocorrelation function depends on both times Tnϭ(nϪ1/2)T⍀ . Furthermore, the strength P1 of the tϩ␶ and t. However, in real experiments t represents first peak can be easily calculated by integrating N(T) 1 1 the time set for the trigger in the data acquisition proce- over an interval ͓( 2 Ϫ␣),( 2 ϩ␣)͔T⍀ , with 0Ͻ␣р1/4. dure. Typically, the averages implied by the definition of Skipping the details of the integration, one realizes that the autocorrelation function are taken over many sam- P1 is a function of the ratio ⍀/rK and attains its maxi- pling records of the signal x(t), triggered at a large num- mum for rKӍ2␯⍀ as illustrated in the inset of Fig. 5(b). ber of times t within one period of the forcing T⍀ . In this section we detailed the symmetric two-state Hence, the corresponding phases of the input signal, model as an archetypal system that features stochastic ␪ϭ⍀tϩ␸, are uniformly distributed between 0 and 2␲. resonance. We profited greatly from the analytical study This corresponds to averaging ͗x(tϩ␶)x(t)͘as with re- of McNamara and Wiesenfeld (1989). The two-state spect to t uniformly over an entire forcing period, model can be regarded as an adiabatic approximation to whence any continuous bistable system, like the overdamped ͗͗x͑tϩ␶͒x͑t͒͘͘ quartic double-well oscillator of Eqs. (2.1)–(2.3), provided that the input-signal frequency is low enough for 2 2 1 A0xm 4rK the notion of transition rates [Eq. (3.4)] to apply. ϭx2 exp͑Ϫ2r ͉␶͉͒ 1Ϫ m K ͫ 2 ͩ D ͪ 4r2 ϩ⍀2ͬ In general, the difficulty lies in the derivation of time- K dependent transition rates in a continuous model. A sys- 2 2 2 xm A0xm 4rK tematic method consists of finding the unstable periodic ϩ cos͑⍀␶͒, (3.10) 2 ͩ D ͪ 4r2 ϩ⍀2 orbits in the absence of noise, since they act as basin K boundaries in an extended phase-space description where the outer brackets ͗ ...͘ stay for (Jung and Ha¨nggi, 1991b). Rates in periodically driven T⍀ (1/T⍀)͐0 ͓ ...͔dt. systems can be defined as the transition rates across

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 234 Gammaitoni et al.: Stochastic resonance those basin boundaries and correspond to the lowest- Grigolini and Marchesoni, 1985) to arrive at the periodi- lying Floquet eigenvalue of the time-periodic Fokker- cally modulated Langevin equation Planck operator (Jung, 1989, 1993)—see also the Sec. VII.D.3. Depending on the degree of approximation ␥x˙ ϭf͑x͒ϩA0 cos͑⍀tϩ␸͒ϩͱ2␥D␰͑t͒. (4.3) needed, the intrawell dynamics may become signiﬁcant With the choice f(x)ϭ(axϪbx3)/m, where aϾ0, bϾ0, and more sophisticated formalisms may be required. we recover the bistable quartic double-well potential V(x)ϭϪ(1/2)ax2ϩ(1/4)bx4 of Fig. 1. On making use IV. CONTINUOUS BISTABLE SYSTEMS of the rescaled variables: ¯ ¯ A two-state description of stochastic resonance is of ¯x ϭx/xm , t ϭat/␥, A0ϭA0 /axm , limited use for a number of reasons. First, the dynamics D¯ϭD/ax2 , ⍀¯ϭ␥⍀/a, (4.4) is reduced to the switching mechanism between two m metastable states only. Thus it neglects the short-time where Ϯxmϭͱa/b locate the minima of V(x), the rel- dynamics that takes place within the immediate neigh- evant Fokker-Planck equation takes on a dimensionless borhood of the metastable states themselves. Moreover, form. Dropping, for the sake of convenience, all over- our goal is to describe both the linear as well as the bars one recovers the Smoluchowski limit of Eq. (4.2); nonlinear stochastic resonance response in the whole re- i.e., in terms of an operator notation we obtain gime of modulation frequencies, extending from expo- ץ -nentially small Kramers rates up to intrawell frequen p͑x,t;␸͒ϭ ͑t͒p͑x,t;␸͒ϵ͓ ϩ ͑t͔͒p͑x,t;␸͒. t L L0 Lextץ cies, and higher. Put differently, a more elaborate approach has to model the nonadiabatic regime of driv- (4.5) ing in the whole accessible state space of the dynamical Here, the Fokker Planck operator process x(t). This goal will be presented within the class 2ץ ץ ,of continuous-state random systems (Stratonovitch ϭϪ ͑xϪx3͒ϩD (4.6) x2ץ xץ Ha¨nggi and Thomas, 1982; Risken, 1984; van Ka- L0 ;1963 mpen, 1992), which can be modeled in terms of a describes the unperturbed dynamics in the rescaled Fokker-Planck equation. bistable potential A. Fokker-Planck description 1 1 V͑x͒ϭϪ x2ϩ x4, (4.7) As a generic system modeling stochastic resonance we 2 4 shall consider a Brownian particle of mass m that moves with barrier height ⌬Vϭ 1 . The operator (t) de- 4 Lext in a bistable potential V(x) and is subjected to thermal notes the gradient-type perturbation noise ␰(t) of the Nyquist type at temperature T. More- ץ over, we perturb the particle with a periodically varying ͑t͒ϭϪA cos͑⍀tϩ␸͒ . (4.8) xץ force, i.e., we start from the Langevin equation Lext 0

mx¨ϭϪm␥x˙ϪVЈ͑x͒ϩmA0 cos͑⍀tϩ␸͒ 1. Floquet approach ϩͱ2m␥kT␰͑t͒. (4.1) The inertial, as well as the overdamped Brownian dy- Here ␰(t) denotes a Gaussian white noise with zero av- namics in Eqs. (4.2) and (4.5) describe a nonstationary erage and autocorrelation function ͗␰(t)␰(s)͘ϭ␦(tϪs). Markovian process where the symmetry under time The external forcing term is characterized by an ampli- translation is retained in a discrete manner only. Since the Fokker-Planck operators in Eqs. (4.2) and (4.5) are tude A0 , an angular frequency ⍀, and an arbitrary but invariant under the discrete time translations t tϩT , fixed initial phase ␸. The statistically equivalent descrip- → ⍀ tion for the corresponding probability density where T⍀ϭ2␲/⍀ denotes the modulation period, the p(x,vϭx˙ ,t;␸) is governed by the two-dimensional Floquet theorem (Floquet, 1883; Magnus and Winkler, Fokker-Planck equation 1979) applies to the corresponding partial differential equation. For a general periodic Fokker-Planck operator such as (t)ϭ (tϩT ), defined on the ץ ץ ץ p͑x,v,t;␸͒ϭ Ϫ vϩ ⍀ v multidimensionalL spaceL of state vectorsץ xץ ͭ tץ X(t)ϭ(x(t);v(t)ϭx˙(t);...), one finds that the rel- ϫ͓␥vϩf͑x͒ϪA0 cos͑⍀tϩ␸͔͒ evant Floquet solutions are functions of the type 2 (p͑X,t;␸͒ϭexp͑Ϫ␮t͒p ͑X,t;␸͒ (4.9 ץ ϩ␥D p͑x,v,t;␸͒, (4.2) ␮ v2ץ ͮ with Floquet eigenvalue ␮ and periodic Floquet modes where we introduced f(x)ϭϪVЈ(x)/m and the diffu- p␮ , sion strength DϭkT/m. For large values of the friction p ͑X,t;␸͒ϭp ͑X,tϩT ;␸͒. (4.10) coefficient ␥ we can simplify the above inertial dynamics ␮ ␮ ⍀ through adiabatic elimination of the velocity variable The periodic Floquet modes ͕p␮͖ are the (right) eigen- x˙ ϭv (Marchesoni and Grigolini, 1983; Risken, 1984; functions of the Floquet operator

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 235

ץ ͑t͒Ϫ p ͑X,t;␸͒ϭϪ␮p ͑X,t;␸͒. (4.11) tͬ ␮ ␮ץ Lͫ

Here the Floquet modes ͕p␮͖ are elements of the product space L (X) , where is the space of func- 1 T⍀ T⍀ tions that are periodic in time with period T⍀ , and L1(X) is the linear space of the functions that are inte- grable over the state space. In view of the identity

exp͑Ϫ␮t͒p␮͑X,t;␸͒ϭexp͓Ϫ͑␮ϩik⍀͒t͔p␮͑X,t;␸͒ ϫexp͑ik⍀t͒

ϵexp͑Ϫ␮ˆ t͒pˆ ͑X,t;␸͒, (4.12) ␮ FIG. 6. The spectral amplification ␩ for stochastic resonance in where ␮ˆ ϭ␮ϩik⍀, kϭ0,Ϯ1,Ϯ2,..., and the symmetric bistable quartic double well is depicted vs the dimensionless noise strength D at a fixed modulation ampli- pˆ␮(X,t;␸)ϭp␮(X,t;␸)exp(ik⍀t)ϭpˆ ␮(X, tϩT⍀ ;␸), we tude A0ϭ0.2 for three different values of the frequency ⍀. The observe that the Floquet eigenvalues ͕␮n͖ can be defined only mod(i⍀). Likewise, we introduce the set of results were evaluated with the nonadiabatic Floquet theory Floquet modes of the adjoint operator †(t), that is for the corresponding time-periodic Fokker-Planck equation in L Eq. (4.5). After Jung and Ha¨nggi (1991a). ץ †͑t͒ϩ p† ͑X,t;␸͒ϭϪ␮p† ͑X,t;␸͒. (4.13) tͬ ␮ ␮ periodically driven stochastic systems: With ␪ϭ⍀tϩ␸ץ Lͫ we could as well embed a periodic N-dimensional † Here the sets ͕p␮͖ and ͕p␮͖ are bi-orthogonal, obey- Fokker-Planck equation into a Markovian ing the equal-time normalization condition (Nϩ1)-dimensional, time-homogeneous Fokker-Planck

T equation by noting that ␪˙ ϭ⍀. With the corresponding 1 ⍀ † dt dXp␮ ͑X,t;␸͒p ͑X,t;␸͒ϭ␦n,m . stationary probability p (x,␪) not explicitly time de- T ͵ ͵ n ␮m as ⍀ 0 pendent, an integration over ␪ does not yield the ergodic (4.14) probability pas(x,t;␸) in Eq. (4.16) but rather the time- Eqs. (4.11) and (4.13) allow for a spectral representation averaged result ¯p as of Eq. (4.18), not withstanding a of the time inhomogeneous conditional probability claim to the contrary (Hu et al., 1990). P(X,t͉Y,s): With tϾs we ﬁnd Given the spectral representation (4.15) for the con- ϱ ditional probability, we can evaluate mean values and † correlation functions. Of particular importance for sto- P͑X,t͉Y,s͒ϭ ͚ p␮ ͑X,t;␸͒p␮ ͑Y,s;␸͒ nϭ0 n n chastic resonance is the asymptotic expectation value

ϫexp͓Ϫ␮ ͑tϪs͔͒ ͗X͑t͒͘asϭ͗X͑t͉͒Y0 ,t0 Ϫϱ͘, (4.19) n → where X(t)͉Y ,t is the conditional average ϭP͑X,tϩT⍀͉Y,sϩT⍀͒. (4.15) ͗ 0 0͘ ͗X(t)͉Y0 ,t0͘ϭ͐dXXP(X,t͉Y0 ,t0). With P(X,t͉Y0 ,t0 With all real parts Re͓␮n͔Ͼ0 for nϾ0, the limit s Ϫϱ Ϫϱ) approaching the asymptotic time-periodic prob- of Eq. (4.15) yields the ergodic, time-periodic probabil-→ → ability, the relevant asymptotic average ͗X(t)͘as is also ity periodic in time and thus admits the Fourier series representation pas͑X,t;␸͒ϭp␮ϭ0͑X,t;␸͒. (4.16) ϱ The asymptotic probability pas(X,t;␸) can be expanded ͗X͑t͒͘asϭ Mn exp͓in͑⍀tϩ␸͔͒. (4.20) into a Fourier series, i.e., nϭϪ͚ϱ ϱ The complex-valued amplitudes MnϵMn(⍀,A0) de- pas͑X,t;␸͒ϭ am͑X͒exp͓im͑⍀tϩ␸͔͒. (4.17) m͚ϭϪϱ pend nonlinearly on both the forcing frequency ⍀ and the modulation amplitude A0 . Within a linear-response With the arbitrary initial phase being distributed uni- approximation (see Sec. IV.B), only the contributions formly, i.e., with the probability density for ␸ given by Ϫ1 with ͉n͉ϭ0,1 contribute to Eq. (4.20). Nonlinear contri- w(␸)ϭ(2␲) , the time average of Eq. (4.17) equals butions to the stochastic resonance observables, both for the phase average (Jung and Ha¨nggi, 1989, 1990; Jung, M1 and higher-order harmonics with ͉n͉Ͼ1 have been 1993). Hence evaluated numerically by Jung and Ha¨nggi (1989, 1991a) 1 2␲ by implementing the Floquet approach for the Fokker- ¯p ͑X͒ϭ p ͑X,t;␸͒d␸ as 2␲ ͵ as Planck equation of the overdamped driven quartic 0 double-well potential. The spectral ampliﬁcation ␩ of 1 T⍀ Eq. (2.9), i.e., the integrated power in the time-averaged ϭ p ͑X,t;␸͒dtϭa ͑X͒. (4.18) ¨ T ͵ ␮ϭ0 0 power spectral density at Ϯ⍀ (Jung and Hanggi, 1989, ⍀ 0 1991a) [note also Eq. (4.24) below], is expressed in At this stage it is worth pointing out a peculiarity of all terms of ͉M1͉, i.e.,

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 236 Gammaitoni et al.: Stochastic resonance

2 2͉M1͉ ␩ϭ . (4.21) ͩ A ͪ 0 Its behavior versus the noise strength D is depicted for different angular driving frequencies in Fig. 6. We observe that for a fixed modulation amplitude A0 the stochastic resonance behavior of the spectral power amplification ␩ decreases upon increasing the forcing frequency ⍀. The behavior of ␩ versus increasing ⍀ at fixed noise strength D is generally that of a monotonically decreasing function. An exception occurs in a symmetric bistable potential composed of two square wells and a square barrier with one well depth modulated periodi- FIG. 7. The spectral amplification ␩ versus the noise intensity cally. For this case the SNR versus ⍀ has been shown to D at a fixed modulation frequency ⍀ϭ0.1 is depicted for four be nonmonotonic (Berdichevsky and Gitterman, 1996). values of the driving amplitude AϵA0 . The result of the linear response approximation in Eq. (4.51) is depicted by the The dependence on the modulation amplitude A0 at fixed forcing frequency ⍀ is depicted in Fig. 7. We note dotted line. From Jung and Ha¨nggi (1991a). that the maximum of the spectral amplification decreases with increasing amplitude A0 . Hence nonlinear re- complex-valued amplitudes ͕Mn͖ bring in an additional sponse effects tend to diminish the stochastic resonance ¯ phase lag ␾n for each Fourier component (see Sec. IV.B phenomenon. For a small, fixed noise strength D (so below). that the driving frequency ⍀ exceeds the Kramers rate This oscillatory, asymptotic long-time behavior yields rK), the spectral amplification ␩ exhibits, however, a in turn sharp ␦ spikes at multiples of the driving angular maximum as a function of the forcing amplitude frequency ⍀ for the power spectral density of K¯ (␶). De- A0—note the behavior in Fig. 7 below Dϳ0.15, and Fig. pending on the symmetry properties of the Floquet op- 36 in Sec. V.C.5. erator, one finds that some of the amplitudes Mn assume The analog of the correlation function of a stationary vanishing weights (Jung and Ha¨nggi, 1989; Ha¨nggi et al., process is the asymptotic time-inhomogeneous correla- 1993). In particular, for a symmetric double well, all tion even-numbered amplitudes M2n assume zero weight; likewise a multiplicative driving xA0 cos(⍀t) in Eq. (4.2) in a symmetric double well yields identically vanishing ͗X͑t͒X͑tЈ͒͘asϭK͑t,tЈ;␸͒ϭ XYP͑X,t͉Y,tЈ͒ ͵͵ weights for all nϭ0,Ϯ1,Ϯ2,... . ϫp ͑Y,tЈ;␸͒dXdY, (4.22) Before we proceed by introducing the linear-response as theory (LRT), we also point out that the result for the where tϭtЈϩ␶, with ␶у0 and tЈ ϱ. An additional av- corresponding conditional probability (4.15) for zero → eraging procedure (indicated by the double brackets) forcing A0ϭ0 boils down to the time-homogeneous con- over a uniformly distributed initial phase ␸ for ditional probability density, i.e., with ␶ϭ(tϪs)Ͼ0 K(t,tЈ;␸) (or equivalently, a time average over one ϱ modulation cycle) yields a time-homogeneous, station- P0͑X,␶͉Y,0͒ϭ ͚ ␺n͑X͒␸n͑Y͒exp͑Ϫ␭n␶͒. (4.25) ary correlation function nϭ0

1 Here, for A0 0 the set ͕␮n͖ (with kϭ0) reduces to the ¯ → K͑␶͒ϭ͗͗X͑t͒X͑tЈ͒͘͘asϵ K͑t,tЈ;␸͒d␸. (4.23) set of eigenvalues ͕␭ ͖ of , the set ͕p (X,t)͖ reduces 2␲ ͵ n 0 ␮n L † to the right eigenfunction ͕␺n͖ of 0 , and ͕p (Y,s)͖ to In terms of the Fourier amplitudes ͕M ͖ of Eq. (4.20), L ␮n n the right eigenfunctions ͕␸ (Y)͖ of † , respectively. the long-time limit of K¯ (␶) assumes the oscillatory ex- n L0 pression B. Linear-response theory ␶ ϱ ¯ → ¯ K͑␶͒ —— ϵKas͑␶͒ϭ͗͗X͑tϩ␶͒͘as͗X͑t͒͘as͘ → As detailed in the Introduction, the prominent role of ϱ the stochastic resonance phenomenon is that it can be 2 ϭ ͉Mn͉ exp͑in⍀␶͒ used to boost weak signals embedded in a noisy environ- nϭϪ͚ϱ ment. Thus the linear-response concept, or more gen- ϱ eral, the concept of perturbation theory (see Appendix) 2 ϭ2 ͚ ͉Mn͉ cos͑n⍀␶͒. (4.24) for spectral quantities like the Floquet modes and the nϭ1 Floquet eigenvalues as discussed in the previous section In the last equality we used the fact that M0ϭ0 for a are adequate methods for studying the basic physics that reﬂection-symmetric potential. Note that this asymptotic characterizes stochastic resonance. Both concepts have result is independent of the initial phase ␸ (no phase lag been repeatedly invoked and investigated in stochastic here!). This is in contrast with ͗X(t)͘as , where the resonance studies by several research groups (Fox, 1989;

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 237

McNamara and Wiesenfeld, 1989; Presilla et al., 1989; For ␦x(t)ϭx(t)Ϫ͗x(t)͘0 this can be recast as Dykman et al., 1990a, 1990b; Hu et al., 1990; Jung and d ¨ Hanggi, 1991a, 1993; Dykman, Mannella, et al., 1992; ␹͑t͒ϭϪH͑t͒ ͗␦x͑t͒␨(x͑0͒)͘0 . (4.31) Hu, Haken, and Ning, 1992; Dykman, Luchinsky, et al., dt 1995). Here we shall focus on the linear-response con- This result is intriguing: the linear-response function can cept, which also emerges as a specific application of per- be obtained as the time derivative of a stationary, gen- turbation theory. In doing so, we shall rely on the linear- erally nonthermal correlation function between the two response theory pioneered by Kubo (1957, 1966) for unperturbed fluctuations ␦x(t) and ␨(x(t)). From the equilibrium systems—and extended by Ha¨nggi and Tho- spectral representation of the time-homogeneous condi- mas (1982) to the wider class of stochastic processes that tional probability (4.25), it follows immediately that (on admit also nonthermal, stationary nonequilibrium states. assuming that the eigenvalue ␭0ϭ0 is not degenerate) This extension is of particular relevance because many ϱ prominent applications of stochastic resonance in opti- ␹͑t͒ϭH͑t͒ ͚ gn␭n exp͑Ϫ␭nt͒. (4.32) cal, chemical, and biological systems operate far from nϭ1 thermal equilibrium. Without lack of generality, we con- The coefficients ͕gn͖ are given by fine the further analysis to a one-dimensional Markovian observable x(t) subjected to an external weak periodic gnϭ͗␦x␺n͑x͒͘0͗␨͑y͒␸n͑y͒͘0 . (4.33) perturbation. Following Ha¨nggi (1978) and Ha¨nggi and The corresponding Fourier transform Thomas (1982), the long-time limit of the response ϱ ͐0 exp(Ϫi␻␶)␹(␶)d␶ will be denoted by ␹(␻), with ͗x(t)͘as due to the perturbation A(t)ϭA0 cos(⍀t), i.e., ␹(␻)ϭ␹Ј(␻)Ϫi␹Љ(␻). Generally, the eigenvalues of we set ␸ϭ0, assumes up to first order the form the real-valued operator are complex valued and oc- L0 t cur by the pair, e.g., ␭n and ␭n* with the corresponding ͗x͑t͒͘ ϭ͗x͑t͒͘ ϩ ds␹͑tϪs͒A cos͑⍀s͒, as 0 ͵ 0 eigenfunctions ␺n(x) and ␾n(x) introduced above. Ϫϱ Hence the contribution of each pair of complex conju- (4.26) gate eigenvalues is a real-valued quantity, thus yielding where ͗x(t)͘0 denotes the stationary average of the un- an overall real expression for ␹(t) in Eq. (4.32). Upon perturbed process. The memory kernel ␹(t) of Eq. substituting Eq. (4.32) into Eq. (4.26) we find the linear- (4.26) is termed, hereafter, the response function. For an response approximation external perturbation operator of the general form ϱ A0 ext͑t͒ϵA0 cos͑⍀t͒⌫ext , (4.27) ͗␦x͑t͒͘ϭ͗x͑t͒͘asϪ͗x͑t͒͘0ϭ ͚ ␭ngn L 2 nϭ1 ␹(t) is expressed by ei⍀t eϪi⍀t ϫ ϩ . (4.34) ͫ␭ ϩi⍀ ␭ Ϫi⍀ͬ ␹͑t͒ϭH͑t͒ ͵͵͵dxdydzP0͑x,t͉y,0͒x⌫ext͑y,z͒p0͑z͒. n n (4.28) Moreover, on Fourier transforming Eq. (4.32) we derive the spectral representation of ␹(t), i.e., H(t) denotes the (Heaviside) step function expressing causality of the response, p (z) is the stationary prob- ϱ 0 ␭ngn ability density of the corresponding unperturbed, gener- ␹͑␻͒ϭ␹Ј͑␻͒Ϫi␹Љ͑␻͒ϭ ͚ . (4.35) nϭ1 ␭nϩi␻ ally nonthermal equilibrium process, P0(x,t͉y,0) denotes the conditional probability density, and ⌫ext(x,y) Therefore, the result of Eq. (4.34) can be cast into the denotes the kernel of the operator ⌫ext that describes form the perturbation in the master operator (either an inte- ¯ gral operator or a differential operator such as in the ͗␦x͑t͒͘ϭ2͉M1͉cos͑⍀tϪ␾͒, (4.36)

Fokker-Planck case of Sec. IV.A, e.g., ⌫ext(y,z) where the spectral amplitude ͉M1͉ is given by ϭ␦Ј(zϪy) for Eq. (4.8)). An appealing form of the re- A0 sponse function can be obtained by introducing the fluc- ͉M ͉ϭ ͉␹͑⍀͉͒, (4.37) tuation ␨(x(t)) defined by 1 2 and the retarded positive phase shift ␾¯ reads dy⌫ext͑x,y͒p0͑y͒ϭϪ dz 0͑x,z͒␨͑z͒p0͑z͒, ͵ ͵ L ␹Љ͑⍀͒ (4.29) ␾¯ϭarctan . (4.38) ͭ ␹Ј͑⍀͒ͮ where 0(x,z) is the kernel of the unperturbed Fokker- PlanckL operator. We note that ␨(x(t)) is indeed a fluc- The above results are valid for a general nonthermal stationary system. The fluctuation ␨(x(t)) can be evalu- tuation, i.e., it satisfies ͗␨(x(t))͘0ϭ0. The response function (4.28) can then be expressed through the fluctuation ated in a straightforward manner for all one-dimensional theorem systems modeled by a Fokker-Planck equation. Ex- amples include the stochastic resonance for absorptive d optical bistability, Sec. V.A.3, or that for colored noise- ␹͑t͒ϭϪH͑t͒ ͗x͑t͒␨(x͑0͒)͘ . (4.30) dt 0 driven bistable systems in Sec. VI.D.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 238 Gammaitoni et al.: Stochastic resonance

Ϫ1 For the case of the quartic double-well potential [Eqs. p0͑x͒ϭZ ͑x͒exp͑Ϫ⌽0͑x͒/D͒. (4.44) (2.1)–(2.3)], where the unmodulated system admits ther- The relevant intrawell relaxation rates in the two wells mal equilibrium, the perturbation operator ext(t) L located at xϭx0 are estimated as the real part of the is of the gradient type: from Eq. (4.8), ext(t) 1,2 L two smallest eigenvalues that describe the equilibration -x͔. This, in turn, implies that the reץ/ץϭA0 cos(⍀t)͓Ϫ sponse function obeys the well-known ﬂuctuation- of the probability density in the vicinity of the two stable dissipation theorem known from classical equilibrium states xm , mϭ1,2, respectively. For small noise intensi- statistical mechanics (Kubo, 1957, 1966), i.e., ties, these eigenvalues can be approximated as

d ␭2ϭ⌽0Љ͑xϭx1͒ (4.45) ␹͑t͒ϭϪ͓H͑t͒/D͔ ͗␦x͑t͒␦x͑0͒͘0 , (4.39) dt and where the corresponding fluctuation ␨ reads ␨(x(0))ϭ␦x(0)/D. Note that this result holds true irre- ␭3ϭ⌽0Љ͑xϭx2͒. (4.46) spective of the detailed form of the equilibrium dynam- Note that, here, the indices of ␭2 and ␭3 have been cho- ics. sen for later convenience and do not necessarily coincide with the index ordering of the Fokker-Planck spectrum ͕␭n͖. Given these three dominant time scales, the response at weak noise is cast as the sum of three terms, 1. Intrawell versus interwell motion i.e., for a driving phase ␸ϭ0 we have the weak-noise approximation Given the spectral representation (4.35) of the re- i⍀t Ϫi⍀t sponse function ␹(t), we can express the two stochastic A0 e e ͗␦x͑t͒͘ϭ ͚ ␭mgm ϩ , resonance quantifiers, namely the spectral amplification 2 mϭ1,2,3 ͫ␭ ϩi⍀ ␭ Ϫi⍀ͬ m m ␩ of Eqs. (2.9), Eq. (4.21), and the signal-to-noise ratio (4.47) of Eq. (2.13) in terms of the spectral amplitude ͉M1͉. From Eq. (4.36) we find for the spectral amplification yielding corresponding estimates for ␹(⍀) and the sto- within linear response chastic resonance quantifiers ␩ and the SNR. The weights g can be evaluated from the corresponding ap- 2 2 m ␩ϭ͑2͉M1͉/A0͒ ϭ͉␹͑⍀͉͒ . (4.40) proximate eigenfunctions (Ha¨nggi and Thomas, 1982; In view of the unperturbed power spectral density Dykman, Haken, et al., 1993), or from a three-term ex- S0 (⍀) of the fluctuations ␦x(t), i.e., ponential ansatz for the response function (Jung, 1993). N For the overdamped, symmetric quartic double-well ϱ 0 Ϫi␻␶ dynamics [Eq. (4.7)], the spectral amplification given by SN͑␻͒ϭ e ͗␦x͑␶͒␦x͑0͒͘0d␶, (4.41) ͵Ϫϱ Eq. (4.40) has been evaluated in the literature by means of Eq. (4.37) to give (Jung and Ha¨nggi, 1991a, 1993) the linear response result for the SNR reads 2 2 2 2 ϭ 2 0 ϭ 2 2 0 4g1rK g ␣ SNR 4␲͉M1͉ /SN͑⍀͒ ␲A0͉␹͑⍀͉͒ /SN͑⍀͒. ␩ϭDϪ2 ϩ (4.42) ͫ 4r2 ϩ⍀2 ␣2ϩ⍀2 K Both stochastic resonance observables possess a spectral 2 4g1g␣rK͑2␣rKϩ⍀ ͒ representation via the spectral representations of ͉␹(␻)͉ ϩ 2 2 2 2 . (4.48) 0 ͑4r ϩ⍀ ͒͑␣ ϩ⍀ ͒ ͬ K and SN(␻). In the following we shall explicitly assume that the where ␭2ϭ␭3ϵ␣, with ␣ϭ2, and g2ϭg3ϵg/2. The rel- noise strength D is weak. This implies that for a general evant weights g for D 0 read bistable dynamics there exists a clear-cut separation of n → Ϫ1 2 time scales. These are the escape time scale to leave the g1Ӎ1Ϫ͑1ϩ␣ ͒DϩO͑D ͒, corresponding wells, i.e., the exponentially large time scale for interwell hopping, and the time scale that char- gϭD/␣ϩO͑D2͒, (4.49) acterizes local relaxation within a metastable state. The and rK is the steepest-descent approximation for the eigenvalue ␭1 that characterizes the intrawell dynamics is always real valued and of the Kramers type (Ha¨nggi Kramers rate et al., 1990), i.e., Ϫ1 rKϭ͑&␲͒ exp͓Ϫ1/͑4D͔͒. (4.50) ␭1ϭ2rKϭrϩϩrϪϵ␭ , (4.43) Upon neglecting the intrawell motion, the leading-order where rϮ are the forward and backward transition rates, contribution in Eq. (4.48) reproduces Eq. (2.7a), i.e., respectively. The rates rϮ depend through the Arrhen- Ϯ 2 Ϫ1 ius factor on the activation energies ⌬⌽0 , where ⌽0(x) 1 ␲ 1 ␩Ӎ 1ϩ ⍀2 exp . (4.51) is the generalized (non-thermal-equilibrium) potential D2 ͫ 2 ͩ 2Dͪͬ associated with the unperturbed stationary probability density This approximation exhibits the typical bell-shaped sto-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 239 chastic resonance behavior as a function of increasing 0 4rK 2g␭2 noise intensity D—see again Fig. 7, and also Figs. 18 and SN͑⍀͒Ӎ 2 2 ϩ 2 2 , (4.52) 4rKϩ⍀ ␭2ϩ⍀ 19 below. Likewise, we can evaluate the SNR for the whence yielding the linear-response result for the SNR potential under study. In the weak-noise limit we have (Hu et al., 1992; Jung, 1993), i.e.,

2 2 2 2 2 2 2 2 2 ␲A0 4g1rK͑␣ ϩ⍀ ͒ϩ͑g␣͒ ͑4rKϩ⍀ ͒ϩ4␣g1rK͑2␣rKϩ⍀ ͒ SNRϭ 2 2 2 2 2 . (4.53) 2D 2g1rK͑␣ ϩ⍀ ͒ϩg␣͑4rKϩ⍀ ͒

This result, when plotted vs D displays a bell-shaped 2. Role of asymmetry behavior for ⍀ not too large (see Fig. 8). Moreover, note In this subsection we study the effect of a potential that the result for the SNR diverges as D 0, propor- asymmetry on stochastic resonance detectability. Here, tional to DϪ1. This is due to the intrawell contributions→ the asymmetry of ⌽0(x) is characterized by the differ- in Eq. (4.53). This feature is in agreement with simula- Ϫ ϩ ence ⑀ϭ⌬⌽0 Ϫ⌬⌽0 between the Arrhenius energies tions (McNamara and Wiesenfeld, 1989). On neglecting for backward and forward transitions. We shall assume intrawell contributions, i.e., by setting g ϭg ϭ0, one 2 3 that ⑀Ͼ0; thus the backward rate rϪ is exponentially finds in leading order the result of Eq. (2.14) (Gammai- suppressed over the forward escape rate. Such an asym- toni, Marchesoni, et al., 1989; McNamara and Wiesen- metry implies also an exponential suppression of the feld, 1989; Presilla et al., 1989; Dykman et al., 1990b), corresponding weight g1ϳ1 exp(Ϫ⑀/D) [see Eq. i.e., (6.3.46) of Ha¨nggi and Thomas,→ 1982]. As a consequence, the spectral amplification suffers an exponential SNRϭ͑␲A2/D2͒r ϭ͓A2/͑&D2͔͒exp͓Ϫ1/͑4D͔͒. 0 K 0 suppression proportional to ͓exp(Ϫ⑀/D)͔2ϭexp(Ϫ2⑀/D), (4.54) while the suppression of the SN ratio is weaker, being We remark that within this interwell approximation the proportional to exp(Ϫ⑀/D). SNR—contrary to the spectral amplification ␩ in Eq. On inspecting the leading order results in Eq. (4.51) (4.51)—is no longer dependent on the angular modula- for ␩ and Eq. (4.54) for SNR, we note that the stochas- tion frequency ⍀! This effective two-state approxima- tic resonance maximum is located in the neighborhood Ϫ2 tion also exhibits a bell-shaped behavior, typical for sto- where the monotonic decreasing function y1ϭD chastic resonance. In contrast to Eq. (4.53) the SN ratio crosses the monotonic, exponentially increasing Arrhen- vanishes for D 0. It is also worthwhile to point out the ius factor y2ϭexp(Ϫ⌬⌽0 /D) for the symmetric barrier → ϩ Ϫ difference in how rK enters the two stochastic resonance with ⌬⌽0 ϭ⌬⌽0 ϭ⌬⌽0 . The suppression caused by observables. The leading order contribution to SNR in asymmetry now modifies y2 into exp(Ϫ⑀/D)y2 . Hence Eq. (4.54) is proportional to rK , while ␩ in Eq. (4.51) is the intersection point of y1 and y2 as functions of D is 2 proportional to rK . moved to larger noise intensities. Both the exponential decrease (induced by the asymmetry in activation barriers in the unperturbed potential) of the peak for ␩ (and likewise for the SNR), as well as the shift to larger noise intensities of the peak position have been confirmed numerically for a nonequilibrium optical bistable system (Bartussek, Jung, and Ha¨nggi, 1993; Bartussek, Ha¨nggi, and Jung, 1994) (see also Sec. V.A.3) and again numerically in an asymmetric rf SQUID loop by Bulsara, In- chiosa, and Gammaitoni (1996). The detailed analysis for an asymmetric quartic bistable well with asymmetry energy ⑀, but identical curvatures, gives for the spectral amplification (Jung and Bartussek, 1996; Grifoni et al., 1996) 1 ⑀ ⍀2 Ϫ1 ␩ϭ cosh4 1ϩ , (4.55) D2 ͫ ͩ 2Dͪͩ 4r2 ͑⑀͒ͪͬ K

with rK(⑀)ϭrK cosh(⑀/2D), while the corresponding re- FIG. 8. The signal-to-noise ratio as function of the noise sult for the SNR in the in presence of an asymmetry ⑀ strength D for A0ϭ0.1 and different driving frequencies ⍀.In reads contrast to the spectral ampliﬁcation ␩—see Fig. 7—we note 2 ␲A0 1 that the signal-to-noise ratio diverges as the noise strength SNR͑⑀͒ϭ r ͑⑀͒. (4.56) D 0. D2 cosh2͑⑀/2D͒ K →

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 240 Gammaitoni et al.: Stochastic resonance

An important feature of Eq. (4.55) is its universal shape for vanishing driving frequencies: In contrast to the symmetric case ⑀ϭ0, where the maximum of the spectral amplification increases with decreasing driving frequencies ⍀ (see Fig. 6 and Fig. 18 below for the optical bistability) the spectral amplification in asymmetric systems (see Fig. 19 below) approaches for ⍀ 0 a limiting curve, with the stochastic resonance maximum→ assumed at a finite noise level. As a result, there exists no obvious time-scale matching condition in asymmetric systems.

3. Phase lag

The asymptotic probability pas(x,t;␸) [see Eq. (4.17)] ¯ depends periodically on the modulation phase FIG. 9. Phase shift ␾ of stochastic resonance in a periodically driven, overdamped quartic double well for a dimensionless ␪ϭ⍀tϩ␸. Moreover, due to the complex-valued ampli- noise strength Dϭ0.05 vs driving frequency ⍀ for increasing tudes am(x), the contribution to pas(x,t;␸) stemming driving amplitudes (1) A0ϭ0.1, (2) A0ϭ0.3, (3) A0ϭ0.4, (4) from the pair of Ϯm introduces each its own additional A ϭ0.5, and (5) A ϭ1. The lines are evaluated within a full ¯ 0 0 phase lag ␾m . For periodically driven (linear) Gauss- nonadiabatic Floquet approach for the overdamped, time- Markov processes, only the terms with mϭϮ1 and periodic Fokker-Planck equation; see Jung and Ha¨nggi (1993). mϭ0 contribute to Eq. (4.17). The corresponding phase ¯ lag ␾1 for the asymptotic probability of a Brownian har- the maximum that characterizes stochastic resonance. monic oscillator has been evaluated explicitly as a func- Put differently, the noise value for the maximum of ␩,or tion of the friction coefficient ␥ by Jung and Ha¨nggi SNR, is in no immediate relationship with the noise (1990). Analogously, in Eq. (4.20) each nonlinear contri- value that characterizes the maximum of ␾¯: The sto- bution to ͗x(t)͘as with power amplitude Mn introduces chastic resonance phenomenon is rooted in a physical ¯ its own phase lag ␾m . synchronization effect between the interwell time scale From the linear-response function ␹(⍀) we obtain the and the period of the modulation signal, which acts here ¯ ¯ phase lag ␾ϵ␾1 of Eq. (4.38). Correspondingly, the as an external ‘‘clock’’ (Jung and Ha¨nggi, 1991a, 1993; linear-response approximation for pas(x,t;␸) in Eq. Fox and Lu, 1993; Gammaitoni, Marchesoni, and San- (4.17) yields such a phase lag in terms of the amplitude tucci, 1995). In contrast, a peak in ␾¯ vs noise intensity D M1 of Eqs. (4.20) and (4.37). Neglecting all the intrawell at small amplitude A0 and small angular frequency ⍀ is terms gn with nу2 in Eq. (4.32) gives the single- due to the competition between hopping and intrawell exponential approximation (2.7b) for the phase lag ␾¯ dynamics—and should not be mistaken for a signature (Nicolis, 1982; McNamara and Wiesenfeld, 1989; Gam- of stochastic resonance. The peak behavior of ␾¯ van- maitoni, Marchesoni, et al., 1991), i.e., ishes with increasing modulation amplitude A0 (Jung ¨ ⍀ and Hanggi, 1993), where no clear-cut time-scale sepa- ␾¯ϭarctan . (4.57) ration for hopping versus intrawell motion occurs. Such ͩ 2rK ͪ a dependence on the modulation amplitude is depicted ␾¯ decreases monotonically from ␲/2 at Dϭ0ϩ to zero in Fig. 9 as a function of the angular driving frequency (for ⍀ 0) as D is made to grow to infinity. Note that ⍀. The characteristic dependence of ␾¯ on the driving → this is the equivalent of the two-state approximation of strength A0 (⍀ held constant) clearly lies beyond the Sec. III. The inclusion of intrawell terms (Dykman, regime of validity of linear-response theory (Jung and Mannella, et al., 1992; Gammaitoni and Marchesoni, Ha¨nggi, 1993; Go´ mez-Ordo´ñez and Morillo, 1994). 1993; Dykman, Mannella, McClintock, and Stocks, 1993) changes this monotonic behavior into a bell-shaped be- C. Residence-time distributions havior, as long as the modulation amplitude A0 remains small. This feature is consistent with the linear-response The residence-time distribution offers another possi- result [Eq. (4.47)], which accounts for the hopping term bility to characterize stochastic resonance. Historically, with rate rK and the two intrawell terms with rates ␭2 residence-time distributions were first employed in the and ␭3 . In particular, in a symmetric bistable potential stochastic resonance literature by Gammaitoni, with D 0, ␾¯ approaches arctan(⍀/␣)—see the defini- Marchesoni, Menichella-Saetta, and Santucci (1989) and tion of ␣→below Eq. (4.48). The presence of the intrawell Zhou and Moss (1990) and then interpreted theoreti- dynamics suppresses at low noise the influence of the cally by Zhou, Moss, and Jung (1990). interwell dynamics on the phase lag. Hence within the Residence-time distributions were mentioned briefly regime of validity of the linear-response approximation, in Sec. II. In this section, we discuss in more detail how the peak of the phase shift marks the turnover between these residence-time distributions can be obtained ap- the regimes dominated by hopping and intrawell mo- proximately and how stochastic resonance is manifested tion. Note that this maximum is not physically related to in their properties.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 241

In the absence of periodic forcing, an escape-time dis- double-well potential upon combining the Kramers ap- tribution is defined as the distribution of times it takes proach with the adiabatic assumption of a slow potential for the system to escape out of a potential well. For Vad(x,t) (Jung, 1989), i.e., weak noise, such a distribution is independent of the 1 initial point apart from a small boundary layer around rϩ͑⍀tϩ␾͒ϭ ͱ͉VadЉ ͑xb͉͒VadЉ ͑xm͒ the basin boundary. In periodically driven systems, 2␲ escape-time distributions depend additionally on the ini- ⌬VϪ͑⍀tϩ␾͒ tial phase of the periodic forcing—they represent there- ϫexp Ϫ , (4.60) ͫ D ͬ fore a conditional escape-time distribution. Residence-time distributions are defined as the distri- where the barrier height ⌬VϪ(⍀tϩ␾) and the curva- bution of time intervals ⌬n between two consecutive es- tures of the adiabatic potential at the barrier top VЉ(xb) cape events, regardless of the phases of the periodic and in the left potential minimum VЉ(xm) can be ob- forcing ␾ϭ⍀t at which these switching events occur. tained for small A0 to give Each time interval ⌬n corresponds to a different switch- 3 ing phase ␾ which in turn depends on the prehistory of r ͑⍀tϩ␾͒Ϸr 1Ϫ A cos͑⍀tϩ␾͒ n ϩ K 4 0 the process. The approximation strategy of Zhou, Moss, ͫ ͬ and Jung (1990) to obtain the residence-time distribu- A0 ϫexp cos͑⍀tϩ␾͒ . (4.61) tions is to first compute the conditional escape-time dis- ͫ D ͬ tribution and then to average over the distribution function of the switching phases ␾: i.e., the temporal The escape rate of the undriven system rK is given in modulation of the potential must be slow. Eq. (2.4). As mentioned above, the applicability of Eq. Although we refer to the quartic double-well potential (4.61) is restricted to the adiabatic regime, i.e., the fre- [Eq. (2.2)] throughout, the conclusions we arrive at are quency ⍀ has to be small compared to the local relax- of general validity since only hopping between the stable ation rate. In our scaled units this means ⍀Ӷ2, as well as states is taken into account—relaxational motion within weak forcing, i.e., A0Ӷͱ4/27. The two-state model rates the potential wells is neglected. We therefore essentially WϮ(t) of Eq. (3.3) can thus be recovered from this adia- resort to a two-state description, discussed in Sec. II. batic theory when prefactor corrections due to the forc- Furthermore, the subsequent analysis holds true only for ing are neglected. low forcing frequencies according to the adiabatic ap- On coming back to Eq. (4.58), we note that the con- proximation; this means that the temporal change of the ditional escape-time distribution ␳e(t;␾) can be written adiabatic potential [Eq. (4.59) below] has to be slow in as comparison to the intrawell relaxation. Such a restric- ␳ ͑t;␾͒ϭϪn˙ Ϫ͑t;␾͒ϭrϩ͑⍀tϩ␾͒ tion is needed here for the definition of the interwell e ⍀t transition rates rϮ(t) to make sense. On the other hand, 1 it is clear (see below) that under such a circumstance, ϫexp Ϫ rϩ͑␪ϩ␾͒d␪ . (4.62) ͫ ⍀ ͵0 ͬ the two-state model is a good approximation to the continuous dynamics of a bistable process. In order to obtain the residence-time distribution we A detailed calculation of both the escape- and have to find an expression for the distribution of the residence-time distributions has been reported in a re- jump phase out of the left well YϪ(␾). It should be cent paper by Choi, Fox, and Jung (1998). To help the noted that this problem has not yet been solved system- reader interpret the results in Figs. 4 and 5, without bog- atically. For driving frequencies much smaller than the ging down in complicated algebraic manipulations, we Kramers rate, there is no preferred phase and thus outline here the simplified approach developed earlier Y(␾)ϭ1/(2␲). For larger frequencies, the following by Zhou, Moss, and Jung (1990), with the caution that self-consistent approximation has been proposed: its validity is restricted to relatively large values of ⍀. 1 A0 The starting point for the calculation of the conditional Y ͑␾͒ϭ exp Ϯ cos͑␾͒ , (4.63) escape-time distribution ␳ (t) out of the left potential Ϯ 2␲I ͑A /D͒ ͩ D ͪ e 0 0 well is the instantaneous rate equation for the popula- with I (x) being a modified Bessel function (Abramow- tion in the left well n (t,␾), i.e., 0 Ϫ itz and Stegun, 1965). The residence-time distribution in n˙ Ϫ͑t;␾͒ϭϪrϩ͑⍀tϩ␾͒nϪ͑t;␾͒. (4.58) the symmetric bistable potential of Eq. (2.2) thus reads

The quasistationary forward rate rϩ(⍀tϩ␾) denotes 2␲ the adiabatic transition rate, obtained for a frozen po- N͑T͒ϭ͗N͑t;␾͒͘ϭ YϪ͑␾͒rϩ͑⍀Tϩ␾͒ ͵0 tential 4 2 1 ⍀T Vad͑x,t͒ϭx /4Ϫx /2ϪA0x cos͑⍀tϩ␾͒. (4.59) ϫexp Ϫ r ͑␪ϩ␾͒d␪ d␾. (4.64) ⍀ ͵ ϩ Initially, the system is in the left well, yielding the initial ͫ 0 ͬ condition nϪ(0,␾)ϭ1. The quasistationary rate Performing a series expansion for small A0 /D in Eq. rϩ(⍀tϩ␾) can be obtained from the weakly driven (4.64) we ﬁnd after some algebra that

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 242 Gammaitoni et al.: Stochastic resonance

FIG. 10. Observable ¯x (arbitrary logarithmic scale) and height FIG. 11. Observable ¯x (arbitrary logarithmic scale) and height of the nth peak Pn with nϭ1,2,3 vs D for ␯⍀ϭ40 Hz and of the nth peak Pn with nϭ1,2,3 vs ␯⍀ for Dϭ0.3⌬V and 1 ϭ ␣ϭ 4 . Data obtained by means of analog simulation of system ␣ 1/4. Other simulation circuit parameters are as in Fig. 10. 4 −1 Ϫ2 of Eqs. (4.3)–(4.7) with A0xmϭ0.5⌬V and aϭ3.2ϫ10 s . Af- An independent measurement yielded ␮Kϭ1.8ϫ10 a. After ter Gammaitoni, Marchesoni, and Santucci (1995). Gammaitoni, Marchesoni, and Santucci (1995).

2 1 A0 troduce the areas under the peaks ϪrKT N͑T͒ϭ 0 1Ϫ cos͑⍀T͒ rKe , (4.65) N ͫ 2 ͩ D ͪ ͬ T ϩ␣T n ⍀ P ϭ N͑T͒dT, (4.67) Ϫ1 1 2 2 n ͵ with ϭ1Ϫ (A x /D) /͓1ϩ(⍀/r ) ͔. To indicate TnϪ␣T⍀ N0 2 0 m K the dimensional dependence we use 1ϭx throughout m with nϭ1,2,... and 0Ͻ␣р1 . The actual value of the the remaining part of this section. The opposite limit, 4 parameter ␣ is immaterial for the behavior of P1 . Let us namely ⌬VӷA0xmӷD, is reported here for completeness, though it will be used only in Sec. IV.D. The focus now on the distribution of Eq. (4.65). In the re- saddle-point approximation for the integral in Eq. (4.64) gime of validity of Eq. (4.65), i.e., rKϽ⍀Ӷ2, the back- yields ground of the distribution N(T) is negligible. The strength Pn of the nth peak is thus a function of the ratio N͑T͒Ӎexp͓Ϫ͑A0 /D͒cos͑⍀T͔͒ rK /⍀ alone. As a consequence, Pn attains its maximum 1/2 by setting either the forcing frequency ␯⍀ to ϫexp͕Ϫ͑rmax/2⍀͒͑2␲D/A0͒ ␯ Ӎ͑2nϪ1͒r /2, (4.68) ¯ n K ϫ͓2nϩ1ϩerf͑ͱA0/2D͑⍀TϪ␲͔͖͒, (4.66) or tuning the noise, with constant ␯⍀ , according to Eq. where erf(x) denotes the error function (Abramowitz (4.68). The picture of SR as a ‘‘resonant’’ synchroniza- ¯ and Stegun, 1965), rmaxϭrK exp(A0 /D) and ⍀T tion mechanism is thus fully established (Gammaitoni, ϭmod(⍀T,2␲). Both limiting expressions (4.65) and Marchesoni, and Santucci, 1995). However, the reader (4.66) for N(T) exhibit a series of peaks centered at should keep in mind that although the weight P1 exhib- Tnϭ(2nϪ1)(T⍀/2) (see Fig. 5). The location of the first its a maximum as a function of the frequency, the under- peak is due to the fact that the clock was triggered at lying mechanism is not a resonance in the sense of dy- Tϭ0, immediately after the system had crossed the bar- namical systems. While a dynamical resonance is due to rier at xϭ0 and half a forcing period before the relevant the interaction of two degrees of freedom when their escape barrier attained a minimum for the first time. time scales agree, the nature of the peak of P1 as a The nth peak corresponds to the event that the system function of ⍀ is merely due to the coincidence of two switches first after nϪ1 entire periods. Such ‘‘wait time scales. For the sake of comparison, in Figs. 10 and loops’’ do not correspond to any subharmonic compo- 11 we show the dependence of Pn on D and ⍀, as pro- nent in the process power spectral distribution as duced by means of an analog simulation. The basic pointed out in Sec. II.B. properties of the synchronization mechanism are clearly We are now in a position to discuss the synchroniza- visible: (i) For D tending to zero at fixed ⍀ (Fig. 10), all tion mechanism that occurs in the bistable potential of Pn approach a constant value independent on n, as ex- Eq. (2.2) subjected to a periodic driving. In order to pected in the nonadiabatic weak-noise limit of Sec. quantify the strength of the nth peak of N(T), we in- IV.D. (ii) Each curve PnϭPn(D) passes through a

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 243

FIG. 13. Simulated residence-time distribution N(T) for the system of Eqs. (4.69) and (4.70) with ␯⍀ϭ40 Hz, A0ϭ1, and different values of B. Other circuital parameters are a 4 −1 ¯ ϭ2ϫ10 s , xmϭ1, and Dϭ0.16⌬V. After Gammaitoni, Marchesoni, and Santucci (1994).

2 ͗f͑t͒f͑tЈ͒͘ϭ2D␦͑tϪtЈ͓͒A0 cos͑⍀t͒ϩB͔ . (4.71) In Fig. 12 we show examples of digitized time series for f(t) in the three typical cases discussed below, i.e., for A0ϭ1 and Bϭ0, for Bϭ0.2 and Bϭ1, and for fixed ¯ 2 2 FIG. 12. Digitized time series for f(t) with ␯⍀ϭ40 Hz, A0ϭ1, effective noise intensity DϭD(B ϩA0/2). and different values of B. The envelope function Contrary to the process of Eqs. (2.1)–(2.3), the pro- Ϯ͉A0 cos(⍀t)ϩB͉ is drawn for convenience. Ordinate units are cess under investigation here is symmetric under parity arbitrary (Gammaitoni, Marchesoni, and Santucci, 1994). transformation x Ϫx at any time. As a consequence, → ͗x(t;␾)͘asϭ0 vanishes identically and no peak is detect- maximum, the position Dn of which shifts progressively able in the power spectrum S(␻). towards smaller values with the index n. (iii) The ob- Evidence of stochastic resonance effects can be de- servables ¯x (D) and P1 peak at different D values, in tected through a synchronization-based analysis (Gam- agreement with the predictions for D , Eq. (2.8), and SR maitoni, Marchesoni, Menichella-Saetta, and Santucci, D , Eq. (4.68). (iv) Contrary to ¯x , which decreases 1 1994). In Fig. 13, residence-time distributions are dis- monotonically with the forcing frequency, the curves played for the same A0 and B values as in Fig. 12. The PnϭPn(␯⍀) of Fig. 11 exhibit a clear-cut resonant-like profile. The positions ␯ of the maxima of P are appar- series of exponentially decaying peaks is apparent. n n Moreover, the dependence of N(T) on the offset pa- ently odd multiples of the fundamental frequency rK/2; (v) Moreover, we notice that the inequality rameter B [see Eq. (4.71)] is also noteworthy. As illustrated in Fig. 12 for BуA0 the modulation period of Pn(␯⍀)ϾPm(␯⍀) holds for nϽm in the whole range of simulated forcing frequencies. Finally, we stress that for f(t)isT⍀, whence the peaks of N(T) are centered at T ϭnT . For Bϭ0, instead, the f(t) modulation pe- ⍀ϳrK or smaller, condition in Eq. (4.68) may be ful- n ⍀ filled by the exponential background of the distribution riod is half the forcing period and, accordingly, the peaks of N(T) are located at T ϭnT /2. In the inter- N(T) alone, even in the absence of peaks at Tn.Of n ⍀ mediate case 0ϽBϽA , two series of higher and lower course, in this limit the quantities Pn provide no charac- 0 terization of the synchronization mechanism. peaks show up with maxima at the even and odd mul- As an application of the residence-time analysis, we tiples of T⍀/2, respectively. now show how stochastic resonance may occur in the When the strength of the first peak is plotted against absence of symmetry breaking. Following Dykman, the noise intensity D, the curves P1(D) pass through a Luchinsky, et al. (1992), we consider the symmetric maximum for D1ϭD1(B). The B dependence of D1 bistable process can be interpreted quantitatively as follows. For A0ӶB the rate performs small oscillations about the unper- x˙ ϭϪVЈ͑x͒ϩf͑t͒, (4.69) 2 2 turbed rate rK(B D), where B D is the effective inten- ¯ where the potential function V(x) is as in Eq. (2.2) and sity D for A0ϭ0. Therefore, according to the stochastic resonance condition (4.68), the synchronization of f͑t͒ϭ͓A cos͑⍀tϩ␸͒ϩB͔␰͑t͒, (4.70) 0 switching events and periodic noise amplitude modula- 2 with A0 , Bу0. On setting ␸ϭ0 for convenience, the tion is maximum for ␯⍀ϭrK(B D1)/2. For A0ӷB, the autocorrelation function of the noise reads characteristic switching rate to compare with is the

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 244 Gammaitoni et al.: Stochastic resonance

Kramers rate rK with effective noise intensity Heading towards the power spectrum of this two-state ¯ 2 process, we subtract the periodic oscillations of the DϭA0D1/2. On remembering that the period of the asymptotic correlation (see Sec. IV.A) by introducing f(t) amplitude modulation is now T⍀/2, we conclude that the relevant stochastic resonance condition is the newly defined correlation function, i.e., with ␸ set 0 2 we write 2␯⍀ϭrK(A0D1/2)/2. Of course, such estimates for D1(B), which fit very closely the simulation results of K͑tϩ␶,t͒ϭ͓͗x͑tϩ␶͒Ϫ͗x͑tϩ␶͔͓͒͘x͑t͒Ϫ͗x͑t͔͒͘͘. Gammaitoni, Marchesoni, Menichella-Saetta, and San- (4.73) tucci (1994), could have been obtained by explicitly cal- ¯ culating the relevant distributions N(T). The phase-averaged correlation function K(␶), defined by averaging K(tϩ␶,t) over one period of t, has been obtained by solving the two-state master equation to yield D. Weak-noise limit of stochastic resonance—power spectra 2 ␶/⌬ϩ1 xm␮ K¯͑␶͒ϭ4 ␮[␶/⌬](1Ϫ͑1Ϫ␮͓͒␶/⌬͔), (4.74) ͑1ϩ␮͒2 The most common approach in investigating stochastic resonance (apart from numerical solutions or analog where ⌬ϭT⍀/2, ␮ϵexp(Ϫ␣), and ͓␶/⌬͔ denotes the in- simulations) is linear-response theory or the perturba- teger part of ␶ in units of ⌬ (Shneidman et al., 1994a, tion theory outlined in the Appendix. The condition for 1994b). The autocorrelation function K¯ (␶) exhibits linear-response theory and perturbation theory to work cusps at each multiple of T⍀/2. While such cusps of well is that the effect of the periodic forcing can be K¯ (␶) look to be a minor artifact of the phase-locking treated as a small perturbation, i.e., A0xmӶD. The re- approximation for D/A0xm 0, the periodicity of their sponse of the periodic driving can then be described in recurrence will affect the corresponding→ power spectral terms of quantities of the unperturbed Fokker-Planck density in a rather peculiar way. The Fourier transform equation, such as its eigenfunctions and eigenvalues ¯ and/or some corresponding unperturbed correlation of K(␶) reads 2 function. 4xm ␮͑1Ϫ␮͒ In this section, we consider the complementary limit S͑␻͒ϭ ⌬ ͑1ϩ␮͒␻2 where A0xmӷD, i.e., linear-response theory is no longer valid. This weak noise limit, sometimes also 1Ϫcos͑⌬␻͒ ϫ . (4.75) termed nonlinear stochastic resonance limit, reveals ͓1Ϫ␮ cos͑⌬␻͔͒2ϩ␮2 sin2͑⌬␻͒ some peculiar properties of the power spectrum that we shall discuss (Shneidman et al., 1994a, 1994b; Stocks, We next discuss this result and its regime of validity. 1995). The starting point for these investigations is, as in (i) S(␻) corresponds to the noise background compo- Sec. III, the two-state master equation Eq. (3.1) for the nent of the power spectral density of the periodically population dynamics. Assuming adiabatic conditions, driven dynamics analyzed in Sec. IV.A. It depends on i.e., the change of the adiabatic potential the forcing frequency ⍀ through a modulation factor and decays according to the same ‘‘universal’’ ␻Ϫ2 V(x,t)ϭV0(x)ϪA0x cos(⍀t) is slower than the thermal relaxation within a potential well (in our units ⍀Ӷ2), power law as the unperturbed system does. Equation the two stable states in the master equation are given by (4.75) was derived by assuming the deltalike transition rates of Eq. (4.72). On increasing the noise intensity or, the local (time-dependent) minima xϮ(t) of the adiabatic potential V(x,t). Fluctuations of the system within equivalently, upon decreasing the forcing frequency, the potential wells will be neglected, or treated as a such an assumption becomes progressively invalid: The small perturbation. switching phase may no longer be locked to the phase of We consider a situation in which the noise strength D the input signal. is the smallest parameter. The driving frequency is large (ii) S(␻) exhibits sharp dips at even multiples of the enough (though adiabatically slow) so that almost all driving frequency. In the asymptotic approximation [Eq. escape events take place when the potential barriers as- (4.75)] the power spectral density vanishes at 2n⍀, that sume their smallest values. Under these conditions, the is S(2n⍀)ϭ0. These zeros in the profile of S(␻) are the fingerprints of the periodic structure of nonanalytical transition probability densities WϮ(t) are sharply peaked at those instants when the escape times are mini- cusps in the correlation function K¯ (␶). mal. For the continuous bistable systems under study, (iii) The experimental observability of the S(␻) dips two adiabatic transition rates rϮ(t) were introduced in is a delicate matter. For finite noise intensities these dips Eq. (4.61). As a starting point, on taking the limit broaden and undergo a ‘‘wash out’’ effect to an extent D/A0xm 0 of Eq. (4.61) for A0xmӶ⌬V, we set that we estimate only at the end of this subsection. Here, → we limit ourselves to noticing that the width of the sharp Wϯ͑t͒ϭ␣␦͓tϪ͑2mϩ␦Ϯ1,1͒͑␲/⍀͔͒. (4.72) peaks in the transition rates rϮ(t) is, in fact, of order 1/2 Ϫ1 The constant ␣, not related to the quantity in Sec. (D/A0xm) ⍀ . It follows that for spectral frequencies 1/2 IV.B.1, is the escape probability (rmax/⍀)(2␲D/ ␻ smaller than ⍀(A0xm /D) the deltalike approxima- 1/2 A0xm) , with rmaxϭrK exp(A0xm /D), and tion (4.72) is sound, and the predictions of the present mϭ0,Ϯ1,Ϯ2, . . . (phase locking approximation). analysis are physically correct. For larger frequencies,

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 245 i.e., a finer temporal resolution, the cusps of K¯ (␶) become increasingly unphysical. The condition of observability for the mth dip is thus 2 ͑2m͒ ӶA0xm /D. (4.76) Note that detecting even the fundamental dip requires a sufficiently large value of the ratio A0xm /D. This explains why dips have not been predicted within the linear-response treatment where A0xm /DӶ1 (see Secs. IV.B and Appendix). The fact that sometimes (at weak noise) the dips were seen experimentally (termed ‘‘un- explained’’ or ‘‘strange’’—see Zhou and Moss, 1990; Bulsara et al., 1991; Kiss et al., 1993), and sometimes (at strong noise) not, has continued to baffle experimental- ists and theorists at the heyday of early stochastic resonance simulations. (iv) In the derivation of S(␻), the modulation of the quasiequilibrium states xϮ(t) has been neglected. This effect is significant at large frequencies, when the system, far from being synchronized, may sojourn many forcing cycles in one well. The approximation xϩ(t)ϪxϪ(t)ϭ2xm employed to derive Eq. (4.74) ought to be improved. Without specializing our analysis to any potential, we observe that the next-to-leading order cor- rection to the difference xϩ(t)ϪxϪ(t) is proportional to 2 2 (A0xm /⌬V) cos (⍀tϩ␸). It follows that only corrections to fourth order in A0xm /⌬V, i.e., proportional to 4 (A0xm /⌬V) cos(2⍀t)cos͓2⍀(tϩ␶)͔, may become important in the computation of the phase-averaged correlation function K¯ (␶) and its Fourier transform S(␻). Shneidman, Jung, and Ha¨nggi (1994a, 1994b) concluded that for 1/2 2 ⍀ӷrmax͑2␲D/A0xm͒ ͑⌬V/A0xm͒ , (4.77) FIG. 14. Unsubtracted power spectral density (p.s.d.) (arbi- a finite-width intrawell peak may become detectable at trary units) of: (a) the full signal; (b) the filtered signal, for 2⍀, and may even dominate over the relevant S(␻) dip. different values of the forcing frequency ␯⍀ϭ⍀/2␲. The cir- 3 3 Furthermore, on accounting for higher-order terms of cuital parameters are ⌬V/Dϭ1.6ϫ10 , A0xm /Dϭ2.3ϫ10 , 4 −1 the time modulation xϮ(t), we can generate a harmonic and aϭ3.2ϫ10 s . After Gammaitoni, Marchesoni, structure of intrawell peaks at all even multiples of the Menichella-Saetta, and Santucci (1995). forcing frequency. In the weak-noise limit, the residence-time distribu- In a recent paper, Gammaitoni, Marchesoni, tion can be calculated analytically without much effort. Menichella-Saetta, and Santucci (1995) used analog We can either proceed as for the two-state model, or simulations for the periodically driven weak-noise limit adopt the approach of the foregoing Sec. IV.C. In both of the quartic double-well dynamics to verify the theo- cases, consistently with the assumptions used above, we retical predictions of Eqs. (4.75)–(4.78). In order to can assume a phase-switch distribution given by build up significant statistics, these authors increased the YϮ(␾)ϭ␦(␾Ϫ␲␦Ϯ1,1). Not surprisingly, the final result amplitude A0 of the input signal close to, but smaller coincides with Eq. (4.66), the low-noise expression for than, the critical value Ac above which bistability is lost N(T) in the adiabatic approximation. It follows imme- when the maximal tilt is assumed. In this way, the ratios 3 diately that the ratio of any two consecutive N(T) peaks of A0xm /D became of the order 10 , and could easily be is given by simulated (strong-forcing regime). As a matter of fact, the condition A x Ӷ⌬V does not enter explicitly in the N͑T ͒/N͑T ͒ϭexp͑Ϫ␣/2͒, (4.78) 0 m nϩ1 n discrete-switching approximation. It was originally intro- with Tnϭ(nϪ1/2)T⍀ , and ␣ defined below Eq. (4.72). duced to simplify rϮ(␪)torKexp͓ϯ(A0xm /D)cos ␪͔ in Thus, as anticipated in Sec. IV.C, the strengths Pn of the Eq. (4.61) and to determine the probability ␣ in Eq. N(T) peaks approach one another in the asymptotic (4.72). Therefore, we expect that in the strong-forcing limit ␣ 0. Moreover, the width of such peaks is of the regime rmax does differ substantially from → 1/2 Ϫ1 order of (D/A0xm) ⍀ , namely the same as that of rK exp(A0xm /D) in Eq. (4.72), but its physical role re- the sharp peaks of the actual transition densities rϮ(t)at mains unchanged. Gammaitoni, Marchesoni, low noise, cf. Eq. (4.61). Menichella-Saetta, and Santucci (1995) verified that in

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 246 Gammaitoni et al.: Stochastic resonance the limit of weak noise and strong forcing the shape of V. APPLICATIONS the nth N(T) peak is approximated well by a Gaussian 1/2 Ϫ1 function with standard deviation ␴Tϭ(D/A0xm) ⍀ A. Optical systems independent of n. Furthermore, the peak height N(Tn) turned out to decay according to the exponential law 1. Bistable ring laser (4.78), whence the estimate of the parameter ␣. Mante- A ring laser (Sargent et al., 1974) consists of a ring gna and Spagnolo (1996) reached the same conclusion in interferometer formed by three or more mirrors and a their experimental investigation of the switching-time laser medium inside the cavity. In two-mode ring lasers, distributions in a periodically driven tunnel diode. the light can travel in a clockwise or counterclockwise Gammaitoni, Marchesoni, Menichella-Saetta, and direction. Bistability with respect to the direction has Santucci (1995) experimentally confirmed the predic- been discussed in large detail—see, for example, Man- tions of Shneidman, Jung, and Ha¨nggi (1994a, 1994b) in del, Roy, and Singh (1981). Random switching of the great detail. In particular, the following. beam intensities, initiated by spontaneous emission in (a) The interwell and intrawell dynamics were sepa- the laser medium and fluctuations in the pump mecha- rated by filtering the output signal x(t) through a two- nism, indicates bistable operation of the ring laser. To demonstrate stochastic resonance, the symmetry be- state filter (xϭϮxm) and contrasting the statistics of the filtered signal with that of the full signal. The nonsub- tween the two modes has to be broken by applying a tracted power spectral densities for both output signals periodic modulation that favors one of the modes. In the are displayed in Fig. 14. In both cases the number of pioneering experiment by McNamara, Wiesenfeld, and resolved dips m is bounded from above by the inequal- Roy (1988), an acousto-optic modulator (Roy et al., 1987) has been used to convert an acoustic frequency to ity in Eq. (4.76). a modulation of the pump parameter. Gage and Mandel (b) The dip structure of the power spectral densities (1988) have alternatively used Faraday and quartz rota- becomes more apparent by filtering x(t). Most notably, tors to control the asymmetry of the modes. Before we the spectral dips tend to disappear with increasing ⍀, discuss the experiment by McNamara, Wiesenfeld, and and their shape is not as sharp as that predicted by Eq. Roy (1988), we briefly review the theoretical description (4.75). of bistable ring lasers. (c) No peak structure due to the intrawell modulation The semiclassical equations for a two-mode ring laser is observable in the power spectral density of the filtered (Sargent et al., 1974) augmented by noise sources ⌫1,2(t) signal. In contrast, for the full signal, broad peaks lo- to account for the effect of spontaneous emission and cated in the vicinity around 2n⍀ can be resolved at rela- pump noise p(t) assume in dimensionless units the form tively high values of ⍀, namely for ␣Ӷ1, in agreement I˙ ϭ2I ͑a ϩp͑t͒ϪI Ϫ␰I ͒ϩͱ2I ⌫ ͑t͒, with Eq. (4.77) for A0xmӍ⌬V. Moreover, such peaks 1 1 1 1 2 1 1 get sharper on further increasing ⍀, and their height de- ˙ ϭ ϩ Ϫ Ϫ ϩ creases with increasing peak index. I2 2I2͑a2 p͑t͒ I2 ␰I1͒ ͱ2I2⌫2͑t͒, (5.1) (d) Dips and peaks of the power spectral density may where I1 and I2 are the scaled dimensionless intensities coexist for the full signal, as suggested by items (a) and of the modes. The terms linear in I1,2 describe the net 2 (b). In such a case, their position deviates slightly from gain of the laser modes, those with I1,2 describe satura- the predicted value 2n⍀, by shifting to the right and, tion and the mixed terms ␰I1I2 represent coupling be- possibly, to the left. At very low ⍀ values, no intrawell tween the laser modes. The fluctuations of the pumping modulation peak is detectable, whereas at high ⍀ values, mechanism are described by the exponentially corre- peaks dominate over dips which, in turn, tend to vanish. lated Gaussian noise p(t), i.e., We additionally remark that these intrawell modulation P 1 peaks should not be mistaken for the delta-like spikes at ͗p͑t͒p͑tЈ͒͘ϭ exp Ϫ ͉tϪtЈ͉ , ␶ ͩ ␶ ͪ (2nϩ1)⍀. c c Finally, we mention that in addition to the detailed ͗p͑t͒͘ϭ0 (5.2) experimental analog simulations (Gammaitoni, Marchesoni, Menichella-Saetta, and Santucci, 1995), the with correlation time ␶c and intensity P, while the structure of the characteristic dips in the time-averaged spontaneous-emission noise terms are assumed to be un- power spectrum S(␻) at even-numbered harmonics correlated: have also been observed in computer simulations of a ͗⌫ ͑t͒⌫ ͑tЈ͒͘ϭ␦ ␦͑tϪtЈ͒, neural network using a model describing the perceptual i j ij interpretation of ambiguous figures (Riani and Simo- ͗⌫i,j͑t͒͘ϭ0. (5.3) notto, 1994, 1995) and even in in situ experiments with The pump parameters a and a are modulated antisym- human observers (Riani and Simonotto, 1995). These 1 2 metrically by noise and a periodic signal (Vemuri and latter experiments yielded results that are in good quali- Roy, 1989) tative agreement with the neural model predictions for ¯ the stochastic resonance power spectra that characterize a1͑t͒ϭ a ϩ⌬a͑t͒ϩr͑t͒, the perceptual bistability in the presence of noise and ¯ weak periodic perturbations. a2͑t͒ϭ a Ϫ⌬a͑t͒Ϫr͑t͒, (5.4)

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 247

FIG. 15. The effective potential of Eq. (5.8) for the intensity of a single laser mode in a bistable ring laser is plotted as a function of the intensity at the pump strength ¯a ϭ60 and the coupling ␰ϭ2 for different amplitudes ⌬aϭA0 cos(⍀t) of the external field (i.e., during one period of the external field). It is seen that the effective barrier height for a transition between the ‘‘ON’’ state and the ‘‘OFF’’ state becomes periodically modulated. After Vemuri and Roy (1989). FIG. 16. The signal-to-noise ratio, obtained from the time de- with r(t) being white Gaussian (injected) noise pendence of the intensity of one laser mode, shown as a function of the injected noise strength. After McNamaura, Wiesen- ͗r͑t͒r͑tЈ͒͘ϭ2R␦͑tϪtЈ͒, feld, and Roy (1988). ͗r͑t͒͘ϭ0, (5.5) In the experiment by McNamara, Wiesenfeld, and and Roy (1988), the intensity of one of the two modes has been extracted from the ring laser. The time series for ⌬a͑t͒ϭA0 cos͑⍀t͒. (5.6) the intensity has subsequently been compressed into a In the absence of the periodic modulation binary pulse train that contains only information on (⌬a=const.), pump noise, and injected noise, the station- whether the mode was ‘‘ON’’ or ‘‘OFF.’’ This procedure ary probability Pst(I1 ,I2) can be obtained analytically has been repeated for a large number of samples to ob- (Sargent et al., 1974). Integration over the intensity I2 tain the sample-averaged power spectrum. The power yields the stationary probability density of I1 , which can spectrum consists of a smooth background, a sharp peak be written as at the frequency of the modulating field ⍀, and smaller ϱ 1 peaks at multiples of ⍀. Most significantly, the intensity P ͑I ͒ϭ P ͑I ,I ͒dI ϭ exp͓ϪV͑I ͔͒, (5.7) of the peaks at the driving frequency first increases with st 1 ͵ st 1 2 2 Z 1 0 increasing injected noise strength, passes through a with the effective potential maximum and then decreases again. The signal-to-noise ratio, shown in Fig. 16, reflects this characteristic bell- 1 1 1 2 2 ¯ shaped behavior. V͑I1͒ϭϪ ͑␰ Ϫ1͒I ϩ a͑␰Ϫ1͒Ϫ ⌬a͑␰ϩ1͒ I1 4 1 ͫ2 4 ͬ For the archetypal quartic bistable potential in Sec. IV, the power spectrum contains peaks only at odd mul- 1 1 1 Ϫln erfc ␰I Ϫ ¯a ϩ ⌬a , (5.8) tiples of the driving frequency. In the power spectrum ͭ ͫ2 1 2 4 ͬͮ and erfc being the complementary error function (Abramowitz and Stegun, 1965). For slow and weak periodic driving, the potential V(I1) undergoes a periodic change obtained by substituting ⌬a by A0 cos(⍀t), but it remains bistable. The minimum corresponding to the ON state (high intensity) rocks up and down—see Fig. 15. This situation looks similar to the quartic double- well potential, discussed in Sec. IV. There are, however, some differences:

(1) The potential V(I1) is only an effective potential. For its construction, the pump noise and injected noise had been neglected. The intuitive picture of a particle (here the intensity) moving in the effective potential can be used only as a heuristic guideline. (2) The original Langevin equations (5.1) exhibit no in- FIG. 17. The light intensity of a single-mode laser with a saturable absorber shown as a function of the pump current. The version symmetry. hysteresis loop indicates optical bistability.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 248 Gammaitoni et al.: Stochastic resonance here, however, a peak at twice the driving frequency can pump current are taken into account by modulating the be observed, reflecting the lack of inversion symmetry of amplifier strength A, i.e., A A(t)ϭAϩF cos(⍀t) the two-mode laser Langevin equations (5.1), or the ef- ϩ␨(t), with → fective potential (5.8). ␨͑t͒␨͑tЈ͒ ϭ2q␦͑tϪtЈ͒, Digital simulations of Eq. (5.1) in Vemuri and Roy ͗ ͘ (1989) are in qualitative agreement with the experimen- ͗␨͑t͒͘ϭ0. (5.10) tal results. One finally arrives at the equation of motion for the field intensity IϵE*E

A¯ A ␨͑t͒ϩF cos͑⍀t͒ 2. Lasers with saturable absorbers I˙ϭϪI 1ϩ Ϫ Ϫ ͩ 1ϩaI 1ϩI 1ϩI ͪ A laser with a saturable absorber is a quantum device consisting of a laser cavity where an amplifying as well ϩͱI⌫͑t͒, (5.11) as an absorbing medium are placed. Bistability has been with the real-valued zero-mean Gaussian noise ⌫(t), observed in these devices by Arimondo and Dinelli characterized by the correlation function (1983) and Arimondo et al. (1987). Within a certain range of the pump intensity, the output of the laser can ͗⌫͑t͒⌫͑tЈ͒͘ϭ2Q␦͑tϪtЈ͒. (5.12) be in two different modes, depending on its previous Again, in contrast to the quartic double-well system in history. Sec. IV.A, the equation of motion (5.11) does not have In Fig. 17, the laser intensity (Fioretti et al., 1993) is inversion symmetry. The stationary intensities in the ab- shown as a function of the pump current, which was sence of fluctuations and modulation, are determined by slowly increased until the output switched to the high ¯ intensity level, and then decreased again. The observed the zeros of F(I)ϭI(1ϩA/(1ϩaI)ϪA/(1ϩI)). For hysteresis loop is an indication of bistability. aϾA/(AϪ1), the dynamical system exhibits a subcriti- In the experiment by Fioretti et al. (1993), the dc cal pitchfork bifurcation at AϭA¯ ϩ1, which explains the pump current is chosen in order that the laser operates hysteretic behavior (see Fig. 17), bistability, and stochas- in the bistable regime. The pump current is modulated tic resonance observed in the experiment when the around this dc value by a small periodic signal plus an pump current A is ramped slowly up and down. external Gaussian noise source whose intensity can be controlled. The periodic signal is too small to cause switching by itself. In the presence of pump noise, however, switching 3. Model for absorptive optical bistability takes place. The intensity of the laser light has been re- Consider a ring interferometer with a passive medium corded as a function of time and subsequently filtered by placed in it. Light is coupled into the interferometer a two-bit filter, which only detects ‘‘ON’’ and ‘‘OFF’’ through a semipermeable mirror and, likewise, light is information. The extracted signal-to-noise ratios exhibit transmitted at another mirror. Measuring the intensity maxima as a function of noise strength Q (see below). of the transmitted wave against the intensity of the inci- The laser with a saturated absorber is described on a dent wave, one finds an S-shaped curve; e.g., for some semiclassical level by a system of three ordinary differ- values of the intensity of the incident beam the intensity ential equations (Zambon et al., 1989) of the transmitted wave can have a small and a large 1 A¯ intensity. There are different mechanisms that can be E˙ ϭϪ Dϩ ϩ1 Eϩ␰͑t͒, responsible for this behavior. One of them is due to non- 2 ͩ 1ϩa͉E͉2 ͪ linear absorption in the passive medium. ˙ 2 A model for purely absorptive optical bistability in a DϭϪ␥͑DϩAϩD͉E͉ ͒Ϫc1͑SϪD͒, cavity was introduced by Bonifacio and Lugiato (1978). ˙ SϭϪ␥1͑SϪD͒, (5.9) For the scaled dimensionless amplitude y of the input light and the scaled dimensionless amplitude of the where E is the complex field amplitude and D is the transmitted light x, they have derived the equation of difference between the population of the upper and motion lower energy level of the amplifier medium relevant for lasing. The quantity S describes the influence of other 2cx x x˙ ϭyϪxϪ ϩ ⌫͑t͒, (5.13) energy levels coupled to the populations of the lasing 1ϩx2 1ϩx2 levels; A and A¯ describe the strength of the amplifying with ⌫(t) denoting Gaussian fluctuations of the inver- and absorbing medium, respectively. Quantum fluctua- sion tions are modeled by zero-mean white Gaussian noise ␰(t) with ͗␰*(t)␰(tЈ)͘ ϭ 4q␦(tϪtЈ). In order to sim- ͗⌫͑t͒⌫͑tЈ͒͘ϭ2D␦͑tϪtЈ͒, plify the laser equations, S and D are adiabatically ⌫͑t͒ ϭ0. (5.14) eliminated by assuming the field strength E to be small ͗ ͘ and much slower varying in time than S and D. Periodic The dimensionless parameter c is proportional to the modulation and (Gaussian) fluctuations ␨(t) of the population difference in the two relevant atomic levels.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 249

FIG. 20. Transmitted intensity I vs incident light intensity I FIG. 18. The numerical results for the spectral amplification ␩ t 0 for a laser-illuminated sample of the optical bistable semicon- are shown by the solid lines for the ‘‘symmetric case’’ ductor CdS. The pulse length of the incident light intensity y ϭ6.72584 at cϭ6 and A ϭ10Ϫ4. Different lines labeled ac- 0 0 determines the rate at which the hysteresis loop is run through. cording to ‘‘n’’ correspond to the external angular frequency ⍀ϭ10Ϫn. The dotted lines correspond to results within the linear-response approximation. They can be distinguished Planck equation corresponding to the Langevin equa- from the numerical results only for frequencies larger than tion (5.13) with the modulated parameter y has been Ϫ about 10 2. The vertical dashed lines indicate the position solved numerically by Bartussek, Ha¨nggi, and Jung DSR of the maxima determined by the argument of matching (1994) by using the matrix-continued fraction technique, time scales discussed in Sec. II, and also in Bartussek, Ha¨nggi, and alternatively with linear-response theory for weak and Jung (1994). modulation A0 . According to Sec. IV, the spectral amplification ␩ [see Eq. (4.21)] of the periodic modulation For a large enough population difference c, the sta- has been constructed from the asymptotic long-time so- tionary transmitted light amplitude is an S-shaped func- lution of the Fokker-Planck equation. tion of the amplitude of the injected light, thus indicat- The following discussion is restricted to the bistable ing bistability. regime. Here, it is important to distinguish a symmetric In the presence of a small, periodic perturbation of case from an asymmetric case. In the symmetric case the the incident light, the parameter y in Eq. (5.13) has to be stationary probability density in the absence of periodic modified by y y(t)ϭyϩA0 cos(⍀t). The Fokker- driving has two peaks with the same heights in the zero- → noise limit D 0. For all other cases (asymmetric cases), the peaks have→ different probabilistic weights at weak noise. In the symmetric case, one observes stochastic resonance very much like in the quartic bistable double-well potential (see Fig. 18), i.e., a peak in the amplification of the modulation when the sum of the mean escape times out of both stable states equals the period of the driving (the values of the noise strength D where we have such a time-scale matching are indicated by dashed lines in Fig. 18). In the asymmetric case (Fig. 19), the spectral amplification is suppressed at weak noise, because—in contrast to the symmetric case—the contribution of the hopping FIG. 19. The numerical results for the spectral amplification ␩ motion to the response of the system disappears expo- are shown by the solid lines for the ‘‘asymmetric case’’ y0ϭ6.8 nentially for small noise; cf. Sec. IV.B.2. As a conse- Ϫ4 at cϭ6 and A0ϭ10 . Different lines labeled according to quence, the maximum of the spectral amplification does Ϫn ‘‘n’’ correspond to the external frequency ⍀ϭ10 . The not indicate a time-scale matching as in the symmetric curves for ⍀Ͻ10Ϫ3 are not distinguishable from the curve for Ϫ3 case. For small driving frequencies, the maximum of the ⍀ϭ10 . The dotted lines correspond to results within the spectral amplification becomes independent of the driv- linear-response approximation. They can be distinguished ing frequency. from the numerical results only for frequencies larger than about 10Ϫ2. The rightmost dashed vertical line indicates the position DSR of the maximum determined by the argument of 4. Thermally induced optical bistability in semiconductors matching time scales between the period of driving T⍀ and the sum of corresponding two escape times at ⍀ ϭ 10Ϫ1; see It has been shown that semiconductors exhibit a ther- Bartussek, Ha¨nggi, and Jung (1994). The angular driving fre- mally induced optical bistability facilitated by the ther- quencies corresponding to the vertical dashed lines from right mal shift of the fundamental band edge (Lambsdorff to left are ⍀ ϭ 10Ϫ1, ⍀ ϭ 10Ϫ2, ⍀ ϭ 10Ϫ3, ⍀ ϭ 10Ϫ4, and et al., 1986; Grohs et al., 1989). The semiconductor is al- ⍀ ϭ 10Ϫ7. most transparent at low intensities of the incident light.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 250 Gammaitoni et al.: Stochastic resonance

the overshoot has been done by Grohs et al., 1991, and Jung et al., 1990). Thermally optically bistable semiconductors have been discussed for the design of optical parallel computers (Grohs et al., 1989). Experiments on stochastic resonance have been performed (Grohs et al., 1994) with a semiconductor (CdS). The CdS crystal had a thickness of about 6 ␮m and shows thermally induced optical bistability (Lambsdorff et al., 1986; Grohs et al., 1989) under illumination with an Ar+ laser (␭ϭ514.5 nm). A two-beam setup has been used for the experiments, with both beams incident on the same spot of the crystal and having an equal diam- eter of approximately 100 ␮m. The transmission state of the crystal is read by a constant beam with an intensity that is too small to induce nonlinear behavior by itself. FIG. 21. The signal-to-noise ratio of the transmitted light in- The intensity of the second beam consists of a constant tensity as a function of the noise strength (in arbitrary units) part to hold the system at the working point (i.e., in the for different amplitudes A (in arbitrary units) of the modula- bistable regime), and of a weak periodic signal. More- tion of the incident light intensity. The lines are to guide the over, the weak periodic signal is perturbed by additional eye. injected noise ␰ with a controllable amplitude. The transmitted intensity has been measured as a function of Increasing the intensity, the absorbed fraction of light time and the power spectrum has been measured. In Fig. heats up the probe, which in turn induces a stronger 21, the signal-to-noise ratio is shown as a function of the light absorption—a nonlinear effect. For large intensities strength of the injected noise. It shows the bell-shaped of the incoming light, the transmitted intensity is there- curve, which is characteristic of stochastic resonance. fore small. Ramping down the intensity of the incoming light, the transmitted intensity becomes larger again, but 5. Optical trap describes a hysteresis loop (see Fig. 20) if the absorbed fraction of light is a steep function of the temperature. In the experiment by Simon and Libchaber (1992) a For increasing ramping speed, the hysteresis loop spatial bistable potential was generated by optical smears out, but its area increases (a quantitative study of means. The experimental setup was based on the fact

FIG. 22. Escape-time distributions of a particle in an optical trap. The time is measured in units of the mean escape time TK (from one potential minimum to the other one). (a) The period of the forcing in units of the mean escape time was chosen as T/TKϭ3.08Ͼ1; (b) the period is given by T/TKϭ0.76Ͻ1. While in (b), the peaks are clearly located at odd multiples of half the forcing period, the second peak in (a) is shifted to the left (as far as the accuracy allows for such an interpretation).

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 251 that in the presence of an electrical field gradient, a dielectric object (here a 1 ␮m glass sphere) moves towards the region of highest field strength. The electric field of a laser beam has typically a transverse Gaussian intensity profile, so that dielectric objects are pulled into the beam axis. The two double wells are created by two Gaussian beams obtained from one beam of an Ar laser by utilizing beam splitters. In order to observe stochastic resonance, the depths of the potential wells have to be modulated periodically, which is achieved by modulating the intensity of the two partial beams. The experimental setup was positioned under a microscope so that the motion of the glass sphere in water was directly observable. The pictures were recorded on video and electronically analyzed. Simon and Libchaber measured the distribution of times the dielectric sphere stayed in one well before it FIG. 23. Functional block scheme for simulating a heavily was kicked into the other one. In the absence of modu- damped particle moving in a quartic double-well potential; see lations, they obtained an exponentially decaying distri- Eq. (2.1). The triangles denote operational amplifiers. bution. In the presence of the periodic forcing, they observed a sequence of peaks at odd multiples of the half driving period, with exponentially decaying peak height non-bistable standard np semiconductor diode (Jung as shown in Fig. 22. The time is measured in units of the and Wiesenfeld, 1997). Further experimental evidence of stochastic resonance driven by externally time- mean escape time TK out of a potential well in the ab- ˜ modulated magnetic fields (magnetic stochastic reso- sence of periodic modulation, i.e., t ϭt/TK . The period nance) was reported by Spano, Wun-Fogle, and Ditto of the modulation in Fig. 22 is measured in the same (1992) in magnetoelastic ribbons, and by I and Liu units. They are given by T/TKϭ3.08 (a) and T/TKϭ0.76 (1995) in weakly ionized magnetoplasmas. Finally, mag- (b). The first peaks are observed at ˜t Ϸ1.54 (a) and netic stochastic resonance was predicted theoretically by ˜t Ϸ0.38 (b), i.e., at the half period of the driving. The Grigorenko, Konov, and Nikitin (1990) and Grigorenko other peaks are located at odd multiples of the half pe- and Nikitin (1995) for the interdomain magnetization riod of the driving as predicted by the theory in Sec. tunneling in uniaxial ferromagnets, by Raikher and IV.C. Stepanov (1994) for single-domain uniaxial superpara- In the case that the dwell time equals half the period magnetic particles, and by Pe´rez-Madrid and Rubı´ T an optimal synchronization occurs, leading to the con- (1995) in an assembly of single-domain ferromagnetic centration of the escape within the first period; thus an- particles dispersed in a low-concentration solid phase. In ticipating the notion of stochastic resonance in symmet- recent experiments, stochastic resonance has been dem- ric, bistable systems as a ‘‘resonant’’ synchronization onstrated in Bi-substituted ferrite-garnet films (Grig- phenomenon (see Sec. IV). orenko et al., 1994) and also in yttrium-iron garnet spheres, where the noise-free, chaotic spin-wave dynamics alone induces stochastic resonance in the presence of B. Electronic and magnetic systems an external modulation (Reibold et al., 1997).

In this subsection we review some applications of sto- 1. Analog electronic simulators chastic resonance to electronic and magnetic systems. We recall that the very first experimental verification of As mentioned in Sec. II.C, electronic circuits have stochastic resonance was realized in an electronic circuit, been widely employed in the study of nonlinear stochas- a simple Schmitt trigger (Fauve and Heslot, 1983). Since tic equations (for reviews, see McClintock and Moss, then, stochastic resonance has been observed in a vari- 1989; Fronzoni et al., 1989). Initially (Debnath et al., ety of more or less complicated electronic devices, 1989; Gammaitoni, Marchesoni, et al., 1989; Gammai- mostly constructed with the purpose of building flexible toni, Menichella-Saetta, Santucci, Marchesoni, and and inexpensive simulation tools. A rather peculiar elec- Presilla, 1989; Zhou and Moss, 1990, Hu et al., 1991), the tronic device that exhibits stochastic resonance is the stochastic resonance studies via analog simulations con- tunnel diode, a semiconductor device with a bistable centrated on the simulation of the standard quartic characteristic I-V curve, whose dynamics can be con- double-well system of Eq. (2.1). The temporal behavior trolled by tuning the operating voltage. Due to the fast of the stochastic process x(t) can be reproduced by us- switching dynamics between stable states (a few tenths ing a voltage source v(t) that obeys Eq. (2.1): the coef- of a ns), stochastic resonance in a tunnel diode has been ficients for the dynamical flow are realized by means of a observed for forcing frequencies as high as 10 KHz suitable combination of passive analog components, like (Mantegna and Spagnolo, 1994, 1995, 1996). More re- resistors and capacitors. Active elements (transistors, cently, stochastic resonance has been reported also in a signal generators, etc.) are employed to simulate more

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 252 Gammaitoni et al.: Stochastic resonance

FIG. 24. Circuits for the functional blocks. (a) Miller integrator; (b) adder-amplifier; (c) multiplier. complicated potential functions. The block scheme of for a multiplier. The block scheme of Fig. 23 can thus be the circuit corresponding to Eq. (2.1) is drawn in Fig. 23. translated into the electronic circuit scheme of Fig. 25. The signal in A is assumed to represent the derivative The underdamped dynamics of Eq. (4.1), with V(x) de- x˙ (t) of the variable of interest at time t.InB, after the fined in Eq. (4.7), can be simulated by a modified ver- integration block, the signal x(t) is fed into three input sion of the previous circuit—see Fig. 26. Usually both terminals: the terminal of the amplifier block a, to obtain the stochastic force ␰(t) and the periodic modulation the signal ax, and the terminals of two multiplier blocks A0 cos(⍀t) are fed into the simulation circuits by means 2 3 X, which yield x after the first block, and x after the of suitable voltage generators. The output voltage v0(t) second block. The latter signal is then inverted and am- is sampled at regularly spaced times tm , with plified through two cascaded amplifying blocks. Finally, tmϩ1Ϫtmϭ⌬t, digitized as v0,mϭv0(tm) and then stored the three signals in C are added together through block into a digital memory unit. The sequence (tm ,vm)isfi- ⌺ to give the starting signal x˙ (t). nally analyzed, mostly off line, by means of standard The realization of an electronic simulation circuit re- data-analysis algorithms. quires the design of specific electronic devices which op- The first, and probably the simplest circuit of this type erate as the single components of the block scheme de- employed to investigate stochastic resonance dates back picted in Fig. 23. Nowadays, such a purpose is best to Fauve and Heslot (1983). It consists of a Schmitt trig- served by the use of operational amplifiers (Millman, ger, a two-state hysteretic device, that can be easily re- 1983): versatile electronic analog devices characterized alized by means of an operational amplifier with positive by high input impedance, low output impedance, and feedback (Fig. 27). If the input voltage vi is lower than a wide frequency-response range. In Fig. 24 we show the threshold value Vϩ , the output voltage v0 takes on the main devices employed to simulate Eq. (2.1). The rel- constant value v0ϭV. On increasing vi through Vϩ , the evant output voltages v0 are output voltage v0 switches to v0ϭϪV and stays negative as long as v is larger than a second threshold value V 1 t i Ϫ v0͑t͒ϭ vi͑␶͒d␶ (5.15) with VϪϽVϩ . Vice versa, the output transition from RC ͵0 ϪV to V occurs at viϭVϪ . The amplitude of the hys- for a Miller integrator, teresis cycle is given by VϩϪVϪ . The stochastic resonance phenomenon follows from the periodic modula- R R R tion of the position of the center of the hysteresis cycle v0͑t͒ϭ v1ϩ v2ϩ v3 (5.16) R1 R2 R3 around the mean value (VϩϩVϪ)/2. Such a modulation is reminiscent of the periodic tilting of the wells of a for an adder amplifier, and bistable potential model. As a matter of fact, Eqs. (2.6) ͑x1Ϫx2͒͑y1Ϫy2͒ and (2.7) apply to the output x(t)ϵv0(t) of a symmetric v0͑t͒ϭ ϩ͑z1Ϫz2͒ (5.17) 10 Schmitt trigger (where, say, VϩϭϪVϪϭVb) with minor

FIG. 25. Simulator circuit for the overdamped dynamics of Eq. FIG. 26. Simulator circuit for the underdamped dynamics of (4.3). Eq. (4.1).

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 253

FIG. 28. Block scheme of an EPR system. Block S: EPR spec- FIG. 27. Operational amplifier in the Schmitt trigger configu- trometer; block C: measurement devices; block D: data- ration: (a) transfer characteristic; (b) electronic scheme. acquisition system; block F: feedback system; block ⌺: adder. changes. For a Gaussian, exponentially autocorrelated measure the power reflected from the cavity. Block D 2 input signal vi(t) with zero mean, variance ␴i , and cor- represents the data acquisition system. relation time ␶i , the parameters in Eqs. (2.7) read In such a device, stochastic resonance was observed 2 Ϫ1 xmϭV, Dϭ␴i ␶i , and rKϭ͓TK͔ . The time T0(Vb) first by monitoring the working frequency ␯ (Gammai- denotes the mean-first-passage time for vi(t) to diffuse toni, Marchesoni, et al., 1991; Gammaitoni, Martinelli, from ϪVb up to Vb ; in the subthreshold limit ␴iӶVb it et al., 1991, 1992). A polypyrrole paramagnetic sample 2 2 may be approximated to ␶iͱ2␲(␴i /Vb)exp(Vb/2␴i ) was placed in the resonant cavity. The reflection coeffi- (Melnikov, 1993; Shulgin et al., 1995). Note that when cient of the cavity, measured as the ratio between the the amplitude of the hysteresis cycle is forced to zero, reflected to the incoming power, was monitored by vary- the two thresholds overlap and we end up with a single ing the working frequency. Under proper conditions threshold system (see Sec. VII.A.1). (strong coupling) the reflection coefficient, as a function By now, stochastic resonance has been studied in a of the frequency, showed two separated minima, both variety of analog circuits: symmetric and asymmetric stable working points for the spectrometer. The dynami- quartic double-well potentials (Debnath et al., 1989; cal behavior of the frequency ␯ is driven by the feedback Dykman, Luchinsky, et al., 1992; Gammaitoni, system. Under stationary conditions, ␯ fluctuates around Marchesoni, and Santucci, 1994; Gammaitoni, the frequency of one of the two minima of the reflection Marchesoni, et al., 1994); the Hopfield neuron potential coefficient due to the action of the internal noise of the (Bulsara, Jacobs, Zhou, Moss, and Kiss, 1991); the system. When such noise intensity grows appreciable Fitzhugh-Nagumo neuron model (Wiesenfeld et al., compared to the height of the barrier that separates the 1994), to mention only a few. two minima, random switches are observed. Under such circumstances the EPR system shows a noise-driven op- 2. Electron paramagnetic resonance erating frequency. Its dynamics can be described by the approximate stochastic differential equation: An EPR system consists of a paramagnetic sample placed in a microwave cavity. A microwave generator ␯¨ ϭϪ␥␯˙ϪVЈ͑␯͒ϩ␰͑t͒, (5.18) irradiates the sample while a feedback electronic circuit where V(␯) denotes the effective bistable potential re- locks the oscillator frequency to the resonant frequency lated to the reflection coefficient. The same dynamics ␯c of the cavity. A static magnetic field H0 is applied to can be observed by measuring the error voltage signal the cavity and slowly modulated in order to vary the generated by the feedback system. On inserting in the Larmor frequency ␯0ϭ␥H0/2␲ of the sample. Here ␥ is feedback loop an external signal made of a periodic and the gyromagnetic factor. The cavity response, deter- a random component and varying the intensity of the mined by measuring the reflected microwave power, injected noise, behavior typical of stochastic resonance usually exhibits a single minimum at ␯ϭ␯c ; however, in was detected and measured. the presence of a strong coupling between the cavity and the spin system (a high number of paramagnetic cen- 3. Superconducting quantum interference devices ters), one observes a splitting of the resonance frequency into two frequencies and the cavity response ex- The basic components of a SQUID are a supercon- hibits two distinct minima separated by a local ducting loop and a Josephson junction. For practical maximum. The block scheme of an EPR experiment is purposes, a SQUID can be envisioned as an electromag- shown in Fig. 28: an electronic adder (block ⌺); a stan- netic device that converts a magnetic flux variation into dard microwave spectrometer (block S) made of a mi- a voltage variation and, as such, it has been successfully crowave generator, a resonant cavity, and the relevant employed in monitoring small magnetic field fluctua- circuitry; and the feedback system (block F) that locks tions. One of the major limitations to a wide use of these the microwave source frequency to the maximum ab- devices is their extreme sensitivity to environmental sorption of the cavity. Block C contains the measure- noise. Recently, two different groups (Hibbs et al., 1995; ment instrumentation, mainly a frequency counter, to Rouse et al., 1995) succeeded in operating SQUIDs un- monitor the working frequency, and a power meter to der stochastic resonance conditions with the aim of in-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 254 Gammaitoni et al.: Stochastic resonance

toni, Marchesoni, Menichella-Saetta, and Santucci, 1989; Zhou et al., 1990). In this section, we report on the relevant neurophysiological experiments and describe how stochastic resonance enters naturally into standard models for neuronal dynamics. By now, stochastic resonance is a well accepted paradigm in the biological and neurophysiological sciences, and several recent reviews on neurophysiological applications of stochastic resonance FIG. 29. Block scheme of the SQUID system. Block A: test are available (Moss, 1994; Moss et al., 1994; Moss and SQUID, driven by the external signal; block B: measurement Wiesenfeld, 1995a, 1995b; Wiesenfeld and Moss, 1995). SQUID; block C: inductive coupling between the SQUIDs Thus we keep this subsection—though there is a prodi- and with the outside circuitry; block D: low-temperature gious potential for future applications—somewhat tight. shield. 1. Neurophysiological background creasing low magnetic-field detection performance. The There is a large variety of types of neurons in the dynamics of the magnetic flux ⌽ trapped in a standard nervous system of animals and humans with variations rf. SQUID can be generally described in terms of a in structure, function, and size. Let us restrict ourselves second-order differential equation: here to a canonical neuron (Amit, 1989), which presents LC⌽¨ ϭϪ␶ ⌽˙ ϪVЈ͑⌽͒ϩ⌽ ͑t͒, (5.19) the underlying functional skeleton for all neurons. The L e canonical neuron is divided into three parts, an input where part (the dendritic arbor), a processing part (the soma), V͑⌽͒ϭ⌽ϩ͑␤/2␲͒sin͑2␲⌽͒ (5.20) and a signal transmission part (the axon). A neuron communicates via synapses, which are the plays the role of an effective potential and ⌽e(t) repre- interfaces between its dendrites and axons of presynap- sents an externally induced flux variation. Here, the flux tic neurons, i.e., neutrons that talk to the considered ⌽ is measured in units of the fundamental flux quantum neuron. There are a number of dendritic trees entering ⌽0ϭh/2e; L is the inductance of the loop, while C is the the soma of the neuron. Usually, one axon leaves the junction capacitance and ␶LϭL/Rj , where Rj is the soma and then, downstream, it branches repeatedly to normal-state resistance of the junction. The quantity communicate with many postsynaptic neurons, i.e., the ␤ϭ2␲Lic /⌽0 , with ic being the loop critical current, is neurons the specified neuron is talking to. The funda- the adjustable parameter that allows us to set the shape mental process of neural communication is based upon of the effective potential. With a proper choice of ␤ and the following sequence: for small ⌽ values, the system undergoes bistable dynamics. The SQUID loop can be shunted by a low resis- (1) The neural axon is in one of two possible states. In tance in order to reduce the capacitance and discard in- the first state, it propagates an action potential ertial effects in the flux [Eq. (5.19)]. based on the result of the processing in the soma. The block scheme of a generic SQUID system is The shape and amplitude of the propagating action shown in Fig. 29. Blocks A and B represent two potential—the potential difference across the cell SQUIDs. The first one is driven with the external signal membrane—is very stable, and is replicated at the branching points in the axon. The amplitude is of and the second one is used to pick up the signal output. Ϫ1 The inductive coupling between the two loops and with the order of 10 mV. In the second state of the the outside circuitry is also shown (block C). The system axon, i.e., the resting state, no action potential is is shielded by a low-temperature cage (block D). Sto- propagated along the axons. chastic resonance was observed both in the external (2) When the propagating action potential reaches the noise-injected configuration (Hibbs et al., 1995) and in endings of the axons it triggers the secretion of neu- the thermally driven configuration (Rouse et al., 1995), rotransmitters into the synaptic cleft. where the switching of the magnetic flux was driven by (3) The neurotransmitters travel across the synapse and the internal noise inherent in the SQUID. In both cases reach the membrane of the postsynaptic neuron. a low-frequency periodic signal was injected externally. The neurotransmitters bind to receptors that cause the opening of channels, allowing the penetration of ionic currents into the postsynaptic neuron. The C. Neuronal systems amount of penetrating current per presynaptic spike is a parameter that specifies the efficiency of the syn- The development of stochastic resonance took a large apse. There are different ion channels for different leap forward when its potential relevance for neuro- ions. To open, say, a potassium channel, a specific physiological processes had been recognized. Longtin, neurotransmitter substance is required. Bulsara, and Moss (1991) observed that interspike inter- (4) In the absence of a neurotransmitter the resting po- val histograms of periodically stimulated neurons exhibit tential of the membrane of the postsynaptic neuron a remarkable resemblance to residence-time distribu- is determined by the balance of the resting fluxes of tions of periodically driven bistable systems (Gammai- ions such as sodium and potassium. The resting

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 255

membrane voltage is typically slightly above the the period of the stimulus is also encoded in the tempo- low-lying potassium voltage. The opening of say a ral sequence of the action potentials (spikes). In other sodium channel disturbs the equilibrium and trig- words, there is a correlation between the temporal se- gers a postsynaptic potential close to the high so- quence of neuronal discharge and the time dependence dium voltage. The membrane voltage is bound be- of the stimulus. In the neurophysiological literature this tween the lower potassium voltage and the higher correlation is termed phase locking. Earlier work report- sodium voltage. The fact that the rest state is very ing the limitation of the phase locking of the neural dis- close to the lower limit leads to a rectification of charge to the stimulus to small frequencies (Ͻ6 kHz) external stimulus in sensory neurons and is of par- (Rose et al., 1967) has been suggested to be too pessi- ticular importance for the study of stochastic reso- mistic by Teich, Khanna, and Guiney (1993). These au- nance in neurons. thors argue that the synchronization holds up to 18 kHz. (5) The postsynaptic potential diffuses in a graded man- They further argue that information about the period of ner towards the soma. It loses thereby around 80% the stimulus is encoded in the temporal sequence of the of its amplitude. Here, in the processing unit of the action potentials over virtually the complete band of neuron, the inputs from all presynaptic neurons (of the order 104) are summed. The individual postsyn- acoustical perception. aptic potentials are about 1 mV in amplitude. These The resemblance of interspike interval histograms and inputs may be excitatory—depolarizing the mem- residence-time distributions of noisy driven bistable brane of the postsynaptic neuron, increasing the systems—see Sec. IV.D—has connected stochastic reso- probability of a neuronal discharge event (spike), or nance research with neuronal processes. In Longtin, they may be inhibitory—hyperpolarizing the Bulsara, and Moss (1991), Longtin (1993), and Longtin postsynaptic membrane, thereby reducing the prob- et al. (1994), the interspike interval histograms of a sinu- ability of a spike. The high connectivity allows for soidally stimulated auditory nerve from a cat have been two kinds of summation processes, temporal and compared with return-time distributions of the periodi- spatial summation. Both summation processes are cally driven quartic bistable potential, i.e., 3 used in nature. Having a serial input of a train of x˙ ϭxϪx ϩ␰͑t͒ϩA0 cos͑⍀t͒, (5.21) incoming pulses at one synapse, local summation and a soft bistable potential can take place. The typical separation of the pulses is, however, of the same order as the typical decay x˙ ϭϪxϩb tanh͑x͒ϩ␰͑t͒ϩA0 cos͑⍀t͒, (5.22) time of a postsynaptical potential (leakage rate). with the Gaussian noise Spatial summation of incoming events from many different synapses does not suffer from the leakage D ͉tϪtЈ͉ ͗␰͑t͒␰͑tЈ͒͘ϭ exp Ϫ , rate, but requires a spatial distribution of (even very ␶ ͩ ␶ ͪ c c little) information throughout a local neural network. ͗␰͑t͒͘ϭ0. (5.23) (6) If the sum of postsynaptic potentials arriving within The left potential well corresponds to a neuron that is a short period of time exceeds a certain threshold, quiescent; the right potential well corresponds to the fir- the probability for the emission of a spike becomes ing state. The noise correlation time ␶c has been ad- very large. This threshold is of the order of tens of justed to a typical value of the decay of a membrane milliseconds and it therefore requires quite a num- Ϫ4 potential, i.e., ␶cϭ10 in the same dimensionless units ber of inputs to produce a spike. as in Eqs. (5.21) and (5.22). In Fig. 31, the return-time distributions, i.e., the density of time intervals T it takes for the system to be first kicked from one stable state to 2. Stochastic resonance, interspike interval histograms, the other and back again (this second process simulates and neural response to periodic stimuli a reset mechanism), are compared with interspike inter- Over the last 50 years a large body of research has val histograms taken from the cat’s primary auditory been carried out to understand the encoding of acoustic nerve. With only one fitting parameter, it was possible to information on the primary auditory nerve of mammali- achieve excellent agreement. In particular, the sequence ans (see, for example, Teich et al., 1993, and references of peaks in the return-time distributions as well as those therein). Rose, Brugge, Anderson, and Hind (1967) in the interspike interval histograms decay exponentially measured the interspike interval histogram of sinusoi- for large return times. dally stimulated auditory nerve fibers. A few typical ex- The key question is whether neurons also exhibit sto- amples from a squirrel monkey are shown in Fig. 30. On chastic resonance. To this end, Moss and collaborators the vertical axes, the numbers of intervals of length ␶ set up an experiment to study the neural response of (horizontal axes) between two subsequent spikes (taken sinusoidally stimulated mechanoreceptor cells of cray- from a long spike train) are shown. The first peak is fish (Douglass, Wilkens, Pantazelou, and Moss, 1993). located at the period T⍀ of the stimulus and the follow- This experiment has allowed detailed studies of inter- ing peaks are located at multiples of T⍀. In contrast to spike interval histograms to be carried out for a wide the conventional theory of auditory information encod- range of values of the amplitude and frequency of the ing, the results above indicate that the information of stimulation. As in Longtin, Bulsara, and Moss (1991),

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 256 Gammaitoni et al.: Stochastic resonance

FIG. 30. Distributions of interspike intervals when pure tones of different frequencies activate the neuron. Stimulus frequency in cycles per second (cps) is indicated in each graph. Time on the abscissa is in milliseconds. The dots below the abscissa indicate multiples of the period for each frequency employed. On the ordinate, the number of intervals in one bin is plotted (1 bin=100 ␮s). N is the total number of interspike intervals in the sample. N is given as two numbers: the ﬁrst indicates the number of intervals plotted; the second is the number of intervals whose values exceeded the maximal value of the abscissa. After Rose et al. (1967). the interspike interval histograms have been reproduced times. In the presence of periodic driving, Moss et al. by return-time distributions of periodically driven (1993) found—as in the earlier experiments—a multi- bistable systems. Without a stimulus, the interspike in- peaked structure with exponentially decaying peak terval histograms decay exponentially for large return heights. In order to identify stochastic resonance, the

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 257

FIG. 31. Return-time distributions for a sinusoidally stimulated auditory nerve: (a), (b) experimental interspike-interval histogram data from a cat primary auditory nerve with an 800-Hz stimulus. The amplitude of the stimulus is 60 dB in (a) and 30 dB in (b). The full curves in (c) and (d) are results obtained from analog simulation of the standard quartic model of Eqs. (5.21)–(5.23). The parameters A0 and ␶c have been chosen fixed, while D has been fitted to yield best agreement with the interspike-interval histograms in (a) and (b). The best fit in (c) is obtained at a smaller noise level D than in (d). The ratio of driving and noise level in (c) is thus higher than in (d). This is in agreement with the higher stimulus in (a) than in (b). After Longtin, Bulsara, Pierson, and Moss (1994). noise level had to be changed. To change the intrinsic periodically stimulated neurons actually exhibits sto- noise level is not straightforward and requires some chastic resonance. These results, obtained as in situ ex- more involved experimental procedures (Pei et al., perimental results, are quite satisfactory. The present 1996). Douglass, Wilkens, Pantazelou, and Moss (1993) theory, however, based on bistable dynamics [Eqs. chose to change the noise level by adding noise exter- (5.21) and (5.22)], does not describe neuronal dynamics nally, i.e., by applying to the neuron a sum of a single very well. This is because the firing state of a neuron is tone and some noise. The spectral properties of the se- not a stable state. After a neuron has fired, it becomes ries of action potentials were analyzed, yielding a power automatically quiescent after a refractory time. For a spectrum typified by background noise plus sharp peaks more realistic modeling of stochastic resonance in neu- at multiples of the frequency of the stimulus. The signal- ronal processes it is therefore necessary to study differ- to-noise ratio is shown in Fig. 32. The shape of the curve ent, nonlinear (nonbistable) systems. indicates stochastic resonance in a living neuron. An alternative way of describing stochastic resonance is the 3. Neuron firing and Poissonian spike trains dependence of the peak height of the peaks in the interspike interval histograms on the noise level (Zhou et al., Wiesenfeld, Pierson, Pantazelou, Dames, and Moss 1990). The height of the first histogram peak at the pe- (1994) proposed a very elegant approximate theory for riod of the stimulus runs as a function of the noise modeling neuron firing in the presence of noise and a strength through a maximum at a value of the noise that periodic stimulus. The neuron emits uncorrelated, sharp is very close to the peak of the signal-to-noise ratio (see spikes (␦ spikes with weights normalized to unity) at Fig. 32). random times tn . The spiking rate, however, is inhomo- The experiments provide evidence that the firing of geneous, i.e., sinusoidally modulated. This sort of pro-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 258 Gammaitoni et al.: Stochastic resonance

ϱ ¯ 2 S͑␻͒ϭ r ϩ2␲ ͉rn͉ ␦͑␻Ϫn⍀͒, n͚ϭ1

1 T rnϭ r͑t͒exp͑Ϫin⍀t͒dt. (5.25) T ͵0 At this point, the only assumption made is that there are no correlations between the spikes. It is remarkable that spike-spike correlations yield only an additional term to the spectral density (Jung, 1995), but otherwise leave the result [Eq. (5.25)] invariant. To analyze further the expression (5.25), some approximations need to be made. For r(t), an expression has been chosen, motivated by the rate theory for noise-induced barrier crossing in the presence of periodic external forces, i.e. [cf. Eq. (4.60)],

⌬V A0xm r͑t͒ϭ␯ exp Ϫ Ϫ cos͑⍀t͒ , (5.26) ͫ D D ͬ where ⌬V is the barrier height in the absence of the forcing, D is the noise strength, A0 is the amplitude and ⍀ is the frequency of the periodic forcing, xm is a scale factor, and the prefactor ␯ depends on details of the rate process. The expression (5.26) is limited to slow and weak periodic forcing, i.e., ⍀ is small compared to the local (intrawell) relaxation rate and A0 is small enough that its effect on the rate can be treated as a perturbation, i.e., A0xm /D has to be small. In leading order 2 (A0xm /D) , the signal-to-noise ratio (SNR) is given by the ratio of the intensity of the ␦ peak of S(␻)at⍀, i.e.,

⌬V A0xm r ϭ␯I ͑A x /D͒exp Ϫ Ϸ␯ 1 1 0 m ͩ D ͪ 2D ⌬V ϫexp Ϫ for A x /DӶ1, (5.27) ͩ D ͪ 0 m to the noise background in the absence of the periodic driving FIG. 32. Stochastic resonance in mechanoreceptor cells of the ⌬V S0 ͑⍀͒ϭ¯rϭ␯ exp Ϫ , (5.28) tail fan of a crayfish: (a) The signal-to-noise ratio obtained N ͩ D ͪ from power spectra and (b) from the peak height of the first peak in the interspike-interval histogram as a function of the yielding externally applied noise level. 2 2 2 4␲͉r1͉ ␲xmA0 ⌬V SNRϭ Ϸ exp Ϫ . (5.29) ¯r D2 ͩ D ͪ cess is described by the theory of inhomogeneous Pois- sonian point processes—see, for example, chapter 6 of Note that corresponding expression in Wiesenfeld, Pier- Stratonovich (1963). Given the time-dependent spiking son, Pantazelou, Dames, and Moss (1994) differs in the prefactor, because they used a different definition of the rate r(t), the probability to find s events in the time spectral density. interval T is given by [cf. Eq. (6.33) in Stratonovich The signal-to-noise ratio shows the characteristic fea- (1963)] ture of stochastic resonance, i.e., a peak as a function of 1 T s T the noise strength D. Comparison with data obtained Psϭ r͑t͒dt exp Ϫ r͑t͒dt . (5.24) from spike sequences of a mechanically modulated s! ͫ ͵0 ͬ ͭ ͵0 ͮ mechanoreceptor of a crayfish (see Fig. 33) shows quali- The (phase-averaged) spectral density of the spike tative agreement. The decrease of the SNR at large train consists of a frequency-independent term (white noise levels, however, is overestimated by this theory. A shot noise), given by the time-averaged spiking rate ¯r , nonadiabatic theory based on threshold crossing dynam- and a sum of ␦ peaks at multiples of the stimulus fre- ics (see Sec. V.C.5) predicts a slower decrease. Stochas- quency, where the intensities of the peaks are given by tic resonance was also demonstrated in an experiment the Fourier coefficients ␣n of the periodic spiking rate that uses a sensory hair cell of a cricket (Levin and r(t), i.e., Miller, 1996).

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 259

FIG. 34. The first-passage time distribution function g(t)vs FIG. 33. Signal-to-noise ratio in crayfish mechanoreceptors time measured in units of the normalized time t/T⍀ (note that (squares) compared to the electronic Fitzhugh-Nagumo simu- in the figure t denotes the first passage time, and TϭT⍀ is the lation (diamonds) (Wiesenfeld et al., 1994) and the theory of forcing period). The figure contains two sets of curves, one Sec. V.C.3 (solid curve). The horizontal axis represents the with solid lines and another with dashed lines. The set with intensity of externally applied noise: hydrodynamic noise in solid lines corresponds to a current i0ϭ0.065 A0. The smooth the case of the mechanoreceptors, and electronic noise in the solid curve (without multiple peaks) represents the distribu- case of neuron models. The crayfish data do not decrease rap- tion function without periodic stimulus (A0ϭ0) while the solid idly for small noise because of the residual internal noise of the curve with the peaks corresponds to A0ϭ0.03. The smooth neuron. Figure provided by Professor Moss. dashed curve corresponds to A0ϭ0 and the multipeaked dashed curve corresponds to A0ϭ0.03. The other parameters 4. Integrate-and-fire models (for all curves) are bϭ20, ⍀ϭ0.1, Dϭ0.2. A very common model for neuronal dynamics is the so-called integrate-and-fire model. The model works as ϱ follows: the input i(t) of the neuron consists of a spike y͑t͒ϭ͚ h͑tϪtn͒ϭ ͵ yd͑tϪs͒h͑s͒ds, train (cortical neurons) or a continuous signal (sensory n Ϫϱ neurons). The membrane voltage u(t) is obtained by integrating the input i(t). A capacitance C together yd͑t͒ϭ ␦͑tϪtn͒, (5.32) with an ohmic resistance R across the membrane leads ͚n to an exponential decay of the membrane voltage. With where h(t) describes the (fixed) shape of a neuronal additional noise ␰(t), the equation of motion for the spike. Given the distribution function of the times t ,or membrane voltage u reads n equivalently that of the interspike intervals ⌬ ϭt Ϫt , the spectral properties of the spike train 1 n n nϪ1 u˙ ϭϪ uϩi͑t͒ϩ␰͑t͒, (5.30) can be computed by means of the theory of random ␶RC point processes [see, for example, Stratonovich (1963)]. In the case of a ‘‘perfect’’ integrator 1/(RC)ϭ0 , the where ␶RCϭRC. Due to the linearity of Eq. (5.30), the ͓ ͔ noise ␰(t) can consist of a sum of two contributions Fokker-Planck equation for the membrane voltage, stemming from inherent (correlated) fluctuations of the equivalent to Eq. (5.30), reads for a sinusoidal input membrane potential and noise in the input. Here, we i(t)ϭi0ϩA0 cos(⍀t) consider only noisy input and assume that ␰(t) is Gauss- P͑u,t͒ץ P͑u,t͒ץ ,.ian white noise, i.e ϭϪ͓i ϩA cos͑⍀t͔͒ uץ t 0 0ץ ͗␰͑t͒␰͑tЈ͒͘ϭ2D␦͑tϪtЈ͒, 2P͑u,t͒ץ ϩD . (5.33) u2ץ (␰͑t͒͘ϭ0. (5.31͗

When the membrane voltage reaches a critical value u0 The distribution function of the interspike intervals t (the threshold), the neuron fires, then is reset, and the ϭ ⌬ is given by the mean-first-passage time distribution whole procedure can start all over again. ␳MFPT(t) [see, for example, Ha¨nggi et al. (1990)], which Denoting by tn the times at which the neuron fires, the is obtained by solving Eq. (5.33) with absorbing bound- neuron exhibits a spike train of the form ary conditions at the threshold b, i.e., P(uϭb,t)ϭ0, and

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 260 Gammaitoni et al.: Stochastic resonance subsequent differentiation with respect to time. This A0␶RC task was carried out first analytically in the absence of uϱ͑t͒ϭi0␶RCϩ sin͑⍀tϪ␸RC͒, (5.34) 1ϩ⍀2␶2 the periodic stimulus A0ϭ0 [see also the corresponding ͱ RC solution for a finite leakage rate 1/(RC) in Goel and where tan(␸ )ϭ⍀/␶ . The threshold is larger than the Richter-Dyn (1974)], and then numerically in the pres- RC RC maximum of uϱ(t), i.e., we assume a subthreshold ence of the stimulus by Gerstein and Mandelbrot (1964). stimulus. In the presence of the noise ␰(t), the mem- More recently, Bulsara, Lowen, and Rees (1994) and brane voltage u(t) will randomly cross the threshold. In Bulsara, Elston, Doering, Lowen, and Lindenberg contrast to the integrate-and-fire model, the membrane (1996) used an approximate image-source method to voltage is not reset after a threshold crossing in the mod- solve the Fokker-Planck equation in the presence of a weak and slowly varying sinusoidal stimulus. The fea- els being discussed here. The threshold-crossing models tures of their result follow: are relying on a stochastic self-resetting due to the noise itself. (1) The first-passage time distribution shows in the pres- The simple picture we are drawing is the following: ence of the periodic stimulus a multipeaked struc- the input of our threshold trigger consists of the sum of ture (see Fig. 34). For sufficiently large stimulus, the a sinusoidal signal with amplitude peaks are located at tn ϭnT , with T ϭ2␲/⍀ be- 2 2 max ⍀ ⍀ A0 A0␶RC /ͱ1ϩ⍀ ␶RC and random noise ␰(t) which ing the period of the stimulus [see also Gerstein and occasionally→ crosses the reduced threshold Mandelbrot (1964)]. b bϪi0␶RC . Whenever the threshold b is being (2) The peak heights decay exponentially for increasing → crossed (at times tn), a spike is created. This yields a intervals t. stochastic spike train given in Eq. (5.32). To keep things (3) The peak heights run through a maximum as a func- simple, we assume a ␦-shaped spike with area normal- tion of the noise strength D. ized to unity, i.e., h(t)ϭ␦(t) in Eq. (5.32). In specific This behavior resembles very closely the behavior of terms, we assume Gaussian-colored noise ␰(t) with a return-time distribution of the bistable models described zero mean and the correlation function in Sec. V.C.2. As yet, the theory above is based on a number of unrealistic assumptions; moreover, it contains D ͗␰͑t͒␰͑0͒͘ϭ 2 2 ͓␶2 exp͑Ϫt/␶2͒ technical difficulties that have yet to be overcome: ␶2Ϫ␶1 (1) The phase of the sinusoidal stimulus has been reset Ϫ␶1 exp͑Ϫt/␶1͔͒, (5.35) after each firing event to the same initial value. This approximation is unrealistic from a physiological where ␶1 and ␶2 are characteristic time scales. Using the point of view, since a large amount of information fundamental work of Rice (1948), one finds for the about the coherence of the stimulus is eliminated. A threshold crossing rate the central result (Jung, 1995) theory of first-passage time distributions in the pres- 1 (1ϪA¯ sin͑⍀t͒)2 ence of a periodic forcing that explicitly avoids this r͑t͒ϭ exp Ϫ 2␲ͱ␶ ␶ ͩ 2¯␴ 2 ͪ assumption has not yet been put forward. 1 2 (2) Since the resting voltage of the membrane of a neu- A¯ 2⑀2 cos2͑⍀t͒ ron is very close to the potassium voltage, being a ϫ exp Ϫ lower bound for the variation of the membrane volt- ͫ ͩ 2¯␴ 2 ͪ age, an originally sinusoidal stimulus becomes strongly rectified. It is therefore not realistic simply 1 2␲ A¯ ⑀ cos͑⍀t͒ ¯ to add the sinusoidal stimulus to the membrane volt- ϩ A⑀ cos͑⍀t͒erfc Ϫ 2 ͱ¯␴ 2 ͩ ͱ2¯␴ 2 ͪͬ age in the integrate-and-fire model without taking into account rectification. 1 (3) Strictly speaking, the method of image sources is ap- ϵ f͑A¯ ,¯␴,⑀͒, (5.36) plicable only to diffusion processes that are homo- ͱ␶1␶2 geneous in space and time variables. The error made with the scaled parameters A¯ ϭA /b, ¯␴ 2ϭ␴2/b2, by using this method (as an approximation) in time- 0 ⑀2ϭ⍀2␶ ␶ . The periodicity reflects the encoding of the inhomogeneous equations such as Eq. (5.33) has not 1 2 periodic input signal in the spike train. The power spec- been estimated mathematically. trum of the spike train has the same form as in Eq. (5.25) but has an additional term describing the spike- spike correlation function. Computing the Fourier coefficient r1 of the periodic threshold crossing rate r(t), 5. Neuron firing and threshold crossing one finds for the spectral amplification The threshold-crossing model for neuronal spiking is 4͉r ͉2 1 1 ¯ ¯ ¯2 motivated from the leaky integrate-and-fire model as ␩ϭ 2 ϭ ␩͑A,␴ ,⑀͒. (5.37) A0 ␶1␶2 follows: the input i(t) consists of a constant i0 and a sinusoidal modulation A0 sin(⍀t). In the absence of The spectral amplification ␩ describes the encoding gain noise, the solution of Eq. (5.30) reads for large times of the periodicity of the input signal in the stochastic

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 261

¯ FIG. 35. The scaled spectral amplification ␩, of Eq. (5.37), is FIG. 36. The scaled spectral amplification ¯␩ shown as a func- ¯ shown as a function of the variance of the noise (a) for Aϭ0.1, tion of the amplitude of the sinusoidal signal A¯ (a) for ¯ ¯ (b) Aϭ0.5, and (c) Aϭ1. The time-scale ratio ⑀ was chosen as ¯␴ 2ϭ0.1,(b) ¯␴ 2ϭ0.5, and (c) ¯␴ 2ϭ1. The time-scale ratio ⑀ was ⑀ϭ1. Note that due to the scaling relation for ␩ in Eq. (5.37), chosen as ⑀ϭ1. the actual spectral amplification can become much larger than unity (real spectral amplification) if the cutoff frequencies 1/␶1 and/or 1/␶2 are chosen large enough. the amplitude, as in the quartic double-well system (Sec. IV). In conclusion, stochastic resonance has been demon- ¯ spike train. The scaled spectral amplification ␩ is shown strated experimentally in neuronal systems although as a function of the variance of the noise in Fig. 35. For these systems are not bistable. In the course of develop- A0 /bϽ1, i.e., in the subthreshold regime, the spectral ing a theoretical understanding of these experimental amplification first increases with increasing noise, results, the notion of stochastic resonance has been gen- ¯2 reaches a maximum at ␴ maxϷ1/2, and then decreases eralized to include excitable systems with threshold dy- again. For small time scales ␶1,2 , the spectral amplifica- namics. In this latter context we refer also to related tion becomes large and describes a real encoding gain work of Collins and collaborators on aperiodic stochas- facilitated by random noise. The position of the maxi- tic resonance (Collins et al., 1995a, 1995b, Collins, mum is obtained from Eq. (5.36) by expanding for Chow, et al., 1996; Collins, Imhoff, and Grigg, 1996; A0␶RC /DӶ1, i.e., Heneghan et al., 1996), and the recent developments 1 1 ␲ aimed at detecting stochastic resonance in nondynamical ␩ϭ r2͑A ϭ0͒ ϩ⑀2 . (5.38) systems with no intrinsic sharp thresholds (Fulin´ ski, 0 4 2 2 ͩ¯␴ 2¯␴ ͪ 1995; Barzykin and Seki, 1997; Bezrukov and Vodyanoy, The first term on the right-hand side of Eq. (5.38) has 1997; Jung, 1997; Jung and Wiesenfeld, 1997). been obtained within the adiabatic theory of Gingl, Kiss, and Moss (1995), and also by the approach of Wiesen- VI. STOCHASTIC RESONANCE—CARRIED ON feld et al. (1994) (see Sec. V.C.3), while the other term represents nonadiabatic corrections (Jung, 1995). For A. Quantum stochastic resonance large variances of the noise, the nonadiabatic corrections become very important; they yield a decrease of Recently, the question has been posed whether sto- the spectral amplification proportional to 1/¯␴ 2 instead of chastic resonance manifests itself on a quantum scale. In 1/¯␴ 4 in the adiabatic limit. The position of the peak de- particular, recent experiments in a macroscopic quan- viates significantly from that in a driven symmetric tum system, such as a superconducting quantum inter- bistable system. In the limit of vanishing frequency ⍀, ference device (SQUID), established the mechanism of the spectral amplification approaches a limit curve with stochastic resonance in the classical regime of thermal ¯2 1 activation (Rouse et al., 1995; Hibbs et al., 1995). The the maximum at approximately ␴ maxϭ 2 . Increasing the frequency, the peak increases due to periodic jittering experimental work of Rouse, Han, and Lukens (1995) back and forth across the threshold, and shifts towards also addressed nonlinear stochastic resonance, such as the noise-induced resonances, which are elucidated in ¯2 larger values of the variance ␴ . Sec. VII.D.1 below. Because quantum noise persists In Fig. 36, the scaled spectral amplification ¯␩ is shown even at absolute zero temperature, the transport of ¯ ¯2 ¯2 as a function of the amplitude A. For ␴ Ͻ␴ max , the quantum information should naturally be aided by quan- spectral amplification shows a maximum as a function of tum fluctuations as well. Indeed, quantum mechanics

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 262 Gammaitoni et al.: Stochastic resonance

role of quantum tunneling in this regime has been investigated only recently by Grifoni et al. (1996). These authors investigated the dissipative inertial bistable quantum dynamics x(t) at thermal temperature T in a double-well conﬁguration which is modulated by the periodic force A0 cos(⍀t). The asymptotic power spectrum ϱ ¯ Sas(␻)ϭ͐Ϫϱ exp(Ϫi␻␶)Kas(␶)d␶, cf. Eq. (4.24), of the time-averaged, symmetrized autocorrelation function of x(t) is given by ϱ 2 Sas͑␻͒ϭ2␲ ͉Mn͑⍀,A0͉͒ ␦͑␻Ϫn⍀͒. (6.1) nϭϪ͚ϱ

Hereby, we introduced the notation Mn(⍀,A0) for the complex-valued Fourier amplitude to explicitly indicate the dependence on the relevant parameters. The two FIG. 37. The dominant escape mechanisms out of a metastable quantities to exhibit quantum stochastic resonance are potential, and corresponding regimes for stochastic resonance the power amplitude ␩ in the first frequency component depicted vs temperature T. T0 denotes the crossover tempera- of S (␻) and the ratio of ␩ to the unperturbed, equilib- ture below which quantum tunneling dominates over thermally as rium power spectrum S0 (␻)ofx(t) in the absence of activated hopping events. We note that T depends on the N 0 driving, evaluated at the external modulation frequency potential shape and also on the dissipative mechanism (Ha¨nggi et al., 1985). The relative size of the corresponding stochastic ⍀, i.e. the so-called signal-to-noise ratio (SNR): resonance regions hence vary with the dissipation strength. In 2 ␩ϭ4␲͉M1͑⍀,A0͉͒ , the region marked by a question mark, quantum stochastic 2 0 resonance has presently not yet been investigated analytically; SNRϭ4␲͉M1͑⍀,A0͉͒ /SN͑⍀͒. (6.2) a two-level approximation is no longer adequate in that tem- By definition, ␩ has the dimension of a length squared, perature regime. while SNR has the dimension of a frequency. Thus to investigate the interplay between noise and the coherent provides the nonlinear system with an additional chan- driving input, giving rise to the phenomenon of stochas- nel to overcome a threshold. This additional channel is tic resonance, we shall consider two dimensionless quan- provided by quantum tunneling, i.e., a particle can tun- tities: The scaled spectral amplification ˜␩, and the scaled nel through a barrier without ever going over it. As a signal-to-noise ratio SNR. They read matter of fact, we shall see that the classical stochastic 4␲͉M ͑⍀,A ͉͒2 resonance effect can be assisted by quantum tunneling 1 g0 ˜␩ϭ 2 2 , contributions even at finite temperatures. For strongly ͑A0xm/Vb͒ damped systems, such contributions can enhance the 2 0 classical stochastic resonance effect up to two orders of ͓4␲͉M1͑⍀,A0͉͒ /SN͑⍀͔͒/␻b SNRϭ 2 . (6.3) magnitude. With decreasing temperature, quantum tran- ͑A0xm /Vb͒ sitions thus start to dominate over thermally activated Here,gV is the barrier height at the barrier position transitions below a crossover temperature T (Ha¨nggi b 0 x ϭ0, ␻ denotes the corresponding angular barrier fre- et al., 1985), which characterizes the temperature at b b quency, and Ϯx are the positions of the two minima of which activated hopping and quantum tunneling become m the bistable potential. These two quantities that charac- equally important. Its value depends both on the pote- terize stochastic resonance can be evaluated within nial barrier shape and the dissipative mechanism. It is quantum linear-response theory to give for the scaled interesting to note that this crossover temperature can spectral amplification be quite large, reaching in some physical and chemical 2 2 systems values larger than 100 K (Ha¨nggi et al., 1990; Vb 1 ␭ ˜␩ϭ␲ . (6.4) Ha¨nggi, 1993). On the other hand, in Josephson systems ͩ k Tͪ cosh4͑⑀ /2k T͒ ⍀2ϩ␭2 B 0 B (Schwartz et al., 1985; Clarke et al., 1988; Ha¨nggi, 1993) and in mesoscopic, disordered metals (Golding et al., Here, kB is the Boltzmann constant, and ␭ϭrϩϩrϪ is 1992; Chun and Birge, 1993) tunneling dominates in the the sum of the forward and backward quantum rates rϩ cold mK region only. The various escape mechanisms and rϪ , respectively. These unperturbed quantum rates that predominate the physics of stochastic resonance as have been evaluated previously in the literature—see a function of temperature T are depicted in Fig. 37. Sec. IX in the review of Ha¨nggi, Talkner, and Borkovec (1990). A possible difference between the left and the right potential minimum is accounted for by the bias 1. Quantum corrections to stochastic resonance energy ⑀0 . The backward and forward rates are related Let us first focus on the regime TտT0 , where quan- by the detailed-balance condition rϪϭrϩ exp(Ϫ⑀0 /kBT). tum tunneling is not the dominant escape path, but nev- Note that information about the detailed form of the ertheless leads to significant quantum corrections. The potential, and the dissipative mechanism as well, is still

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 263

FIG. 38. The scaled signal-to-noise ratio [within linear re- FIG. 39. Scaled spectral amplification ˜␩ [see Eq. (6.4)] in a sponse; see Eq. (6.5)] SNR of semiclassical, inertial dynamics symmetric double well vs dimensionless temperature (cf. Fig. in a symmetric double well vs the dimensionless temperature 38) for different driving frequencies ⍀ (solid lines). For com- T/T0 , with T0 denotingg the crossover temperature (see text), parison, the dashed lines give the results for the classical sto- for ohmic friction ␥ of strength ␣ϭ␥/2␻bϭ50. The solid line chastic resonance spectral amplification. The dimensionless gives the dimensionless semiclassical signal-to-noise ratio with ohmic friction strength is ␣ϭ␥/2␻bϭ50. The inset depicts the tunneling corrections. The dashed line gives the corresponding ratio between the total (forward and backward) rate ¯⌫ ϵ␭ classical result without quantum corrections. The ratio be- and ⍀ at the temperature T␩* where ˜␩ assumes its maximum. tween semiclassical quantum signal-to-noise ratio, SNR, and The stochastic resonance maximum is thus approximately de- the corresponding classical result, SNRcl, is depicted in the termined by the condition that twice the escape rate, i.e. g ¯ inset. Note that the tunneling contribution can enhance sto- ⌫(T␩*), approximately equals the external driving frequency chastic resonance up to two orders ofg magnitude. From Grifoni ⍀. After Grifoni et al. (1996). et al. (1996). competing effects of an increasing Arrhenius factor and 2 contained in the total quantum rate ␭. Likewise, one a decreasing factor (kBT) with increasing noise tem- finds the result for the scaled SNR, i.e., perature. These two quantities characteristically rule the 2 stochastic resonance effects—see Eqs. (4.51) and (4.54). ␲ Vb ␭/␻b SNRϭ 2 g . (6.5) 2 ͩkBTͪ cosh ͑⑀0 /2kBT͒ 2. Quantum stochastic resonance in the deep cold Bothg the zero-point energy fluctuations and the dissipa- The situation changes drastically when we proceed to- tive tunneling across the barrier near the barrier top re- wards the extreme cold. Here, we shall focus on the sult in a characteristic enhancement of the SNR˜, or the deep quantum regime, where tunneling is the only chan- scaled spectral amplification ˜␩. The enhancement can nel for barrier crossing. In this low-temperature regime, reach values up to two orders of magnitude as compared periodic driving induces several new interesting, coun- to a prediction based solely on a classical analysis. In terintuitive physical phenomena, such as ‘‘coherent de- Fig. 38 we depict the quantum-tunneling-corrected, struction of tunneling’’ (Grossmann et al., 1991), the scaled SNR for zero bias, i.e., ⑀0ϭ0, together with the ‘‘stabilization of dissipative coherence’’ with increasing enhancement over the corresponding classical study (see temperature (Dittrich et al., 1993; Oelschla¨gel et al., the inset). 1993), or the effect of driving-induced quantum coher- The temperature dependence at different driving fre- ence (Grifoni et al., 1995). Quantum stochastic reso- quencies ⍀ of the scaled spectral amplification is de- nance within the regime of incoherent tunneling transi- picted in Fig. 39 for quantum stochastic resonance in a tions at adiabatic driving frequencies has been symmetric (⑀0ϭ0) double well subjected to ohmic quan- investigated first by Lo¨ fstedt and Coppersmith (1994a, tum friction. In presence of tunneling, the role of both 1994b) in the context of impurity tunneling in ac-driven temperature and dissipation must be treated simulta- mesoscopic metals (Golding et al., 1992; Chun and neously in a manner consistent with the fluctuation- Birge, 1993; Coppinger et al., 1995). Linear response for dissipation theorem (Grifoni et al., 1996). In particular, quantum stochastic resonance as well as nonlinear quan- with strong damping the effects of quantum fluctuations tum stochastic resonance has been investigated in the on stochastic resonance can extend well above the cross- whole parameter range by Grifoni and Ha¨nggi (1996a, over temperature T0 . As depicted in the inset of Fig. 39, 1996b); their studies encompass adiabatic and nonadia- the stochastic resonance peak is dominated by the two batic driving frequencies, as well as the role of both an

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 264 Gammaitoni et al.: Stochastic resonance incoherent (i.e., rate-dominated relaxation) and coherent (i.e., oscillatory-dominated damped relaxation) tunneling dynamics. In the regime below the crossover regime, i.e., the regime marked by a question mark (?) in Fig. 37, there exist at present no analytical studies of quantum stochastic resonance. This is due mainly to the fact that in this regime the dissipation-driven tunneling dynamics in- volve many states. At very low temperatures, however, the dynamics are ruled mainly by two tunnel-split levels only. Thus in the deep quantum regime the investigation of quantum stochastic resonance reduces to the study of the dynamics of the spin-boson system in the presence of ohmic dissipation which, in addition, is subjected to periodic driving (Grifoni et al., 1993, 1995; Dakhnovskii FIG. 40. Sketch of the different regimes for the driven and Coalson, 1995; Makarov and Makri, 1995; Goychuk, ͓(ប⑀ˆ /a)cos ⍀t͔ dissipative tunneling dynamics for a two-level 2 Petrov, and May, 1996; Makri, 1997). More explicitly, let system subject to weak ohmic coupling ␣ϵ␥(a /2␲ប)(ade- us consider the driven spin-boson Hamiltonian notes the tunneling distance and ␥ denotes the Ohmic viscous HϭH (t)ϩH , i.e., strength). As the temperature T or the driving angular fre- TLS B quency ⍀ are varied in the parameter space of ‘‘TEMPERA- ប ប⑀ˆ TURE’’ and ‘‘FREQUENCY’’ [see the boxes labeled (a), (b), HTLS͑t͒ϭϪ ͑⌬␴xϩ⑀0␴z͒Ϫ cos͑⍀t͒␴z (6.6) (c)], different novel tunneling regimes are encountered. For 2 2 strong dimensionless ohmic coupling ␣, the regimes (a) and (b) represents the driven bistable system in a two-level- extend down to the lowest temperatures. For comparison we ˆ system approximation with (ប⑀ˆ /a)cos (⍀t) being the ap- also depict the static situation, i.e., ⑀ϭ0, in the upper left plied harmonic force. The ␴’s are Pauli matrices, and the panel: in it, we sketch the dissipative tunneling behavior in the (T,␣) parameter space for quantum incoherent and quantum eigenstates of ␴ are the basis states in a localized rep- z coherent (QC) tunneling. resentation, while aϵ2xm is the tunneling distance. The tunneling splitting energy of the symmetric two-level K¯ (␶) acquire in the asymptotic regime the periodicity of system is given by ប⌬ while the bias energy is again ប⑀0 . Within the spin-boson model (Leggett et al., 1987; the external perturbation. Upon expanding the Weiss, 1993), the environment is modeled by a term H asymptotic expectation Pas(t)ϭlimt ϱP(t) into a Fou- B → describing an ensemble at thermal temperature T of rier series, i.e., harmonic oscillators. The term HB in addition includes ϱ ˆ the interaction between the two-level system and the Pas͑t͒ϭ ͚ Mn͑⍀,⑀͒exp͑Ϫin⍀t͒, (6.7) bath via a bilinear coupling in the two-level system-bath nϭϪϱ coordinates. The effects of the bath are captured by the ˆ it is readily seen that the amplitudes ͉Mn(⍀,⑀)͉ deter- spectral density J(␻) of the environment coupling. We mine the weights of the ␦ spikes of the power spectrum make the specific choice of ohmic dissipation in the asymptotic state Sas(␻) via the relation ϭ 2 Ϫ␻/␻c J(␻) (2␲ប/a )␣␻e , where ␣ denotes the dimen- ϱ ˆ 2 Sas(␻)ϭ2␲͚nϭϪϱ͉Mn(⍀,⑀)͉ ␦(␻Ϫn⍀). In particular, sionless ohmic coupling strength and ␻cӷ␻0 is a cutoff frequency. Insightful exact numerical path-integral stud- to investigate nonlinear quantum stochastic resonance, ies of this driven, dissipative spin-boson system have we shall examine the newly scaled power amplitude ␩n in the nth frequency component of S (␻), i.e., been carried out recently by Makri (1997). as ˆ ˆ 2 The relevant theoretical quantity describing the dissi- ␩nϭ4␲͉Mn͑⍀,⑀͒/ប⑀͉ . (6.8) pative dynamics under the external perturbation is the For a quantitative study of quantum stochastic reso- expectation value P(t)ϭ͗␴z(t)͘. On the other hand, the quantity of experimental interest for quantum stochastic nance, it is necessary to solve the asymptotic dynamics resonance is the time-averaged power spectrum S(␻), of the nonlinearly driven dissipative bistable system. In defined as the Fourier transform of the correlation func- doing so, we shall take advantage of novel results for the tion driven dynamics obtained by use of a real-time path- integral approach (Grifoni et al., 1993; 1995). At weak ⍀ 2␲/⍀ 1 ¯ and strong ohmic coupling, driving distinctly alters the K͑␶͒ϭ dt ͗␴z͑tϩ␶͒␴z͑t͒ϩ␴z͑t͒␴z͑tϩ␶͒͘. 2␲ ͵0 2 qualitative tunneling dynamics: see Fig. 40. The incoherent dynamics can still be modeled by rate equations, The combined influence of dissipative and driving forces however. At low driving frequencies, these rate equa- renders an evaluation of the full correlation function tions are intrinsically Markovian—see region (a) in Fig. K¯ (␶) extremely difficult (and hence of the power spec- 40. As the external frequency ⍀ is increased and/or trum). Matters simplify for times t,␶ large compared to when the temperature is lowered, quantum coherence the time scale of the transient dynamics, where P(t) and and/or driving-induced correlations render the

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 265

cess towards thermal equilibrium is maximal. On the other hand, the minimum in ␩ appears in the temperature region where driving-induced coherent processes are of importance. This latter feature is a nonlinear quantum stochastic resonance effect, which linear- response theory clearly cannot describe. In addition, this implies that the power amplitude ␩ plotted versus frequency shows resonances when ⍀Ϸ⑀0 /n (nϭ1,2,...); correspondingly, the driven dissipative dynamics are intrinsically non-Markovian! For arbitrary values of the ohmic coupling strength, one has to resort to approximate solutions of the dissipative dynamics. For strong coupling ␣Ͼ1, or weak coupling ␣Ͻ1 and high enough FIG. 41. The spectral amplification ␩1 , Eq. (6.8), for the peri- temperatures, the bath-induced correlations between odically driven (driving strength ⑀ˆ ), ohmic damped spin-boson tunneling transitions may be treated within the noninter- system depicted vs temperatures T at a fixed bias energy acting blip approximation (Leggett et al., 1987; Weiss, ប⑀0ϭ10 and at an angular driving frequency ⍀ϭ5. The dimen- 1993). A set of coupled equations for the Fourier coef- sionless parameters are defined in the text. For the smallest ficients Mn can then be derived for any strength and driving strength ⑀ˆ ϭ5 we additionally depict the linear- frequency of the driving force. In particular, driving- response theory approximation (dashed line). The solid lines induced correlations may result in an highly coherent give the nonlinear quantum stochastic resonance for the ex- dynamics, leading to resonances in the power spectrum. actly solvable case of ohmic coupling strength at ␣ϭ1/2 (cf. In this coherent regime, quantum stochastic resonance Fig. 40). The data are taken from Grifoni and Ha¨nggi (1996b). never occurs: the power amplitudes ␩n always show a monotonic decay as the temperature is increased (Gri- asymptotic dynamics intrinsically non-Markovian—note foni and Ha¨nggi, 1996a, 1996b). It is only in the low- regions (b) and (c) in Fig. 40. frequency regime ប⍀Ӷ␣kBT that—to leading order— Let us focus first on some characteristics of quantum driving-induced non-Markovian correlations do not stochastic resonance as they emerge from the study of contribute. The asymptotic dynamics, within the nonin- the case ␣ϭ1/2 of the ohmic strength. For the special teracting blip approximation, are intrinsically incoherent 1 ˙ value ␣ϭ 2 , exact analytical solutions for quantum sto- and governed by the rate equation Pas(t)ϭϪ␭(t) chastic resonance are possible (Grifoni and Ha¨nggi, ϫ͓Pas(t)ϪPeq(t)͔, with time-dependent rate 1996a, 1996b). The resulting fundamental power ampli- ␭(t)ϭRe ⌺͓␧(t)͔ and equilibrium value Peq(t) ˆ tude ␩1ϵ␩ is plotted in Fig. 41 as a function of the ϭtanh͓ប␧(t)/2kT͔. Here, ␧(t)ϭ⑀0ϩ⑀ cos ⍀t plays the temperature for different driving strengths ⑀ˆ . There, and role of a time-dependent adiabatic asymmetry, and the in Fig. 42, frequencies are in units of the rate ␭(t)(␣Ͻ1) is obtained as the real part of bath-renormalized tunneling splitting ⌬eϭ⌬(⌬/ 1Ϫ2␣ ⌬e ␤⌬e ⌫͑␣ϩiប␤␧͑t͒/2␲͒ ␻ )␣/(1Ϫ␣)͓cos(␲␣)⌫(1Ϫ2␣)͔1/(2Ϫ2␣), ␣Ͻ1. For ␣ϭ 1 , ⌬ ⌺͓␧͑t͔͒ϭ , c 2 e ␲ ͩ 2␲ ͪ ⌫͑1Ϫ␣ϩiប␤␧͑t͒/2␲͒ 2 reduces to ␲⌬ /2␻c . For ␣Ͼ1, relevant energy scale is (6.9) set by ⌬ren , which equals the previous expression ⌬e without the term in the square brackets (Legget et al., 1987; Weiss, 1993; Lo¨ fstedt and Coppersmith, 1994a). The temperatures are in units of ប⌬e /kB . Note that the Ϫ2 spectral amplification is measured in units of (ប⌬e) . ˆ For highly nonlinear driving fields ⑀Ͼ⑀0 , the power amplitude decreases monotonically as the temperature increases (uppermost curve). As the driving strength ⑀ˆ of the periodic signal is decreased, a shallow minimum, followed by a maximum, appears when the static asymme- ˆ try ⑀0 equals, or slightly overcomes, the strength ⑀ (in- termediate curves). For even smaller external amplitudes, quantum stochastic resonance can be studied within the quantum linear-response theory (dashed FIG. 42. Quantum noise-induced resonances for the third- curve in Fig. 41). In the linear-response region the shal- ˆ ˆ 2 order amplitude ␩3ϭ4␲͉M3(⍀,⑀)/ប⑀͉ vs temperature T in low minimum is washed out, and only the principal the regime of adiabatic incoherent tunneling. The different maximum survives. It is now interesting to observe lines are for three different driving frequencies ⍀. The noise- that—because the undriven two-level system dynamics induced suppression which characterizes the ‘‘resonance’’ 1 (which comprises linear-response theory) for ␣ϭ 2 is al- sharpens with decreasing driving frequency ⍀. The results are ˆ ways incoherent down to Tϭ0—the principal maximum for a bias energy ⑀0ϭ20, driving strength ⑀ϭ10, and a coupling arises at the temperature T at which the relaxation pro- strength of ␣ϭ0.1. Data are from Grifoni and Ha¨nggi (1996b).

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 266 Gammaitoni et al.: Stochastic resonance

␤ប⑀0 where ⌫(z) denotes the gamma function, and ␤ is rϩϭe rϪ . It is now interesting to note that formally the inverse temperature (1/kBT). the same expression for the incoherent susceptibility The rate equation can then be solved in terms of (and hence for ␩) holds true for the classical case, with quadratures, and the nonlinear low-frequency power rϩ and rϪ denoting the corresponding classical forward spectrum can be investigated. Quantum stochastic reso- and backward Kramers rates. Thus in classical stochastic nance indeed occurs in this incoherent tunneling regime. resonance, the maximum arises because of competition As for the case ␣ϭ 1 , the quantum stochastic resonance between the thermal Arrhenius dependence of these 2 Ϫ1 maximum appears only when the static asymmetry ⑀ rates and the algebraic factor (kBT) that enters the 0 linear susceptibility, and this maximum occurs (roughly) overcomes the external strength ˆ . Moreover, Fig. 42, ⑀ at a temperature that follows from the matching be- which shows the behavior of the third scaled spectral tween the frequency scales of the thermal hopping rate amplification ␩3 versus temperature, reveals another and the driving frequency. Detailed investigation reveals striking effect: As the driving frequency is decreased, a that quantum stochastic resonance characteristically oc- noise-induced suppression of higher harmonics occurs in curs when incoherent tunneling contributions dominate correspondence to the stochastic resonance maximum in over coherent tunneling transitions. Moreover, in clear the fundamental harmonic. In this regime one finds the contrast to classical stochastic resonance, and also to quantum analogue of noise-induced resonances that semiclassical stochastic resonance near and above T0 characterize classical nonlinear stochastic resonance (see (see above), quantum stochastic resonance in the deep Sec. IV.A). A numerical evaluation shows that the cold occurs only in the presence of a finite asymmetry noise-induced suppression indeed appears when ⑀0Þ0 between forward and backward escape paths. ⍀Ӷmin␭(t), so that the quasistatic expression holds, Thus while classical and semiclassical stochastic reso- i.e., nance is maximal for zero bias [see Eqs. (4.55), (4.56)] the quantum stochastic resonance phenomenon vanishes 1 2␲ ˆ in the deep quantum regime when the symmetry be- Mnϭ dx tanh͓ប␤͑⑀0ϩ⑀ cos x͒/2͔cos͑mx͒. 2␲ ͵0 tween forward and backward dissipative tunneling tran- (6.10) sitions is approached. What is the physics that governs In contrast to classical stochastic resonance, where the this behavior? Clearly, with decreasing temperature the enhancement is maximal for symmetric bistable systems, thermal, exponential-like Arrhenius factor no longer we find that a nonzero bias is necessary for quantum dominates the escape rates; rather, its role is taken over stochastic resonance. To understand this behavior, one by the action of the tunneling paths that govern adia- can investigate the predictions for quantum stochastic batic and nonadiabatic tunneling—see Sec. IX in resonance within a linear-response approach. In this Ha¨nggi, Talkner, and Borkovec (1990). This non- Arrhenius action term possesses a rather weak tempera- case, only the harmonics 0,Ϯ1ofPas(t) in Eq. (6.7) are ture dependence as compared to the Arrhenius depen- different from zero. In particular, P0 becomes the ther- dence. Hence the crucial factor in quantum stochastic mal equilibrium value Peq of the operator ␴z in the ab- resonance is not this exponential action part governing sence of driving, and M ϭប⑀ˆ ␹(Ϯ⍀) is related by Ku- Ϯ1 the quantum rate behavior but rather the Arrhenius-like bo’s formula to the linear susceptibility detailed balance factor relating the forward rate to the ˜ 2 ␹ xx(⍀)ϭa ␹(⍀) for the particle position xϭ(a/2)␴z , backward rate [see below Eq. (6.12)]. This exponential where detailed balance factor contains the energy scale i ϩϱ (ប⑀0/kBT); thus it is this exponential dependence that ˜␹ ͑⍀͒ϭ d␶ exp͑Ϫi⍀␶͒H͑␶͒ crucially competes with the algebraic factor (k Ϫ1 xx ប ͵ BT) Ϫϱ that enters the linear susceptibility. Whenever Ӷ ϫ͓͗x͑␶͒,x͑0͔͒͘␤ . (6.11) ប⑀0 kBT, the energy levels are essentially equally oc- cupied; hence with ⑀0ϭ0 no quantum stochastic reso- Here, H(␶) is the Heaviside function, [...,...] de- nance peaks occur! notes the commutator, and ͗ ...͘␤ the thermal statistical The second consequence is that over a wide range of average of the full system in the absence of the external driving frequencies the stochastic resonance maximum ˆ ϭ periodic force (⑀ 0). In the regime where incoherent arises at a temperature obeying kBTӍប⑀0 . Similar transitions dominate, the dynamical susceptibility is ex- qualitative results, together with the occurrence of plicitly obtained in the form noise-induced suppression are obtained also in the pa- 1 1 ␭ rameter region of low temperatures kTрប⌬ and weak ␹͑⍀͒ϭ . (6.12) coupling ␣Ӷ1, where overdamped quantum coherence 4k T cosh2 ⑀ /2k T ␭Ϫi⍀ B ͑ 0 B ͒ occurs. In this regime, the noninteracting blip approxi- The quantity ␭ϭRe⌺(⑀0), ␣ Ͻ 1 [for ␣>1, see Ha¨nggi, mation fails to predict the correct long-time behavior. Talkner, and Borkovec (1990), or Weiss (1993)] is the This is so because the neglected bath-induced correla- sum of the forward and backward (static) quantum rates tions contribute to the dissipative effects to first order in out of the metastable states, rϩ and rϪ , respectively. the coupling strength. Nevertheless, a perturbative treat- 2 The factor 1/cosh (ប⑀0 /2kBT) expresses the fact that the ment allows an investigation of quantum stochastic reso- two rates are related by the detailed balance condition nance even in this regime.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 267

In conclusion, quantum noise in the presence of peri- The stationary solutions ⌽(x) in the absence of the odic driving can substantially enhance or suppress quan- noise, obeying tum stochastic resonance. Note also that all of this dis- d2⌽ cussion of quantum stochastic resonance constitutes a 3 m⌽Ϫ⌽ ϩ␬ 2 ϭ0, (6.17) situation where inertial effects in stochastic resonance dx (i.e., finite ohmic friction strengths ␣) are always implic- with von Neumann boundary conditions itly accounted for. Of particular relevance is the occurrence of noise-induced suppression within nonlinear d⌽͑0͒ d⌽͑L͒ ϭ ϭ0, (6.18) quantum stochastic resonance. This phenomenon of sig- dx dx nal suppression at higher harmonics can be used for a distortion-free spectral amplification of information in extremalize the functional V͓⌽͔. Equation (6.17) can be quantum systems. Experimental candidates to observe interpreted as a Newtonian equation of motion in the 1 4 1 2 these novel quantum stochastic resonance effects are the inverse double-well potential UN(⌽)ϭϪ 4 ⌽ ϩ 2 m⌽ above-mentioned mesoscopic metals (Golding et al., with x as the time variable. The relevant homogeneous 1992; Chun and Birge, 1993; Lo¨ fstedt and Coppersmith, stationary solutions (stationary solution in the Newton- 1994a, 1994b; Coppinger et al., 1995), ac-driven SQUID ian picture) with boundary conditions (6.18) are given systems (Hibbs et al., 1995; Rouse et al., 1995), as well as by ac-driven atomic force microscopy (Eigler and Schweitzer, 1990; Louis and Sethna, 1995), or ac- ⌽ϮϭϮͱm, modulated proton tunneling (Grabert and Wipf, 1990; ⌽ ϭ0, (6.19) Benderskii et al., 1994). 0 with 1 B. Stochastic resonance in spatially extended systems V͓⌽ ͔ϭV͓⌽ ͔ϭϪ m2L, ϩ Ϫ 4 So far we mainly investigated stochastic resonance in V͓⌽ ͔ϭ0. (6.20) systems with only one degree of freedom, such as a par- 0 ticle moving in a potential under the influence of an The first two solutions (the whole string is sitting in external driving force and noise. In this section we de- one of the potential minima) are stable and the third scribe how stochastic resonance manifests itself in spa- one (the whole string is sitting on the barrier top) is tially extended systems such as a string moving in a unstable. There is also a class of stable inhomogeneous bistable potential under the influence of noise and exter- solutions, the multi-instanton solutions ⌽(k), that obey nal forcing, or in a two-dimensional medium forming the boundary conditions (6.18) and have k zeros in the spatiotemporal patterns in the presence of noise. interval ͓0,L͔. For the potential energies V͓⌽(k)͔ one finds the inequalities 1. Global synchronization of a bistable string V͓⌽͑0͔͒ϽV͓⌽͑1͔͒ϽV͓⌽͑2͔͒ ..., (6.21) In this section, we consider a one-dimensional bistable (0) with ⌽ ϭ⌽Ϯ . In the presence of noise, the string can medium in the presence of noise and isotropic periodic escape out of the stable homogeneous states ⌽Ϯ .A forcing. The model is described by the one-dimensional generalization of Ventsel and Freidlin’s theory (Benzi Ginzburg-Landau equation (Benzi et al., 1985): et al., 1985) yields for the mean exit times in the weak- 2⌽͑x,t͒ noise limit DӶ⌬Vץ ⌽͑x,t͒ץ ϭm⌽͑x,t͒Ϫ⌽3͑x,t͒ϩ␬ x2 2⌬Vץ tץ T ϭC exp , (6.22) Ϯ ͩ D ͪ ϩA0 cos͑⍀t͒ϩ␰͑x,t͒, (6.13) (2) where ␰(x,t) is white Gaussian noise in both time and with C independent of D. Here, ⌬VϭV͓⌽ ͔ϪV͓⌽Ϯ͔ space, i.e., denotes the smallest V͓⌽͔ barrier that separates the two stable states ⌽Ϯ . This conclusion was verified numeri- ͗␰͑x,t͒␰͑xЈ,tЈ͒͘ϭ2D␦͑tϪtЈ͒␦͑xϪxЈ͒ cally by solving the discretized version of Eq. (6.13) (see below). In the presence of a weak and slow homoge- ͗␰͑x,t͒͘ϭ0. (6.14) 3/2 neous external forcing (AϭA0Ӷm ), Benzi et al. In the absence of driving, Eq. (6.13) can be cast in the (1985) derived analytical expressions for the mean exit form times Ta and Tb , relevant to the transition ⌽ ⌽ in ϩ ϩ ϩ→ Ϫ ,.⌽͑x,t͒ ␦V͓⌽͔ the potential configurations with cos(⍀t)ϭϮ1, i.eץ ϭϪ ϩ␰͑x,t͒, (6.15) a ,t ␦⌽ TϩϭC exp͓2͑⌬VϪA0Lͱm͒/D͔ץ with the functional V͓⌽͔ (Rajaraman, 1982) b TϩϭC exp͓2͑⌬VϩA0Lͱm͒/D͔. (6.23) ⌽ 2ץ L 1 1 ␬ V͓⌽͔ϭ ⌽4Ϫ m⌽2ϩ dx. (6.16) The system shows stochastic resonance if one of the x ͪ ͬ exit times of Eq. (6.23) is shorter than the half drivingץ ͩ 2 2 4ͫ 0͵

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 268 Gammaitoni et al.: Stochastic resonance

(second plot), the collective coordinate switches almost periodically between ͱm and Ϫͱm, i.e., between states where the entire string is either in the right or in the left potential well—the noise has globally synchronized the hopping along the bistable string. A similar conclusion has been reached recently by Lindner et al. (1995), who simulated numerically the same discretized Ginzburg- Landau equation [Eq. (6.13)]. Here, the different notation ␬/(⌬x)2 g and D/⌬x ⑀ hides the underlying Ginzburg-Landau→ equation. Using→ LϭN⌬x, the scaling relations gϰN2 and ⑀ϰN, which were derived in Marchesoni, Gammaitoni, and Bulsara (1996), follow immediately. These authors also noticed that, while jumping back and forth between the stable configurations ⌽Ϯ, a long string develops a remarkable spatial periodicity, which attains its maximum at resonance. FIG. 43. The collective coordinate u ϭ ͐␾(x,t)dx/L plotted The relevant spatial correlation length can be easily es- against time for D/⌬xϭ0.002 (upper plot), D/⌬xϭ0.07 timated within the thermal nucleation theory. Related to (middle plot), and D/⌬xϭ0.11 (lower plot). The parameters these studies of stochastic resonance in extended sys- are Lϭ1, Nϭ20, mϭ0.25, Aϭ0.0125, ␻ϭ0.01␲/3, and tems is the recent study by Wio (1996) on stochastic Bϭ1/64. These parameters imply a barrier height of 1/64 and resonance in a bistable reaction-diffusion system, or the an Arrhenius factor of 4.457. investigation of stochastic resonance in weakly perturbed Ising models (Ne´da, 1995a, 1995b; Brey and Pra- period and the other is longer, so that on average the dos, 1996; Schimansky-Geier and Siewert, 1997). exit times are of the order the half driving period (the factor of 2 stems from the fact that the string has to 2. Spatiotemporal stochastic resonance in excitable media escape twice within one period of the driving). This yields an upper and a lower bound for the optimal noise Pattern formation in excitable media is an important strength with arithmetic mean paradigm with many applications in biology and medicine such as contraction waves in cardiac muscle, slime 2⌬V mold aggregation patterns, and cortical depression DSRϭ . (6.24) ln͑␲/C⍀͒ waves, to name only a few (for an overview, see Murray, A more refined analysis of stochastic resonance in a 1989). While most theoretical and experimental work on modulated string has been derived recently by excitable media focuses on the propagation of spiral Marchesoni, Gammaitoni, and Bulsara (1996) within the waves, the role of fluctuations for pattern selection and framework of the theory of thermal nucleation in one- propagation has been studied only recently (Jung and dimensional chains (Ha¨nggi, Marchesoni, and Sodano, Mayer-Kress, 1995) by using a stochastic cellular model. 1988). The ensuing theoretical predictions have been One of the many interesting features of noisy media is verified experimentally by Lo¨ cher, Johnson, and Hunt that spatiotemporal structures and coherence can (1996). strongly vary with the noise level. Spatiotemporal sto- A full numerical investigation of the discretized chastic resonance describes the enhancement of a spa- Ginzburg-Landau equation has been carried out by tiotemporal pattern (externally applied or intrinsic) by Benzi et al. (1985). The discretized Ginzburg-Landau an optimal dose of noise. equation [Eq. (6.13)] reads The model of Jung and Mayer-Kress consists of a square array of excitable threshold elements with lattice ␬ constant a. Each element e can assume three states: the ␺˙ ͑t͒ϭm␺ Ϫ␺3 ϩ ͓␺ ͑t͒ϩ␺ ͑t͒ ij n n n ͑⌬x͒2 nϩ1 nϪ1 quiescent state, the excited state, and a subsequent refractory state. The state of each element eij is controlled Ϫ2␺n͑t͔͒ϩA0 cos͑⍀t͒ϩͱD/⌬x␰n͑t͒, by an input xij(t). If the input xij(t) is below a threshold (6.25) b, the element is quiescent. If xij(t) is crossing a threshold from below, the element switches into the excited with discretization step ⌬x, string sites ␺ (t)ϭ⌽(x ,t), n n state, i.e., it fires. The inputs x (t) are coupled to a ho- and ij mogeneous thermal environment, i.e., the time depen- ͗␰n͑t͒␰m͑tЈ͒͘ϭ2␦nm␦͑tϪtЈ͒, dence is described by the Langevin equation 2 ͗␰n͑t͒͘ϭ0. (6.26) x˙ ijϭϪ␥xijϩͱ␥␴ ␰ij͑t͒, (6.27) 2 2 In Fig. 43, the string collective coordinate with ͗xij͘ϭ␴ and zero-mean, uncorrelated noise in L uϭ(1/L)͐0 ⌽(x)dx is shown as a function of time at space and time ͗␰ij(t)␰kl(tЈ)͘ϭ2␦(ij),(kl)␦(tϪtЈ). The three different levels of the homogeneous noise inten- excitable elements communicate via pulse coupling. sity. For a properly chosen value of the noise level DSR When an element ekl fires, it emits a spike that is re-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 269

ceived by an element eij with an intensity depending on the distance r(ij),(kl) between ekl and eij . Element eij integrates the incoming spikes from all firing elements yielding after one time step ⌬t (the smallest time scale of our system) the additional input fij 2 r͑ij͒,͑kl͒ fijϭK͚ exp Ϫ␭ 2 , (6.28) kl ͩ a ͪ which is added to xij . The parameter ␭ describes the inverse range of the interaction and the parameter K the coupling strength. The medium is updated synchro- nously in time steps of the smallest time scale ⌬t, the time interval of firing. All other time scales are measured in units of ⌬t. The proper normalization of this model is given by ¯␴ 2ϭ␴2/b2, ¯␥ ϭ␥⌬t, K¯ ϭK/b. The time step and the threshold are therefore normalized to unity. The dissipation constant ¯␥ defines the typical time scale of the temporal evolution of a single element. For large dissipation (¯␥ Ͼ1), the element forgets its prehistory within one time step of temporal evolution, while for small dissipation ¯␥ Ӷ1, the system—as a whole—can build up a long memory. It has been demonstrated that this model shows for large coupling K¯ (in the absence of noise) the typical excitation patterns of excitable media, i.e., rotating spiral waves or target waves, usually described in terms of reaction diffusion equations with two species (Murray, 1989). In the presence of noise, the typical excitation patterns can still be observed, but they exhibit rough wave fronts and—depending on the noise level—more serious imperfections such as breakup of wave fronts and collisions with noise-nucleated waves. The overall picture in the strong-coupling regime is the coexistence of multiple finite-sized cells with coherent patterns. For weak coupling K¯ , however, the discrete nature of the model becomes important and different phenomena can be observed. To maintain a firing pattern, the cou- ¯ ¯ pling K has to exceed a critical value K0 , which is estimated for small ␭ and negligible curvature effects as follows: an infinite front of firing elements reduces the firing threshold of an element next to the front by an amount S0 which is the sum of the contributions from all firing elements along the front. The element, however, is ¯ precharged by the sum of the contributions S pre of firing elements of the front at earlier times (and larger spatial distances). At the critical coupling, the sum of the pre- ¯ ¯ charge S pre and S 0 of an element right before the front is unity (the normalized threshold), i.e., FIG. 44. The excitable medium (200ϫ200 elements) driven by ␭ 1 ¯ a single wave front of slightly increased excitability, A0 K0͑␥͒Ϸͱ ϱ 2 . ␲ exp͑Ϫ␭͒ϩexp͑␥͚͒nϭ2 exp͑Ϫ␭n Ϫn␥͒ ϭ 0.3. The wave front is a single horizontal line in our array (6.29) moving from bottom to top. The position of the wave front at the instant of time we took a snapshot is marked by a pointer Spatiotemporal stochastic resonance can be observed on the right margin. The diamonds denote firing elements. We for coupling strengths below the critical coupling, which show three snapshots at three different noise levels; (a) we define as the subthreshold regime. The excitable me- ¯␴ 2ϭ0.1, (b) ¯␴ 2ϭ0.16, and (c) ¯␴ 2ϭ0.2. The other parameters dium (in the subthreshold regime) is driven by a single ¯ ϭ ϭ ¯ϭ wave front (a line in the array) from bottom to top, are K 0.121, ␭ 0.1, ␥ 0.5.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 270 Gammaitoni et al.: Stochastic resonance

FIG. 45. The mean excess firing rate shown a function of the ¯2 ¯ variance of the noise ␴ for ␭ϭ0.1, ␥ ϭ0.5, and A0ϭ0.3 at three different values of the coupling K¯ . along which the excitability is slightly increased; i.e., the threshold b is reduced: b b Ϫ A0. In Fig. 45, the response of the medium, characterized→ by the correlation between the driving pattern and the firing pattern (see Fig. 44), is shown. The correlation is defined here as the mean excess firing rate along the front of the driving wave, i.e., the number of excess firing events along the front in comparison to the average number of firing FIG. 46. The probability density p(x)of106iterates x shown events along all other lines (a small layer around the n in the absence of modulation (A0ϭ0) (a) below the crisis at front had been excluded). For vanishing coupling K¯ ϭ0, aϭ3.57, and (b) above the crisis at aϭ3.62. one finds the maximum correlation according to the re- ¯2 sults for single thresholds at ␴ ϭ1/2. For increasing cou- The effect of improving an image by using stochastic pling, the effective threshold that has to be overcome resonance has been demonstrated very nicely in a recent with the help of noise is reduced. Therefore, the maxi- work by Simonotto, Riani, Seife, Roberts, Twitty, and mum correlation is shifted to smaller values of the noise. Moss (1997) for an array of uncoupled threshold detec- The peak also becomes more pronounced since the fir- tors (in the model above, this corresponds to the case ing activity is synchronized in an area determined by the K¯ ϭ0). interaction range 1/␭. A rough estimate of the optimal value of the variance for the enhancement of spatiotemporal patterns has C. Stochastic resonance, chaos, and crisis been given in Jung and Mayer-Kress (1995) in terms of a mean-field type approximation. It is well known that deterministic chaos resembles The firing elements along the front generate a sto- the features of noise on a coarse-grained time scale. It is chastic field acting on the elements the front is ap- therefore a natural question to ask whether stochastic proaching. The main contributions to the field acting on resonance can be observed in dynamical systems in the the element eij stem from firing elements along the front absence of noise. Two different approaches to this prob- close to eij. Assuming that all of these elements are ac- lem have been put forward in the recent literature. Car- tually firing, we approximate the sum of these contribu- roll and Pecora (1993a, 1993b) substitute the stochastic tions (for small ␭) by a Gaussian integral and obtain for noise by a chaotic source. The chaotic source is applied the mean field to a periodically driven Duffing oscillator in a regime ¯ ¯ where it produces a period-doubled periodic response. f ϭKͱ␲/␭exp͑Ϫ␭͒. (6.30) The chaotic source yields switching between the attrac- The firing threshold is reduced by the mean field, i.e., tors corresponding to the two phase-shifted responses of ¯ ¯ the Duffing oscillator, separated by an unstable period-1 b effϭ1Ϫ f , leading to a renormalized condition for spatiotemporal stochastic resonance orbit. The switching happens at some preferred locations along the orbits, which are being visited periodi- 1 1 cally. It is therefore synchronized with the orbit. This ¯␴ 2 ϭ ¯b2 ϭ ͓1ϪK¯ͱ␲/␭ exp͑Ϫ␭͔͒2. (6.31) opt 2 eff 2 situation resembles the conventional setup for stochastic

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 271

FIG. 47. The distribution of residence times t shown in the FIG. 48. The mean residence time shown in the absence of the absence of forcing (A ϭ0) at aϭ3.6 and aϭ3.62, both above 0 driving as a function of the control parameter aϪa , where a the crisis a ϭ3.59. 0 0 0 is the value of the control parameter at which the crisis occurs. The triangles represent results from a numerical calculation. resonance and yields stochastic resonance as expected. Apart from the resonances (the dips) the mean residence time ␥ A similar study has been published by Kapitaniak (1994) can be fitted very well by a power law Teϰ(aϪa0) , with and by Yang, Ding, and Hu (1995). For the Lorentz sys- ␥ϭ0.576 (solid line). tem with a time-periodic variation of the control parameter operating near the threshold to chaos, stochastic ϱ resonance has been observed by Crisanti, Falcioni, Pala- Teϭ tpe͑t͒dt. (6.34) din, and Vulpiani (1994). ͵0 A conceptually different approach has been put for- The power-law scaling of the mean residence time T ward by Anishchenko, Neiman, and Safanova (1993). e with the distance to the crisis aϪa , i.e., They use the intrinsic chaotic dynamics of a nonlinear 0 Ϫ␥ map in the vicinity of a band-merging crisis to generate a Te͑a͒ϰ͑aϪa0͒ , (6.35) sort of activated hopping process which is then synchro- (see Fig. 48), is—according to Grebogi, Ott, and Yorke nized by a small periodic signal. No external source is (1987)—characteristic of a crisis. There are some dips in necessary to provide the randomness. They use the non- Te(a), e.g., at aϭ0.36405, that correspond to periodic linear periodically driven map windows in the map. 3 xnϩ1ϭ͑aϪ1͒xnϪaxnϩA0 sin͑2␲f0n͒. (6.32) The decrease (at least in the average) of the mean residence time for increasing aϪa0 implies an increasing The complete description of the period-doubling sce- level of stochasticity, i.e., the level of stochasticity can be nario towards chaos is described in Anishchenko, controlled by varying aϪa . Neimann, and Safanova (1993). Most important for the 0 following discussion is a crisis due to the merging of two chaotic bands (xϾ0 and xϽ0) at aϷ3.598ϵa0 . This is demonstrated in Fig. 46 by the invariant measures of the undriven map A0ϭ0ataϭ3.57 (a) and aϭ3.62 (b). The fixed point x1ϭ0 is stable for 0ϽaϽ2 and unstable for aϾ2. Two chaotic bands emerge out of two disjoint Feigenbaum-type period-doubling scenarios at aу3.3. For aϽa0 , these bands are separated by the unstable fixed point x1ϭ0. At aϭa0 the bands merge. The unstable fixed point x1ϭ0 acts in the vicinity of the band- merging point a0 as a repellor allowing the trajectory to traverse between the formerly separated chaotic bands only very rarely, yielding activation-type behavior of the trajectory. The statistical distribution pe(t) of times between two exits, i.e., the residence-time distribution, is shown in Fig. 47. It shows for not too small times the FIG. 49. The distribution of residence times t shown in the typical exponential decay presence of the driving at aϭ3.60, A0ϭ0.01, and f0ϭ0.1. The 1 carets show actual data points, while the solid line has been p ͑t͒ϭ exp͑Ϫt/T ͒, (6.33) added to guide the eye. The locations of the sequence of e T e e exponentially decaying peaks are given by Ϫ1 Ϫ1 Ϫ1 with the mean residence time tnϭ(1/2)f0 ,(3/2)f0 ,(5/2)f0 ,... .

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 272 Gammaitoni et al.: Stochastic resonance

in Sec. IV for the driven bistable dynamics, it often is appropriate to model the noise source by a (white) random force ␰(t). In the physical world, however, such an idealization is never exactly realized. In order to investigate the importance of corrections to white noise, approximate techniques were introduced to compute the effects of small to moderate to arbitrarily large noise correlation times ␶c (Ha¨nggi et al., 1984, 1989; Ha¨nggi and Jung, 1995). Strong color (i.e., a large ␶c value) is not unrealistic for many physical applications. Usually, a strongly correlated noise emerges as the result of coarse graining over a hidden set of slowly varying variables (Kubo et al., 1985), or colored noise is simply applied and monitored externally by the experimenter. FIG. 50. The response to the periodic forcing (more precisely, The effect of color on stochastic resonance may be the intensity of the line in the power spectrum at the driving nontrivial, as suggested by the very characterization of frequency) shown as a function of the control parameter aϪa0 stochastic resonance as a synchronization mechanism. for A0ϭ0.01 and f0ϭ0.01. The noise correlation time ␶c may compete with T⍀ and TK to determine the realization and the magnitude of the resonance phenomenon. We anticipate that stochas- In the presence of a periodic forcing, i.e., for A0Þ0, the residence-time distribution exhibits a series of expo- tic resonance in overdamped systems driven by an addi- nentially decaying peaks, located at odd multiples of the tive exponentially correlated Gaussian noise ␰(t) is gen- half period of the driving (see Fig. 49). In order to iden- erally reduced compared to the case of white noise ␶cϭ0 tify stochastic resonance, one has to vary the stochastic- of equal strength D. The stochastic resonance peak is ity in the presence of the periodic driving and compute shifted to larger noise intensities due to the fact that the intensity of the spike (the signal strength) in the colored noise exponentially suppresses the switching power spectrum of x(t). To single out contributions to rate with increasing ␶c (Gammaitoni, Menichella-Saetta, the response from forced periodic motion or resonances Santucci, Marchesoni, and Presilla, 1989; Ha¨nggi et al., within the chaotic bands, we apply a binary filter process 1993). to monitor whether the system is in the right band (ϩ1) Following the approach developed by Ha¨nggi et al. or left band (Ϫ1). The response of the binary variable (1993), we treat here the archetypal case of a periodi- yϵx/͉x͉ measured in terms of the intensity of the line in cally perturbed double well in the presence of exponen- its power spectrum at the driving frequency is shown in tially colored Gaussian noise (Ornstein-Uhlenbeck noise). In scaled, dimensionless variables, the dynamics Fig. 50. Starting at the crisis, i.e., at aϭa0 , one observes that the signal strength increases until it reaches a maxi- reads explicitly mum, and then decreases again. x˙ ϭϪVЈ͑x͒ϩA0 cos͑⍀t͒ϩ␰͑t͒, (6.36a) The differences to stochastic resonance in a noisy bistable system follow: 1 1 ␰˙ ϭϪ ␰ϩ ⑀͑t͒, (6.36b) (i) There are several peaks plus additional resonances ␶c ␶c where the driving frequency f and the switching fre- 0 where V(x) is the standard quartic double-well poten- quency f ϭ1/T are commensurable, i.e., f /f ϭm/n, s e 0 s tial of Sec. IV.A, i.e., V(x)ϭϪx2/2ϩx4/4, and ␰(t)isan with m,nϭ1,2,3, . . . . Ornstein-Uhlenbeck stochastic process driven by the (ii) Changing aϪa does not change the stochasticity 0 Gaussian white noise ⑀(t) with ͗⑀(t)͘ϭ0 and (noise strength) in a systematic way. To systematically ͗⑀(t)⑀(0)͘ϭ2D␦(t). The stationary autocorrelation compare with stochastic resonance in a noisy bistable function of ⑀(t) is an exponential function with time system, one should first find a mapping between the constant ␶c , noise strength and aϪa0 . (iii) The periodic windows of period M of the unper- ͗␰͑t͒␰͑0͒͘ϭ͑D/␶c͒exp͑Ϫ͉t͉/␶c͒. (6.37) turbed map yield resonances with the external driving In the limit of zero correlation time ␶ 0, Eq. (6.37) ϭ c whenever their periods agree, i.e., f0 1/M. reproduces the white-noise source of Secs.→ II and IV. Similar results have been obtained by Nicolis et al. Within the framework of the linear-response theory of (1993) by studying a one-dimensional intermittent map. Sec. IV.B for small forcing amplitudes, the relevant response function [Eq. (4.30)] assumes the form of a fluc- D. Effects of noise color tuation theorem; i.e., it is given in terms of a stationary correlation of two fluctuations of the unperturbed pro- In many practical situations the finite time scale ␶c cess characterizing the relaxation of the autocorrelation of the noise (i.e., colored noise) is much shorter than the d ␹͑t͒ϭϪH͑t͒ ͗x͑t͒␨(x͑0͒)͘ , (6.38) characteristic time scale of the system. Hence as we did dt 0

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 273

FIG. 52. Signal-to-noise ratio (SNR) vs D for different values of ␶c : ␶cϭ30 ␮s (crosses); ␶cϭ50 ␮s (diamonds); ␶cϭ100 ␮s (squares); ␶cϭ200 ␮s (pluses). Other simulation parameters are ␯⍀ϭ30 Hz, Axmϭ0.5⌬V, xmϭ7.3 V, and aϭ6850 Hz.

FIG. 51. The spectral amplification ␩ (in linear response), as 2 2 ͗x ͘0ϩ␶c͗xVЈ͑x͒͘0 4rK͑␶c͒ predicted by Eqs. (6.43) and (6.44) for the quartic double-well ␩ϭ 2 2 2 , (6.43) potential V(x)ϭϪx2/2ϩx4/4, shown for increasing values of D 4rK͑␶c͒ϩ⍀ the dimensionless noise correlation time ␶ at ⍀ϭ0.1. c with rK(␶c) and ͗•••͘0 computed in the appropriate limits of Eqs. (6.42) and (6.40), respectively. where H(t) denotes a Heaviside step function. In the Prediction (6.43) for both color regimes suggests that opposite limits ␶cӶ1 (i.e., weak color) and ␶cӷ1 (i.e., noise color degrades the observability of stochastic reso- strong color), ␨(x(t)) may be approximated to read nance. Indeed, upon increasing ␶c , the relaxation rate r (␶ ) gets exponentially depressed with respect to 1 K c rK(␶cϭ0). For ⍀ fixed, we therefore must increase D to ␨(x͑t͒)ϭ ͓xϩ␶cVЈ͑x͔͒, (6.39) D match the stochastic resonance condition, which consists of maximizing ␩ (see Sec. IV.B). This results in a shift of whereas the unperturbed averages ͗•••͘0 in Eq. (6.38) must be taken over the relevant (approximate) probabil- the stochastic resonance peak towards higher D values ity density and a corresponding reduction of the peak height. We note that the limiting expressions (6.42) for rK(␶c) 1 V͑x͒ ␶c stem from one unified approximation scheme (Ha¨nggi p ͑x,␶ ͒ϭ N exp Ϫ Ϫ VЈ2͑x͒ st c ͉1Ϫ␶ VЉ͑x͉͒ ͫ D 2D ͬ and Jung, 1995), that is c (6.40a) 1 Ϫ1/2 for ␶cӶ1, and rK͑␶c͒ϭ ͑1ϩ3␶c͒ &␲ pst͑x,␶c͒ϭ 2͉1ϩ␶cVЉ͑x͉͒ N 27 1 ⌬V 1ϩ ␶ ϩ ͑␶ ͒2 V x ␶ 16 c 2 c ͑ ͒ c 2 ϫexp Ϫ . (6.44) ϫexp Ϫ Ϫ VЈ ͑x͒ (6.40b) D 27 ͫ D 2D ͬ ͫ ͩ 1ϩ 16 ␶c ͪͬ for ␶ ӷ1. Here, and denote the appropriate nor- Correspondingly, the analytical expression obtained by c N1 N2 malization constants. replacing rK(␶c) of Eq. (6.43) with Eq. (6.44) bridges the Within the long-time approximation, the correlation two limiting expressions of ␩ for ␶cӶ1 and ␶cӷ1. In Fig. function ͗x(t)␨(x(0))͘ (see Sec. IV.B) can in leading 51 we display four such curves for ␩ versus D, for in- order be estimated as creasing values of noise color ␶. As expected, noise color suppresses stochastic resonance monotonically with ␶c . ͗x͑t͒␨(x͑0͒)͘0ϳ͗x␨͘0 exp͓Ϫ2rK͑␶c͒t͔ (6.41) This feature is in accordance with the early analog simu- with the colored noise-driven escape rate given as lations by Gammaitoni, Menichella-Saetta, Santucci, Marchesoni, and Presilla (1989), as depicted in Fig. 52. rK͑␶c͒ϭrK͑1Ϫ3␶c/2͒ (6.42a) The suppression of stochastic resonance with increasing for weakly colored noise ␶cӶ1, and noise color has recently been demonstrated experimentally in a tunnel diode (Mantegna and Spagnolo, 1995), r ␶ ͒ϭr exp Ϫ 8/27͒␶ ⌬V/D͒ (6.42b) K͑ c K ͓ ͑ c͑ ͔ and by use of Monte Carlo simulations by Berghaus for strongly colored noise, i.e., ␶cӷ1. Upon inserting et al. (1996). Eqs. (6.41) and (6.39) into the expression for the suscep- Inertial effects, which result in (non-Markovian) tibility in Eq. (6.38) we finally obtain for the spectral memory effects for the spatial coordinate x(t) have amplification ␩ been addressed theoretically for the spectral amplifica-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 274 Gammaitoni et al.: Stochastic resonance tion ␩ by Ha¨nggi et al. (1993), and experimentally for the amplitude quantization can be quantified by intro- the signal-to-noise ratio by Gammaitoni, Menichella- ducing a proper quantization error, ␨ϭyϪx, where x is Saetta, Santucci, Marchesoni, and Presilla (1989). Like- the analog signal before quantization and y is the quan- wise, the role of memory friction, which relates—via the tized signal. It is clear from this definition that if we had fluctuation-dissipation theorem—to internal colored a linear-response characteristic (apart from amplifica- noise, has been studied for overdamped dynamics by tion factors) ␨ would be zero and there would be no Neiman and Sung (1996). The result that inertial effects, distortion at all. A number of studies were performed or equivalently, decreasing finite friction strengths tend over the last thirty years in order to find a way of reduc- to enhance stochastic resonance can similarly be induced ing ␨. The main conclusions follow: with strong color for the memory friction (Neiman and (1) The addition of a proper external signal (called Sung, 1996). dither) to the input x can statistically reduce ␨. (2) The best choice for the dither signal is a random VII. SUNDRY TOPICS dither uniformly distributed. (3) There exists an optimal value of the random dither In this section we present several topics relating to the amplitude, which coincides with the amplitude of the physics of stochastic resonance that have not been fea- quantization step. tured in detail in the previous sections. In doing so, we Hence the quantization error ␨ is minimized and, cor- have confined ourselves to particular examples deter- respondingly, the ADC performances maximized, when mined only by our knowledge and personal taste; thus a noise of a proper intensity is added to the input signal. this selection is necessarily incomplete. The similarity with stochastic resonance, where an optimal strength of the added noise maximizes the output A. Devices signal-to-noise ratio, is apparent. As a matter of fact, stochastic resonance in this class of threshold systems is 1. Stochastic resonance and the dithering effect equivalent to the dithering effect, as demonstrated by A Schmitt trigger, see Sec. V.B.1, operating in the lim- Gammaitoni (1995a, 1995b). iting case when its two threshold voltages coincide, provides an example of a two-state system, namely a thresh- B. Stochastic resonance in coupled systems old device. There are many examples of this kind of electronic device, the more common class being repre- In this section we discuss the impact of noise and pe- sented by the analog-to-digital converters (ADC). The riodic forcing on an ensemble of coupled bistable sys- basic (1-bit) ADC device consists of a signal comparator tems. In view of a possible collective response of the (an operational amplifier followed by a resistor and two system (especially close to a phase transition), one can back-to-back Zener diodes), the output voltage of which expect that the stochastic resonance effect will be even switches between V and Ϫ V when the input v crosses a i more pronounced than in a single system (Jung et al., reference voltage. Multibit ADCs, realized by a proper 1992). combination of comparators (see, e.g., Millman, 1983), are of common use in digital signal processing (Oppen- heim and Schaffer, 1975), where analog signals are 1. Two coupled bistable systems sampled at discrete times and converted into a sequence The simplest way to study stochastic resonance in of numbers. Since the register length is finite, the con- coupled systems is to consider two coupled overdamped version procedure, termed signal quantization, results in bistable elements in the presence of noise and periodic distortion and loss of signal detail. In order to avoid forcing (Neiman and Schimansky-Geier, 1995): distortion and recover the signal detail, it has become a 3 common practice, since the 1960s, to add a small amount x˙ ϭ␣xϪx ϩ␥͑yϪx͒ϩ␰x͑t͒ϩA0 cos͑⍀t͒, of noise to the analog signal prior to quantization—a y˙ ϭ yϪy3ϩ xϪy ϩ t ϩA cos t , (7.1) technique termed dithering (Bennet, 1948). To under- ␤ ␥͑ ͒ ␰y͑ ͒ 0 ͑⍀ ͒ stand how the addition of a proper quantity of noise can with independent Gaussian white noise terms, but iden- improve the performances of an ADC, we note that the tical periodic forcing, i.e., conversion from a continuous (analog) to a digital signal ␰ ͑t͒␰ ͑tЈ͒ ϭ2D␦ ␦͑tϪtЈ͒. (7.2) consists of two different operations: time discretization ͗ x y ͘ xy and amplitude quantization. Time discretization, if prop- As in the bistable string (see Sec. VI.B.1), stochastic erly applied, can be shown to be error free. The effects resonance in the coupled system has been quantified by of amplitude quantization (finite word length) are in- the linear response for the sum s(t) of the two degrees stead always present and manifest themselves in a num- of freedom s(t)ϭx(t)ϩy(t) due to small periodic ber of different ways. First, due to the presence of a modulations. With the help of digital simulations and nonlinear-response characteristic, signal quantization approximation theory, the following results have been leads to an unavoidable distortion, i.e., the presence of obtained: spurious signals in a frequency band other than the (1) At a given coupling constant, the signal-to-noise original one. There is also a loss of signal detail that is ratio goes through a maximum as a function of the noise small compared to the quantization step. The effects of strength.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 275

(2) Starting from zero coupling (which corresponds to served for two coupled bistable systems (cf. Sec. two independent systems), the signal-to-noise ratio vs VII.B.1) and for coupled-neuron models in Sec. VII.B.3. coupling first increases (i.e., the collective response is These results have also been confirmed in later studies indeed higher than that of two uncoupled systems), runs by Morillo et al. (1995), and Hu, Haken, and Xie (1996). through a maximum, and decreases again for large coupling towards an asymptotic (finite) value. 3. Globally coupled neuron models Another approach to describe the response of globally 2. Collective response in globally coupled bistable systems coupled bistable systems to periodic forcing is the appli- An early study dealing with stochastic resonance for cation of adiabatic elimination of all but one degree of systems with many degrees of freedom (Jung et al., freedom (Bulsara and Schmera, 1993; Inchiosa and Bul- 1992) focused on a large number N of identical, linearly sara, 1995a, 1995b,1995c; Inchiosa and Bulsara, 1996). and homogeneously coupled bistable systems in the The model used is motivated by the dynamics of artifi- presence of periodic forcing. The coupled equations of cial neural networks (Amit, 1989; Krogh and Palmer, motion are given by 1991), namely N N 1 ui 3 C u˙ ϭϪ ϩ J tanh͑u ͒ϩ␰͑t͒ϩA cos͑⍀t͒, x˙ n͑t͒ϭxnϪxnϩ g͑xmϪxn͒ϩ␰n͑t͒ i i ͚ ij j 0 N m͚ϭ1 Ri jϭ1 (7.7) ϩA0 cos͑⍀t͒, (7.3) with Ci and Ri denoting capacitances and resistances of with Gaussian, mutually independent and uncorrelated the membranes. The zero-mean Gaussian noise and the fluctuations periodic signal are assumed to be identical for all elements. The coupling constants J can be chosen arbi- ͗␰ ͑t͒␰ ͑tЈ͒͘ϭ2D␦ ␦͑tϪtЈ͒ ij n m nm trarily. The correlation function of the noise is given by ␰ ͑t͒ ϭ0. (7.4) ͗ n ͘ ͗␰͑t͒␰͑tЈ͒͘ϭ2D␦͑tϪtЈ͒. (7.8) The coupling constant is denoted by g. Systems such as The globally coupled system of Eq. (7.7) has been solved this exhibit spontaneous-ordering transitions (Bruce, numerically and analytically by assuming a separation of 1980; Amit, 1984; Dewel et al., 1985; Valls and Mazenko, time scales of one neuron vs the rest of the neurons 1986). Analytical studies of these phase transitions are acting as a linearized bath, thus allowing for adiabatic possible within a mean-field approximation (Mansour elimination of the bath neurons. The most important re- and Nicolis, 1975; Desai and Zwanzig, 1978; Bruce, 1980; sult of these studies is that, as above, the maximal signal- Shiino, 1987; Van den Broeck et al., 1994; Drozdov and to-noise ratio goes through a maximum as a function of Morillo, 1996; Hu, Haken, and Xie, 1996). The station- the coupling strength (the JЈs); moreover, the signal-to- ary solution of the Fokker-Planck equation in mean- noise ratio between the incoming periodic signal field approximation is not unique below a critical noise A cos(⍀t) and the noise strength D has been shown to strength. There are three solutions: two stable solutions 0 provide an upper bound to the signal-to-noise ratio of with spontaneous symmetry breaking, which represent the output u (t). ferromagnetic ordered states, and an unstable one with i zero magnetization m; here the order parameter m is given by the averaged population difference in the po- C. Miscellaneous topics on stochastic resonance tential wells. At the critical point DϭDc , the system undergoes a phase transition of second order. 1. Multiplicative stochastic resonance Within the mean-field approximation and a two-state description, the response of the order parameter ͗x͘ϵm There exist many cases of physical interest where the to the periodic forcing and thus the spectral amplifica- role of fluctuating control parameters is mimicked by tion ␩ of the order parameter has been obtained as multiplicative noise (Fox, 1978; Schenzle and Brand, 1979; Faetti et al., 1982; Graham and Schenzle, 1982). 2r 2 1Ϫm2 K Gammaitoni, Marchesoni, Menichella-Saetta, and San- ␩ϭ 2 2 , (7.5) ͩ D ͪ ⍀ ϩ⌳ tucci (1994) have analogously simulated the phenom- with the collective relaxation rate given by enon of stochastic resonance in the overdamped bistable system described by the stochastic differential equation 1 g ⌳ϭ2r ͱ1Ϫm2 Ϫ . (7.6) x˙ ϭϪVЈ͑x͒ϩx␰ ͑t͒ϩ␰ ͑t͒ϩA͑t͒, (7.9) K ͩ 1Ϫm2 Dͪ M A where V(x) is the standard quartic double-well poten- The mean value m determined by the transcendental tial and A(t)ϭA cos(⍀t), with A x Ӷ⌬V. The fluctu- equation mϭtanh͓(g/D)m͔. 0 0 m ating parameters ␰ (t), with iϭA,M, are stationary The spectral amplification strongly increases with the i zero-mean valued, Gaussian random processes with au- coupling to exhibit a peak at the critical point tocorrelation functions DϭDcϭg. The maximum spectral amplification attains a maximum at gϭ⌬V, a phenomenon that has been ob- ͗␰i͑t͒␰j͑0͒͘ϭ2Qi␦ij␦͑t͒. (7.10)

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 276 Gammaitoni et al.: Stochastic resonance

limit of low forcing frequency ⍀, the adiabatic approximation pas(x,t;A0)Ӎp0(x;A(t)) suffices to shed light on the nonstationary dynamics underlying the phenomenon of multiplicative stochastic resonance. In the purely multiplicative case QAϭ0 (Fig. 53), the forcing term alone is responsible for x(t) switching back and forth between the positive and the negative half axis. Should the adiabatic approximation hold true for any value of QM , the process x(t) would approach in- stantaneously its most probable value in the vicinity of the peak of p0(x;A(t)). Therefore, the amplitude ¯x of ͗x(t)͘as would be of the order of ¯x m , which is a monotonic decreasing function of QM . However, Fig. 53 shows a dramatic drop of ¯x as QM tends to zero. Such a deviation from the prediction of the adiabatic approximation is due to the fact that, with decreasing QM , the switch time of x(t) between positive and negative values, controlled by A(t) with periodically reversing sign, grows much larger than the forcing period T⍀ϭ2␲/⍀. For instance, assuming that at tϭ0 the variable x is confined to the unstable axis xϽx ӍA /a with A Ͼ0, the FIG. 53. The normalized response amplitude ¯x /xm depicted vs A 0 0 Ϫ1 mean-first-passage time T required by x to escape the dimensionless multiplicative noise strength k ϵ2QM /a A for A0ϭ0.1axm and different values of QA . The potential pa- through xA onto the stable half axis xϾxA diverges rameters are x ϭ2.2 V and aϭ104 s−1. After Gammaitoni, strongly for x /x 0. On increasing Q close to a, m A m→ M Marchesoni, Menichella-Saetta, and Santucci (1994). such a divergence is substantially weakened, so that the adiabatic approximation TAӶT⍀ applies. In this regime the relaxation process is controlled mainly by the modu- The main conclusion of their investigation is that the lated interwell dynamics of x(t) described by process x(t) develops a periodic component x(t) ac- ͗ ͘as p0(x;A(t)) and, as stated above, ¯x approaches ¯x m .In cording to the approximate law (2.6); the amplitude ¯x the opposite limit, TAӷT⍀ (i.e., QMӶa), the steady- ¯ and the phase ␾ of ͗x(t)͘as depend on A0 , QA , and state distribution of x(t) spreads over the entire x axis QM . Most notably, ¯x shows typical stochastic resonance with oscillating local maxima at ϮxmϩA(t)/2(aϪQM) behavior with increasing QM while keeping QA fixed (modulated intrawell dynamics). It follows immediately (multiplicative stochastic resonance). In Fig. 53, the de- that for QMϭ0ϩ the amplitude ¯x is of the order A0/2a, pendence of ¯x 1 on QM is plotted for the most remark- which is much smaller than the value of ¯x m at kϭ1, able case of a purely multiplicative bistable process whence the appearance of the stochastic resonance QAϭ0. peaks of Fig. 53 for ⍀TAϳ1. Accordingly, the stochastic To interpret the outcome of the analog simulation of resonance peaks shift to the left with increasing A0 .In Eq. (7.9), one should solve the relevant time-dependent conclusion, the crossover from intrawell to interwell Fokker-Planck equation modulated dynamics is the basic mechanism responsible .for multiplicative stochastic resonance ץ ץ pϭ VЈ͑x͒ϪQ xϪA cos ⍀t x ͫ M 0ץ tץ ץ ϩ ͑Q ϩQ x2͒ p (7.11) 2. Resonant crossing ͬ x A Mץ In this section, we report on an intricate colored-noise for the probability density pϭp(x,t;A0). In the pres- effect for the residence-time distributions, which takes ence of a static tilting, i.e., for A0Þ0 and ⍀ϭ0, the sta- place when the correlation time of the noise is large tionary solution of Eq. (7.11) reads (Gammaitoni, Marchesoni, et al., 1993). In such a situa-

Ϫ 1/2ϩk͓1ϩ ͑k/2͒͑QA /⌬V͔͒ tion, the system dynamics are characterized by four time QA p ͑x;A ͒ϭ ͑A ͒ x2ϩ scales: the local relaxation rate a within a potential well, 0 0 N0 0 ͩ Q ͪ M the correlation time of the noise ␶c , the forcing fre- 2 quency ⍀, and the transition rate r . The residence- x A0 K ϫexp Ϫk Ϫ , (7.12) time distribution (at small amplitudes of the driving A ) x2 Q ͉x͉ 0 ͩ m M ͪ consists—as shown repeatedly in ﬁgures throughout this where kϵa/2Q and (A ) is a suitable normalization review—of a series of peaks located at odd multiples of M N0 0 constant. In the presence of a periodic tilting; i.e., for the half period of the driving T⍀ϭ2␲/⍀, superimposed ⍀Ͼ0, the process x(t) is no longer stationary and a on an exponential backbone. The periodic part can be time-dependent probability density pas(x,t;A0) is re- extracted from the exponential backbone by a ﬁtting quired to describe its asymptotic state. However, in the procedure.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 277

It has been demonstrated that the amplitude of the This area has stimulated an interesting ongoing dis- periodic part evolves as a function of the driving fre- cussion: Do there exist suitable measures quantifying quency ⍀ through a maximum if the correlation time ␶c stochastic resonance—and what are they—that can be of the noise is large, i.e., a␶cӷ1. The location of the based solely on information theory considerations? A 2 promising approach has been put forward by Heneghan maximum has been estimated by ⍀maxϳa/␶c . In contrast to the time-scale matching condition for stochastic reso- et al. (1996) who consider the so-termed transinforma- tion that quantifies the rate of information transfer from nance, i.e., ⍀ϳ␲rK , this new condition describes a matching between the intrawell time scale and the driv- stimulus to response. They demonstrated that the presence of noise optimizes, via aperiodic stochastic reso- ing frequency ⍀. nance, the information-transfer rate. An attempt to characterize conventional stochastic resonance by means of information theory tools has been put forward by 3. Aperiodic stochastic resonance Schimansky-Geier and co-workers (Neiman et al., 1996; Since most of the studies on stochastic resonance as- Schimansky-Geier et al., 1998), by Bulsara and Zador sume periodic external forcing, it is interesting to ask (1996), and by Chapeau-Blondeau (1997). Considering whether noise can also amplify small aperiodic signals. conditional entropies and Kullback measures, To this end several different studies (in scope and tech- Schimansky-Geier et al. (1998) demonstrated with a Schmitt trigger system, driven periodically at strong, but nique) have been put forward. still subthreshold amplitude strengths, that information Jung and Ha¨nggi (1991a) considered noise due to the measures do exhibit characteristic extrema. These ex- phase diffusion of the external force. This is a realistic trema, however, do not decribe the conventional regime assumption, for instance, if the external field is provided of stochastic resonance for the signal-to-noise ratio, but by a laser where spontaneous emission generates phase they rather seem to mimic the stochastic resonance be- diffusion (Haken, 1970). Instead of a deterministic phase havior in a regime that is in accordance with stochastic ⍀t, they proposed the use of a stochastic phase ␪, i.e., resonance for the spectral amplification ␩. ˙ ␪ϭ⍀ϩ (1/␶d) ␰␪(t), with ␰␪(t) the derivative of a Wiener process. The finite coherence time of the phase D. Stochastic resonance—related topics dynamics leads to a broadening of the peaks in the 1. Noise-induced resonances power spectrum and a suppression of the stochastic resonance effect. In studying stochastic resonance, one looks at the pe- Neiman and Schimansky-Geier (1994) considered the riodic contribution of the output at the same frequency overdamped motion of a particle in a bistable potential as the input. More recently, the general question of the 1 4 1 2 generation of higher harmonics in the presence of noise V(x)ϭ 4 x Ϫ 2 x , driven by white Gaussian noise and has been addressed in a number of studies (Bartussek harmonic noise (Schimansky-Geier and Zu¨ licke, 1990; et al., 1993; Dykman, Mannella, McClintock, Stein, and Dykman, Mannella, McClintock, Stein, and Stocks, Stocks, 1993a; Bartussek, Ha¨nggi, and Jung, 1994; Dyk- 1993b). Harmonic noise y(t) is generated by applying man et al., 1994; Jung and Talkner, 1995; Bulsara, In- white Gaussian noise on a second-order linear filter. The chiosa, and Gammaitoni, 1996; Jung and Bartussek, spectral density of the harmonic noise has a peak at a 1996). In this section we focus on a novel effect (Bar- nonzero frequency ␻p and thus mimics a certain degree tussek, Ha¨nggi, and Jung, 1994), namely the noise- of periodicity. The power spectrum of x(t) exhibits a selective resonance-like suppression of higher harmon- maximum at ␻max , which is located close to ␻p , but ics. These ‘‘noise-induced resonances’’ have been with a small variation as a function of the noise. As in observed using numerical solutions of the Fokker- the case of phase diffusion, the peaks have a finite width. Planck equation in bistable systems (Bartussek, Ha¨nggi, The signal-to-noise ratio shows a relative maximum at a and Jung, 1994) as well as in monostable systems (Jung finite noise strength typical of stochastic resonance. and Bartussek, 1996). Noise-induced resonances have al- In recent years, we witness a prosperous period for ready been observed in experiments with a periodically aperiodic stochastic resonance, which was ushered in by driven SQUID by Rouse, Han, and Lukens (1995). Re- cently, similar resonances have been predicted for quan- addressing the problem of optimizing information trans- tum stochastic resonance by Grifoni and Ha¨nggi (1996a, fer in excitable systems (Collins et al., 1995a, 1995b; De 1996b). Weese and Bialek, 1995; Collins, Chow, et al., 1996; Col- Apart from numerical, adiabatic studies (Bartussek, lins, Imhoff, and Grigg, 1996; Heneghan et al., 1996; Ha¨nggi, and Jung, 1994; Rouse et al., 1995), an analytical Levin and Miller, 1996; Nieman et al., 1997). The above- theory allowing one to predict whether or not a particu- named authors considered the Fitzhugh-Nagumo equa- lar system would exhibit noise-induced resonance has tions driven by white noise and an arbitrary aperiodic been put forward by Jung and Talkner (1995). Their ap- signal. This system was operated below threshold and proach is sketched as follows: we consider here a general the aperiodic signal was not large enough to induce ex- overdamped system subject to additive white noise and citation. Together with the noise, however, excitations periodic forcing, i.e., were possible. Stochastic resonance has been demon- x˙ ϭf x ϩA cos t ϩ t , (7.13) strated for the correlation of the aperiodic signal with ͑ ͒ 0 ͑⍀ ͒ ␰͑ ͒ the excitation rate (the number of excitation events per where ␰(t) is as usual white Gaussian noise with zero unit time). mean and strength D, and f(x) is the forcing function.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 278 Gammaitoni et al.: Stochastic resonance

As shown in Sec. IV.A, the spectral density consists of a broad background and ␦ spikes at multiples of the driving frequency. The weights of the ␦ spikes are given by 2 gnϭ2␲͉Mn͉ , where Mn are the complex Fourier coefficients of the asymptotic (large times) mean value ͗x(t)͘as . Since the nth harmonic is in leading order pro- n portional to A0 (see Jung and Bartussek, 1996), the in- 2n tensities gn are proportional to A0 . We therefore define the characteristic coefficients 2 4␲͉Mn͉ ␥nϵ 2 2n , (7.14) n! A0 to describe the intensities of the harmonics. In the adiabatic approximation, the characteristic coefficients ␥n FIG. 54. Unidirectional probability current ¯J vs noise strength have been obtained by Jung and Talkner (1995) in their D at fixed driving amplitude A0ϭ0.5 in a rocking ratchet with 2n respective leading order A0 as asymmetric periodic potential VR(x)ϭϪ͓sin(2␲x) 1 2 2 ϩ 4 sin(4␲x)͔/2␲. The various lines correspond to different 4␲͉Mn͉ 4␲ Knϩ1 ␥ ϭ Ϸ , (7.15) driving angular frequencies ⍀: adiabatic driving: ⍀ϭ0.01 (solid n n!2A2n ͑2D͒2n ͩ n! ͪ line); and nonadiabatic driving: ⍀ϭ1 (dashed), ⍀ϭ2.5 (short- 0 dashed), ⍀ϭ4 (dotted), and ⍀ϭ7 (dashed-dotted). With the where Kn are the cumulants of the stationary probability period L of the ratchet potential equal to unity, the average density of the unperturbed process (A ϭ0). The inten- 0 particle drift ͗x˙ ͘ϭLJ¯equals in this case the probability cur- sity of the basic harmonic ␥1 cannot exhibit noise- ¯ ¯ induced resonance for any system, because the second rent J . A characteristic current reversal (with J passing through zero) occurs in the regime of nonadiabatic driving! order cumulant is strictly positive. The sign of the cumu- The adiabatic theory, see Eq. (7.17), falls on to the line with lants K can change, for example, as a function of the nϾ2 ⍀ϭ0.01. After Bartussek, Ha¨nggi, and Kissner (1994). noise strength D, giving rise to zeros of the intensities ␥n of the higher harmonics, i.e., to noise-induced resonances. such as static external force fields, field gradients of ther- Decomposing the complex amplitude Mn into the mal, chemical, or other origin. Hence the acting non- product Mnϭ͉Mn͉sin ␾n, it can be seen that whenever a equilibrium forces of zero ensemble average are spa- noise-induced resonance occurs, the phase ␾n exhibits a tially uniform, and generally are statistically symmetric. jump of magnitude ␲. Several concrete systems (single The same principle applies if the stationary nonequilib- well, double well, two-state system, etc.) have been dis- rium forces are substituted by a spatially uniform, coher- cussed by Jung and Talkner (1995). ent periodic signal F(t) of zero temporal average (Ma- gnasco, 1993; Ajdari et al., 1994; Bartussek, Ha¨nggi, and 2. Periodically rocked molecular motors Kissner, 1994). These systems are thus closely related in It is generally appreciated that useful work cannot be spirit to the stochastic resonance phenomenon: extracted from thermal equilibrium fluctuations. Such a Fluctuation-induced escape among neighboring states in device would violate the second law of thermodynamics. a periodic, multistable potential supported by weak de- Feynman et al. (1966) discussed this issue by means of a terministic periodic signals is responsible for moving model of a mechanical ratchet—a scheme that was origi- particles forward noisily. In short, a deterministic driving nally devised and elucidated during the heyday of early alone, which exceeds a lower threshold is sufficient to Brownian motion by M. V. Smoluchowski (1912, 1914). create an induced current. Increasing at fixed angular In his articles, which these days still provide delightful frequency ⍀ the driving strength in overdamped, deter- reading, Smoluchowski (1912, 1914) shows that in the ministic ratchet dynamics reveals numerous interesting absence of an intelligent creature, such as a Maxwell features such as a devil’s staircase behavior of current vs demon, no net currents will occur. In the presence of driving strength, current-quantization phenomena, and nonequilibrium forces the situation changes drastically: further peculiar features. In the presence of thermal Ny- now a thermal ratchet system, that is, a periodic struc- quist noise ␨(t), with ͗␨(t)͘ϭ0 and correlation ture with spatial asymmetry subjected to noise, can rec- ͗␨(t)␨(tЈ)͘ϭ2D␦(tϪtЈ) the noisy, periodically driven, tify symmetric, unbiased nonequilibrium fluctuations overdamped rocking ratchet dynamics reads into a fluctuation-induced directed current (Ajdari and d Prost, 1992; Magnasco, 1993; Astumian and Bier, 1994; x˙ ϭϪ V ͑x͒ϩA cos͑⍀t͒ϩ␨͑t͒. (7.16) dx R 0 Bartussek, Ha¨nggi, and Kissner, 1994; Doering et al., 1994; Leibler, 1994; for a comprehensive reviews see Herein VR(x) is the periodic (period L) sawtooth-like Ha¨nggi and Bartussek, 1996; Astumian, 1997; Ju¨ licher ratchet potential VR(x)ϭVR(xϩL) possessing no re- et al., 1997). Put differently, by a ratchet we mean a sys- flection symmetry VR(x)ÞVR(Ϫx). In Fig. 54 we de- tem that is able to move particles with finite macroscopic pict the noise-induced current versus the thermal inten- velocity in the absence of any macroscopic bias forces sity D. Noteworthy in Fig. 54 is the stochastic-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 279 resonance-like feature of the probability current ¯J vs D occur in periodically driven dynamics in a periodic po- characteristics, as well as the phenomenon of current re- tential, where the rate-determining Floquet eigenvalue versal (Bartussek, Ha¨nggi, and Kissner, 1994) that oc- is related to the Floquet eigenvalue at one of the two curs for nonadiabatic driving frequencies ⍀. The mean boundaries of the first Brillouin zone, we refer the ¨ velocity of the particle position is given by ͗x˙ ͘ϭLJ¯, reader to the discussion given in Jung and Hanggi (1991b). A main result is that the rate (or Floquet eigen- where ¯J is the time-averaged probability current. value), which increases proportional to A2, does not ex- Within an adiabatic approximation, i.e., for slow driving 0 hibit any kind of resonance-like behavior! Thus, as re- (⍀ 0) the current ¯J reads explicitly → peatedly demonstrated in this review, the stochastic D 2␲/⍀ resonance phenomenon is not due to a resonance for the ¯J ϭ dt ͕1Ϫexp͓⌽͑L,t͔͖͒Ϫ1 rate of escape in the periodically driven system. In re- 2␲/⍀ ͵ 0 ͭ cent studies by Reichl and her collaborators (Alpatov L L and Reichl, 1994; Kim and Reichl, 1996) the higher- ϫ ͵ dx͵ dy exp͓⌽͑y,t͒Ϫ⌽͑x,t͔͒ order Floquet eigenvalues have been investigated by 0 0 formally mapping the periodically driven overdamped L x Ϫ1 system onto an equivalent quantum dynamics at imagi- Ϫ dx dy exp͓⌽͑y,t͒Ϫ⌽͑x,t͔͒ , (7.17) nary times. As a main result they find that in the regime ͵0 ͵0 ͮ where the quantum system exhibits a transition to chaos where ⌽(x,t)ϭ͓VR(x)ϪxA0 cos(⍀t)͔/D. the spectrum of Floquet eigenvalues shows level repul- In Fig. 54 this approximation for the smallest fre- sion. quency ⍀ϭ0.01 coincides within line thickness with the Yet another quantity related to the rate-determining exact Floquet-theory result (solid line). The effects of Floquet eigenvalue is the diffusion coefficient in a tilted inertia and/or weak friction are also intriguing: In the periodic potential. Here, the interplay of subthreshold absence of thermal noise, the characteristic chaotic mo- static drive and thermal fluctuations results in an en- tion is sufficient to induce a directed current J, which hancement of the ensuing stationary current, with a exhibits multiple current reversals vs the driving ampli- maximum for a certain value of the temperature. Such tude A0 (Jung et al., 1996). an effect, not observable in the overdamped limit (Hu, Another ratchet type that is related closely to the 1993; Gittermann, Khalfin, and Shapiro, 1994; Casado, rocking ratchet in Eq. (7.16) is obtained if one substi- Mejıás, and Morillo, 1995), may be important in weakly tutes the external coherent driving by an oscillating tem- damped systems (Marchesoni, 1997). If the static drive is perature, i.e., replaced by a periodic tilt with suprathreshold amplitude, novel effects can be induced such as an enhance- A cos͑⍀t͒ ␨͑t͓͒1ϩA cos͑⍀t͔͒. (7.18) 0 → 0 ment of the diffusion rate that even exceeds (Hu, Daf- This defines a diffusion ratchet (Reimann et al., 1996), ferstshofer, and Haken, 1996) the rate of free diffusion! which tends to resist carrying a finite current in the Likewise, the role of suprathreshold driving strengths asymptotic limits of fast and slow driving. In this case applied to stochastic resonance systems has recently the current starts only proportional to ⍀2,as⍀ 0 (i.e., been studied for the phenomenon of a noise-induced → a zero net current in the leading-order adiabatic ap- failure mechanism for driven switch transitions (Apos- proximation) and vanishes again proportional to ⍀Ϫ2,as tolico et al., 1997). ⍀ ϱ. → VIII. CONCLUSIONS AND OUTLOOK 3. Escape rates in periodically driven systems In this review we have shown that adding noise to a The problem of activated rates in threshold systems system can sometimes improve its ability to transfer in- that are exposed to noise and periodic perturbations is formation reliably. This phenomenon—known as sto- nontrivial. The phase ⍀tϵ␪ of the periodic driving con- chastic resonance—was originally proposed, almost sev- stitutes an additional dimension that can be used to de- enteen years ago, to account for the periodicity in the fine the escape rate out of a basin of attraction in ex- Earth’s ice ages, but has since been shown to occur in tended space (Jung and Ha¨nggi, 1991b). The basin of many systems. By now, understanding of the phenom- attraction is then shown to be separated by an unstable enon of stochastic resonance has reached a mature level periodic orbit in extended x-␪ space. For a bistable po- that we have attempted to review with this long paper. tential, this rate is related to the smallest nonzero Flo- Numerous contributions to stochastic resonance have quet eigenvalue ␮. Note that this very quantity rules the appeared in most physics journals and can be found scat- long-time relaxation of general statistical quantities such tered through many other scientific journals (e.g., see as time-dependent mean values or time-averaged corre- http://www.umbrars.com/sr), particularly in the fields of lations. This positive-valued, rate-determining eigen- biology and physiology; it thus has reached the level of value, being at weak noise well separated from higher- what we may term an ‘‘industry.’’ order relaxational Floquet eigenvalues, has been Undoubtedly, the neurophysiological applications investigated for a driven, symmetric double well by Jung represent cornerstones in the field of stochastic reso- (1989; see Figs. 4–6 therein). For the pecularities that nance. Such applications have attracted continued inter-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 280 Gammaitoni et al.: Stochastic resonance est from scientists in biology, biomedical engineering, butions paved the way to interpreting a mass of and medicine. A new interdisciplinary field beyond con- physiological data from a new viewpoint; see Secs. IV.C ventional stochastic resonance, with inputs from nonlin- and V.C. ear dynamics, nonequilibrium statistical physics, biologi- Equipped with the basics of stochastic resonance cal and medical sciences has emerged. theory, developed in some detail in Secs. III and IV, we The general feature of a system exhibiting stochastic discussed and interpreted in Sec. V several prominent resonance is its increased sensitivity to small perturba- applications and experiments taken from the fields of tions when an appropriate dose of noise is added; see physics and neurophysiology (a more detailed list of ex- the Introduction and Sec. II. The idea that random noise perimental stochastic resonance demonstrations is given can be beneficial to the formation of ‘‘order’’ sounds in Sec. II.C.3). paradoxical, but must be taken seriously by now. Sto- More recent developments in the field of stochastic chastic resonance simply stands for a new paradigm resonance presently in the limelight of the activities of wherein noise represents a useful tool, rather than a nui- many research groups are discussed in Sec. VI. Both sance. Given the basic three ingredients of stochastic quantum stochastic resonance and spatiotemporal sto- resonance, which are (1) a form of threshold, (2) a chastic resonance have only just begun to be explored. source of ‘‘noise,’’ and (3) a generally weak input Quantum tunneling assists the stochastic resonance ef- source, it is clear that stochastic resonance is generic fect in the semiclassical regime; in the deep cold, how- enough to be observable in a large variety of systems. ever, quantum coherence increasingly spoils the effect; For the benefit of the reader let us summarize here see Sec. VI.A. The notion of stochastic resonance gen- what we think has been achieved in the field of stochas- eralized to spatially extended pattern-forming systems has been the subject of Sec. VI.B: spatiotemporal pat- tic resonance, point out open questions, and finally share terns can be enhanced by adding the proper amount of our views about future perspectives. noise. The notion of deterministic chaos, which intrinsi- For simple physical systems that can be described by cally provides a source of disorder, has been studied by either one of the two generic models introduced in Sec. various groups, and has been reviewed in Sec. VI.C. The III (two-state model) and in Secs. IV.A and IV.B last section, Sec. VI.D, has been devoted to the study of (continuous-state bistable model), the mechanism that the effects of finite correlation times (colored noise) of underpins the stochastic resonance effect is by now well the background noise. For overdamped dynamics the understood. The increased response of the system in the role of colored noise generally results in a reduction of presence of noise is due to the synchronization of noise- the efficiency of stochastic resonance. In contrast, finite induced hopping with the temporal profile of the weak inertia effects, induced by moderate friction, tend to perturbation. This response of the system is ruled by two boost the stochastic resonance response. The physics of competing aspects: starting out from the zero-noise stochastic resonance at extreme weak friction, however, limit, increasing noise allows—correlated with the small still needs to be investigated in greater detail. perturbation—excursions into the neighboring well. This The field of stochastic resonance research has wit- causes an increased response. On the other hand, in- nessed a remarkable flourishing during the last few creasing the noise level counteracts the aforementioned years; needless to say it is no longer possible to present a correlation; thereby reducing the response. These two detailed account on each single contribution. In Sec. VII aspects are encoded mathematically by the susceptibility we have discussed selected contributions that provide ␹, which essentially is made up of two factors: the prod- additional insight. Stochastic resonance and its connec- uct of an Arrhenius factor, describing the activated hop- tion with the dithering effect, globally coupled periodi- ping, and a factor proportional to the inverse noise in- cally modulated bistable elements, or the impact of ad- tensity 1/D, characterizing the degradation of the ditional multiplicative noise on stochastic resonance in response. The result is a bell-shaped curve for the re- the presence of additive noise (see Sec. VII.C.1) are all sponse amplitude vs noise intensity, hence the term sto- topics that relate closely to stochastic resonance. Re- chastic resonance—an expression that for some may ap- cently, research on stochastic resonance in physical sys- pear ill-defined. This physics in turn determines the most tems has diverged into neighboring fields such as the common quantifiers for stochastic resonance: the signal- problem of noise-induced transport in Brownian ratch- to-noise ratio (SNR) of the output, and the spectral ets; see Sec. VII.D.2. The modern topic of nonlinear sto- (power) amplification ␩; see Secs. II.A and IV.B. Like- chastic resonance involves the impact of noise on the wise, the statistical features of the driven residence-time generation and mixing of higher harmonics; see Sec. distributions N(T) (see Sec. IV.C), reflect the synchro- VII.D.1. A peculiar effect, the suppression of higher nization between random hopping and external modula- harmonics at some specific noise strengths, is being re- tion. The multipeaked signatures exhibited by the ported there. This effective elimination of higher har- residence-time distribution at odd multiples of half the monics could be used to the effect of minimizing the driving period are nothing but the fingerprints of this distortion of information transfer in nonlinear systems. synchronization process that occurs in the competition In Sec. VII.C.3 on aperiodic stochastic resonance— between the active driving source and the passive dissi- i.e., the phenomenon being obtained in presence of a pation. It should not go unnoticed that the understand- nonperiodic input signal—we touched on a still open ing of this very concept of driven residence-time distri- problem. Can stochastic resonance be suitably charac-

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 281 terized by means of information theory concepts alone? Saetta and S. Santucci (Perugia). We greatly appreciate In view of quantum stochastic resonance, which intrinsi- insightful discussions with Th. Anastasio, R. Bartussek, cally avoids the notion of joint probability measures, a A. Bulsara, D. K. Campbell, J. Collins, Th. Dittrich, M. unifying answer seems anything but trivial. Dykman, W. Ebeling, J. Grohs, R. F. Fox, H. Frauen- It may be worthwhile to conclude with some specula- felder, G. Hu, M. Grifoni, A. Jackson, L. Kiss, A. Kittel, tions on what the future of stochastic resonance may A. Longtin, R. Mantegna, G. Mayer-Kress, F. Moss, J. look like. What is still lacking from a physics point of Parisi, E. Pollak, C. Presilla, H. Risken, R. Roy, L. view is a detailed, microscopic approach to stochastic Schimansky-Geier, B. Spagnolo, A. Sutera, W. Sung, P. resonance that would account for the mutual interplay Talkner, A. Vulpiani, U. Weiss, K. Wiesenfeld, and C. between the transfer of power among the system x(t), Zerbe. Last but not least, we would like to thank our the bath(s) [or sources of noise ␰(t)], and the external indispensable Mrs. Eva Seehuber for her continuous signal A(t). Another promising area for fruitful further help while preparing this work. research is quantum stochastic resonance. For example, P.H., P.J., and F.M. are deeply indebted to our friend almost nothing is known about the quantum analog of and mentor Hannes Risken, whose wisdom and friend- stochastic resonance in threshold-crossing devices, sto- ship we are missing deeply after his untimely death in chastic resonance in arrays of coupled quantum systems, January 1994. or—last but not least—the difficult problem of modeling quantum stochastic resonance in stationary nonequilibrium systems (i.e., the physics occurring in driven, dissi- APPENDIX: PERTURBATION THEORY pative quantum systems that are far from thermal equilibrium). The observation that classical concepts become For weak external forcing, the time-inhomogeneous increasingly invalid upon crossing the borderline be- term in the Fokker-Planck equation can be treated as a tween the classical and the quantum world, and beyond, small perturbation with the amplitude A0 acting as small is an indication that several surprises and novel stochas- parameter. Perturbation theory (Presilla et al., 1989; Hu tic resonance phenomena are waiting to be uncovered. et al., 1990; Jung, 1993) provides explicit expressions for The same holds true for spatiotemporal stochastic reso- characteristic quantities of the driven systems, such as nance, which yet has to be extended into three- the Floquet eigenvalues, eigenfunctions, and mean val- dimensional structures. ues in terms of the eigenvalues and eigenfunctions of the Clearly, stochastic resonance constitutes an unperturbed Fokker-Planck operator. information-transmitting phenomenon that exploits the The starting point is the Floquet eigenvalue problem, noise in a self-optimizing manner. Therefore, its prom- obtained in Sec. IV.A by inserting the Floquet ansatz ising role in complex systems such as the nervous sys- (4.9) into the Fokker-Planck equation (4.11) with ␸ϭ0 tem, or even the brain have not gone unnoticed in the ץ ץ communities of physiological, biological, and medical ϪA cos͑⍀t͒ Ϫ p ͑x,t;0͒ϭϪ␮p ͑x,t;0͒, tͬ ␮ ␮ץ xץ sciences; see the reviews by Moss, Pierson, and ͫL0 0 O’Gorman (1994) and Wiesenfeld and Moss (1995). For (A1) example, the question of whether extremely low- with frequency electromagnetic fields actually affect biological function via the stochastic resonance phenomenon 2ץ ץ still remains open. What has been achieved so far is the 0ϭϪ f͑x͒ϩD 2 , (A2) xץ xץ successful demonstration of stochastic resonance with L injected external noise in the peripheral nervous system and of crayfish (Douglass et al., 1993; Pei et al., 1996), in crickets (Levin and Miller, 1996), in the human visual f͑x͒ϭxϪx3. (A3) perception (Riani and Simonotto, 1995; Simonotto et al., 1997), and in ion channels (Bezrukov and Vodyanoy, Expanding the Floquet eigenfunctions into a Fourier 1995), to name a few. Without doubt this latter area, series too, is expected to prosper by providing numerous inter- ϱ esting new results and novel insights. p␮͑x,t;0͒ϭ cn͑x͒exp͓in⍀t͔ (A4) nϭϪ͚ϱ ACKNOWLEDGEMENTS yields the hierarchy of coupled ordinary differential equations P.J. and P.H. would like to thank the Deutsche Fors- chungsgemeinschaft for financial support. F.M. and L.G. A0 wish to thank the Istituto Nazionale di Fisica della Ma- ͑ 0Ϫin⍀ϩ␮͒cn͑x͒Ϫ ͓cnЈϩ1͑x͒ϩcnЈϪ1͑x͔͒ϭ0, teria (INFM) and CNR for partial funding. F.M. would L 2 (A5) further like to thank B. I. Halperin for his kind hospital- ity at the Physical Laboratories of the Harvard Univer- with cnЈ(x) ϭ dcn(x)/dx. sity. Most of the analog simulations presented herein For strictly real-valued Floquet eigenvalues ␮, the were carried out in collaboration with E. Menichella- Floquet eigenfunctions are real as well, i.e.,

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 282 Gammaitoni et al.: Stochastic resonance

* cn͑x͒ϭcϪn͑x͒. (A6) ͑1͒ 1 ͑ 0Ϫi⍀ϩ␭l͒c1 ͑x͒ϭ ␺lЈ͑x͒, In order for the perturbation theory to be applicable L 2 to more general situations such as tilted periodic potentials, we do not make use of this assumption. The zeroth ͑1͒ 1 ͑ 0ϩi⍀ϩ␭l͒cϪ ͑x͒ϭ ␺Ј͑x͒. (A15) order perturbation theory for A0ϭ0 L 1 2 l

͑0͒ ͑0͒ Since ␭lϮi⍀ are not eigenvalues of the operator 0 , ͑ 0Ϫin⍀ϩ␮ ͒cn ͑x͒ϭ0 (A7) L L the operators 0ϩ␭lϮi⍀ can be inverted and the func- is solved by (1) L tions cϮ1(x) are obtained formally as

͑0͒ cn ͑x͒ϭ␺l͑x͒, 1 c͑1͒͑x͒ϭ ͑ Ϫi⍀ϩ␭ ͒Ϫ1␺Ј͑x͒, 1 2 L0 l l ͑0͒ ␮ln ϭ␭lϩin⍀, (A8) 1 where the index l of ͕cn(x)͖ has been dropped through- ͑1͒ Ϫ1 † cϪ1͑x͒ϭ ͑ 0ϩi⍀ϩ␭l͒ ␺lЈ͑x͒. (A16) out for convenience. Here, ␺l(x), ␺l (x), and ␭l are the 2 L eigenfunctions and eigenvalues of the unperturbed (ad- Inserting Eq. (A16) into Eq. (A14), multiplying by the joint) Fokker-Planck operator, i.e., † eigenfunction ␺l and then integrating over x, we obtain ␺ ͑x͒ϭϪ␭ ␺ ͑x͒, 0 l l l ϱ ץ ץ † L ͑2͒ 1 ␮l ϭ ␺l ͑x͒ 1 ␺l͑x͒dx xץ x Lץ ␺†͑x͒ϭϪ␭ ␺†͑x͒, 4 ͵Ϫϱ† L0 l l l ץ ץ 1 ϱ ϭ l 1 l , (A17) ʹ xͯץ x Lץͯ ϭ 4 ͳ † ͗l͉m͘ϵ ␺l͑x͒␺m͑x͒dx ␦lm . (A9) ͵Ϫϱ where 0 To order (A0) , all equations with label n have the same form. The eigenvalues differ by a multiple of i⍀. 1 1 Without loss of generality, we can start the perturbation ϭ ϩ . (A18) L1 ϩi⍀ϩ␭ Ϫi⍀ϩ␭ expansion at nϭ0, i.e., L0 l L0 l For the Floquet eigenfunctions one obtains in leading- ͑0͒ ͑0͒ ␮lnϭ0ϵ␮l ϭ␭l , order perturbation theory

0͒ c͑ ͑x͒ϭ␦ ␺ ͑x͒. (A10) A0 n n0 l p ͑x,t͒ϭ␺ ͑x͒ϩ ͕exp͑Ϫi⍀t͓͒ ϩi⍀ϩ␭ ͔Ϫ1 ␮l l 2 0 l The ﬁrst three equations of Eq. (A5) are written down L explicitly as ϩexp͑i⍀t͓͒ Ϫi⍀ϩ␭ ͔Ϫ1͖␺Ј͑x͒. (A19) L0 l l In view of the identity A0 ͑ 0ϩ␮͒c0͑x͒ϭ ͓c1Ј͑x͒ϩcϪЈ 1͑x͔͒, (A11) L 2 ϱ ץ ϱ ץ ϭ † ␺l͑x͒ ͚ ␺q͑x͒ ͵ ␺q͑x͒ ␺l͑x͒dx xץ x qϭ0 Ϫϱץ A0 ͑ 0Ϫi⍀ϩ␮͒c1͑x͒ϭ ͓c2Ј͑x͒ϩc0Ј͑x͔͒, 2 ϱ ץ L ϭ ͚ q l ␺q͑x͒, (A20) ʹ xͯץͯ qϭ0 ͳ A0 ͑ ϩi⍀ϩ␮͒c ͑x͒ϭ ͓cЈ͑x͒ϩcЈ ͑x͔͒. (A12) L0 Ϫ1 2 0 Ϫ2 the Floquet eigenvalues and eigenfunctions can be expressed in terms of the eigenfunctions of the undriven We now seek a solution of the latter system of differential equations in terms of the perturbation expansions Fokker-Planck operator, namely

2 ϱ ץ A ␭ Ϫ␭ 2͒͑ 2 1͒͑ cn͑x͒ϭ␦n0␺l͑x͒ϩA0cn ͑x͒ϩA0cn ͑x͒ϩ••• , 0 l q ␮lϭ␭lϩ 2 2 l q ͯ xץ ͯ ͚ϭ ͑␭ Ϫ␭ ͒ ϩ⍀ 2 q 0 l q ͳ ʹ ␮ ϭ␭ ϩA ␮͑1͒ϩA2␮͑2͒ϩ••• . (A13) ץ l l 0 l 0 l (1) ϫ q l ʹ ͯ xץ ͯ It follows immediately from Eq. (A11) that ␮ van- ͳ 2 ishes. Comparing terms of order A0 and A0 in Eqs. (A11) and (A12) yields ϱ 1 p␮ ͑x,t͒ϭ␺l͑x͒ϩA0 ͚ l qϭ0 Ϫ 2ϩ 2 ͑2͒ ͑2͒ 1 ͑1͒ ͑1͒ ͱ͑␭l ␭q͒ ⍀ ͑ 0ϩ␭l͒c0 ͑x͒ϩ␮l ␺l͑x͒ϭ ͓c1Ј ͑x͒ϩcϪЈ 1 ͑x͔͒, ץ L 2 (A14) ϫ q l cos͑⍀tϩ␣ ͒␺ ͑x͒, (A21) x ͯ ʹ ql qץ ͯ ͳ

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 283

1 ץ with ϱ ϭ ͗x͑t͒͘as A0 ͚ q 0 ͗0͉x͉q͘ 2 xͯ ʹ ͱ␭ ϩ⍀2ץͯ qϭ1 ͳ tan͑␣ql͒ϭ⍀/͑␭lϪ␭q͒. (A22) q

In particular, the asymptotic probability density ϫcos͑⍀tϩ␣q0͒. (A28) x͉0͘ toץ/ץpas(x,t), corresponding to the vanishing Floquet eigen- On approximating ͗0͉x͉1͘ to 1 and ͗1͉ value ␮ ϭ␭ ϭ0, reads 0 0 Ϫ␭1 /D, one eventually recovers Eq. (2.7a) for ¯x (D) (Hu et al., 1990). ץ ϱ pas͑x,t͒ϭp0͑x͒ϩA0 ͚ q 0 ʹ xͯץͯ qϭ1 ͳ REFERENCES 1 ϫ␺ ͑x͒ cos͑⍀tϩ␣ ͒ q 2 2 q0 Abramowitz, M., and I. A. Stegun, 1965, Handbook of Math- ͱ␭qϩ⍀ ematical Functions (Dover, New York). Ajdari, A., D. Mukamel, L. Peliti, and J. Prost, 1994, J. Phys. I ϭp ͑x͒ϩA ͑x͒sin͑⍀t͒ϩA ͑x͒cos͑⍀t͒, 0 s c 4, 1551. (A23) Ajdari, A., and J. Prost, 1992, C.R. Acad. Sci. 315, 1635. Alpatov, P., and L. E. Reichl, 1994, Phys. Rev. E 49, 2630. with Amit, D. J., 1984, Field Theory, The Renormalization Group

ϱ and Critical Phenomena (World Scientific, Singapore). -Amit, D. J., 1989, Modeling Brain Functions (Cambridge Uni ץ ␭q Ac͑x͒ϭA0 ͚ 2 2 q 0 ␺q͑x͒, .(x ͯ ʹ versity, Cambridgeץ ͯ qϭ1 ␭ ϩ⍀ ͳ q Anishchenko, V. S., A. B. Neimann, and M. A. Safanova, 1993, ϱ J. Stat. Phys. 70, 183. ,Anishchenko, V. S., M. A. Safonova, and L. O. Chua, 1994 ץ ⍀ As͑x͒ϭA0 ͚ 2 2 q 0 ␺q͑x͒. (A24) .x ͯ ʹ Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 441ץ ͯ qϭ1 ␭ ϩ⍀ ͳ q Apostolico, F., L. Gammaitoni, F. Marchesoni, and S. San- The spectral amplification [Eq. (4.21)] is obtained by tucci, 1997, Phys. Rev. E 55, 36. inserting Eq. (A23) into the definition Arimondo, E., D. Dangoisse, E. Menchi, and F. Papoff, 1987, ͗x(t)͘asϭ͐xpas(x,t)dx, that is, J. Opt. Soc. Am. B 4, 892. Arimondo, E., and B. M. Dinelli, 1983, Opt. Commun. 44, 277. ϱ 2 Astumian, R. D., 1997, Science 276, 917. ץ ␭n␭mϩ⍀ ␩ϭ 2 2 2 2 n 0 Astumian, R. D., and M. Bier, 1994, Phys. Rev. Lett. 72, 1766. ͯ xץ ͯ ͚ϭ ϩ ϩ n,m 0 ͑␭n ⍀ ͒͑␭m ⍀ ͒ ͳ ʹ Bartussek, R., P. Ha¨nggi, and P. Jung, 1994, Phys. Rev. E 49, 3930. ץ ϫ m 0 ͗0͉x͉n͗͘0͉x͉m͘. (A25) Bartussek, R., P. Ha¨nggi, and J. G. Kissner, 1994, Europhys. .x ͯ ʹ Lett. 28, 459ץ ͯ ͳ ¨ The expression (A25) is exact up to order (A )2. Bartussek, R., P. Jung, and P. Hanggi, 1993, in Noise in Physi- 0 cal Systems and 1/f Fluctuations, edited by P. H. Handel, AIP The expression for ͗x(t)͘ to leading order in A co- as 0 Conf. Proc. 285, 661 (AIP, New York, 1993). incides with the prediction of the thermal-equilibrium Barzykin, A. V., and K. Seki, 1997, Europhys. Lett. 40, 117. linear-response theory of Sec. IV.B. This last statement Benderskii, V. A., D. E. Makarov, and C. A. Wight, 1994, Adv. can be proved explicitly by inserting the completeness Chem. Phys. 85,1. relation into the expression Bennet, W. R., 1948, Bell Syst. Tech. J. 27, 446. Benzi, R., G. Parisi, A. Sutera, and A. Vulpiani, 1982, Tellus .10 ,34 ץ ␹͑t͒ϭ xe 0t Ϫ p ͑x͒dx (A26) xͪ 0 Benzi, R., A. Sutera, G. Parisi, and A. Vulpiani, 1983, SIAMץ ͩ L ͵ (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 565. for the response function ␹(t) of the modulated system Benzi, R., A. Sutera, and A. Vulpiani, 1981, J. Phys. A 14, of Eq. (A.1), whence L453. Benzi, R., A. Sutera, and A. Vulpiani, 1985, J. Phys. A 18, .2239 ץ ϱ ␹͑t͒ϭϪ ͚ eϪ␭qt q 0 ͗0͉x͉q͘. (A27) Berdichevsky, V., and M. Gitterman, 1996, J. Phys. A 29, L447. .x ͯ ʹ Berghaus, C., A. Hilgers, and J. Schnakenberg, 1996, Z. Physץ ͯ qϭ1 ͳ Note that Eq. (A26) follows immediately from the B 100, 157. general definition (4.28) of ␹(t) by substituting Bezrukov, S. M., and I. Vodyanoy, 1995, Nature (London) 378, 362. ϭ (yϪz) for the perturbation kernel, and ⌫ext ␦Ј Bezrukov, S. M., and I. Vodyanoy, 1997, Nature (London) 385, 0t P0(x,t͉y,0)ϭeL ␦(xϪy) for the unperturbed condi- 319. tional probability density of the system under study. On Blake, I. F., and W. C. Lindsey, 1973, IEEE Trans. Inf. Theory further substituting the spectral representation (A27) of IT-19, 295. ␹(t) into Eq. (4.26), we eventually reproduce the per- Bonifacio, R., and L. A. Lugiato, 1978, Phys. Rev. A 18, 1192. turbation theory prediction for ͗x(t)͘as : Brey, J. J., and A. Prados, 1996, Phys. Lett. A 216, 240.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 284 Gammaitoni et al.: Stochastic resonance

Bruce, A. D., 1980, Adv. Phys. 29, 111. Dykman, M. I., G. P. Golubev, I. Kh. Kaufman, D. G. Luchin- Bulsara, A., S. Chillemi, L. Kiss, P. V. E. McClintock, R. Man- sky, P. V. E. McClintock, and E. A. Zhukov, 1995, Appl. nella, F. Marchesoni, G. Nicolis, and K. Wiesenfeld, 1995, Phys. Lett. 67, 308. Eds., International Workshop on Fluctuations in Physics and Dykman, M. I., H. Haken, G. Hu, D. G. Luchinsky, R. Man- Biology: Stochastic Resonance, Signal Processing and Related nella, P. V. E. McClintock, C. Z. Ning, N. D. Stein, and N. G. Phenomena, published in Nuovo Cimento 17D, 653. Stocks, 1993, Phys. Lett. A 180, 332. Bulsara, A., T. C. Elston, C. R. Doering, S. B. Lowen, and K. Dykman, M. I., D. G. Luchinsky, R. Mannella, P. V. E. Mc- Lindenberg, 1996, Phys. Rev. E 53, 3958. Clintock, H. E. Short, N. D. Stein, and N. G. Stocks, 1994, Bulsara, A., and L. Gammaitoni, 1996, Phys. Today 49, No. 3, Phys. Rev. E 49, 1198. 39. Dykman, M. I., D. G. Luchinsky, R. Mannella, P. V. E. Mc- Bulsara, A., M. Inchiosa, and L. Gammaitoni, 1996, Phys. Rev. Clintock, N. D. Stein, and N. G. Stocks, 1995, Nuovo Ci- Lett. 77, 2162. mento D 17, 661. Bulsara, A., E. W. Jacob, T. Zhou, F. Moss, and L. Kiss, 1991, Dykman M. I., D. G. Luchinsky, P. V. E. McClintock, N. D. J. Theor. Biol. 152, 531. Stein, and N. G. Stocks, 1992, Phys. Rev. A 46, R1713. Bulsara, A., S. B. Lowen, and C. D. Rees, 1994, Phys. Rev. E Dykman, M. I., R. Mannella, P. V. E. McClintock, N. D. Stein, 49, 4989. and N. G. Stocks, 1993a, Phys. Rev. E 47, 1629. Bulsara, A., and G. Schmera, 1993, Phys. Rev. E 47, 3734. Dykman, M. I., R. Mannella, P. V. E. McClintock, N. D. Stein, Bulsara, A., and A. Zador, 1996, Phys. Rev. E 54, R2185. and N. G. Stocks, 1993b, Phys. Rev. E 47, 3996. Carroll, T. L., and L. M. Pecora, 1993a, Phys. Rev. Lett. 70, Dykman, M. I., R. Mannella, P. V. E. McClintock, and N. G. 576. Stocks, 1990a, Phys. Rev. Lett. 65, 48. Carroll, T. L., and L. M. Pecora, 1993b, Phys. Rev. E 47, 3941. Dykman, M. I., R. Mannella, P. V. E. McClintock, and N. G. Casado, J. M., J. J. Mejı´as, and M. Morillo, 1995, Phys. Lett. A Stocks, 1990b, Phys. Rev. Lett. 65, 2606. 197, 365. Dykman, M. I., R. Mannella, P. V. E. McClintock, and N. G. Castelpoggi, F., and H. S. Wio, 1997, Europhys. Lett. 38, 91. Stocks, 1992, Phys. Rev. Lett. 68, 2985. Chapeau-Blondeau, F., 1997, Phys. Rev. E 55, 2016. Dykman, M. I., R. Mannella, P. V. E. McClintock, and N. G. Chialvo, D. R., and A. V. Apkarian, J. Stat. Phys. 70, 375. Stocks, 1993, Phys. Rev. Lett. 70, 874. Choi, M., R. F. Fox, and P. Jung, 1998, Phys. Rev. E (in press). Eckmann, J-P., and L. E. Thomas, 1982, J. Phys. A 15, L261. Chun, K., and N. O. Birge, 1993, Phys. Rev. E 48, 11 4500. Eigler, D. M., and E. K. Schweitzer, 1990, Nature (London) Claes, I., and C. Van den Broeck, 1991, Phys. Rev. A 44, 4970. 344, 523. Clarke, J., A. N. Cleland, M. H. Devoret, D. Esteve, and J. M. Faetti, S., P. Grigolini, and F. Marchesoni, 1982, Z. Phys. B 47, Martinis, 1988, Science 239, 992. 353. Collins, J. J., C. C. Chow, A. C. Capela, and T. T. Imhoff, 1996, Fauve, S., and F. Heslot, 1983, Phys. Lett. 97A,5. Phys. Rev. E 54, R5575. Feynman, R. P., R. B. Leighton, and M. Sands, 1966, The Collins, J. J., C. C. Chow, and T. T. Imhoff, 1995a, Phys. Rev. Feynman Lectures on Physics (Addison-Wesley, Reading, E 52, R3321. MA), Vol. I, Chap. 46. Collins, J. J., C. C. Chow, and T. T. Imhoff, 1995b, Nature Fioretti, A., L. Guidoni, R. Mannella, and E. Arimondo, 1993, (London) 376, 236. J. Stat. Phys. 70, 403. Collins, J. J., T. T. Imhoff, and P. Grigg, 1996, J. Neurophysiol. Floquet, G., 1883, Ann. de l’Ecole Norm. Suppl. 12, 47. 76, 642. Fox, R. F., 1978, Phys. Rep. 48, 179. Coppinger, F., J. Genoe, D. K. Maude, U. Gennser, J. C. Por- Fox, R. F., 1989, Phys. Rev. A 39, 4148. tal, K. E. Singer, P. Rutter, T. Taskin, A. R. Peaker, and A. Fox, R., and Y. Lu, 1993, Phys. Rev. E 48, 3390. C. Wright, 1995, Phys. Rev. Lett. 75, 3513. Fronzoni, L., F. Moss, and P. V. E. McClintock, 1989, eds., Crisanti, A., M. Falcioni, G. Paladin, and A. Vulpiani, 1994, J. Noise in Nonlinear Dynamical Systems, Vol. 3 (Cambridge Phys. A 27, L597. University, Cambridge), p. 222. Fulin´ ski, A., 1995, Phys. Rev. E 52, 4523. Dakhnovskii, Yu., and R. D. Coalson, 1995, J. Chem. Phys. Gage, E. C., and L. Mandel, 1988, Phys. Rev. A 38, 5166. 103, 2908. Gammaitoni, L., 1995a, Phys. Rev. E 52, 4691. Dayan, I., M. Gitterman, and G. H. Weiss, 1992, Phys. Rev. A Gammaitoni, L., 1995b, Phys. Lett. A 208, 315. 46, 757. Gammaitoni, L., and F. Marchesoni, 1993, Phys. Rev. Lett. 70, Debnath, G., T. Zhou, and F. Moss, 1989, Phys. Rev. A 39, 873. 4323. Gammaitoni, L., F. Marchesoni, M. Martinelli, L. Pardi, and S. Desai, R., and R. Zwanzig, 1978, J. Stat. Phys. 19,1. Santucci, 1991, Phys. Lett. A 158, 449. De Weese, M., and W. Bialek, 1995, Nuovo Cimento 17D, 733. Gammaitoni, L., F. Marchesoni, E. Menichella-Saetta, and S. Dewel, G., P. Borckmanns, D. Walgraef, 1985, Phys. Rev. A Santucci, 1989, Phys. Rev. Lett. 62, 349. 31, 1983. Gammaitoni, L., F. Marchesoni, E. Menichella-Saetta, and S. Dittrich, T., B. Oelschla¨gel, and P. Ha¨nggi, 1993, Europhys. Santucci, 1990, Phys. Rev. Lett. 65, 2607. Lett. 22,5. Gammaitoni, L., F. Marchesoni, E. Menichella-Saetta, and S. Doering, C. R., W. Horsthemke, and J. Riordan, 1994, Phys. Santucci, 1993, Phys. Rev. Lett. 71, 3625. Rev. Lett. 72, 2984. Gammaitoni, L., F. Marchesoni, E. Menichella-Saetta, and S. Douglass, J. K., L. Wilkens, E. Pantazelou, and F. Moss, 1993, Santucci, 1994, Phys. Rev. E 49, 4878. Nature (London) 365, 337. Gammaitoni, L., F. Marchesoni, E. Menichella-Saetta, and S. Drozdov, A. N., and M. Morillo, 1996, Phys. Rev. E 54, 3304. Santucci, 1995, Phys. Rev. E 51, R3799.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 285

Gammaitoni, L., F. Marchesoni, and S. Santucci, 1994, Phys. Haken, H. 1970, Laser Theory (Springer, Berlin). Lett. A 195, 116. Ha¨nggi, P., 1978, Helv. Phys. Acta 51, 202. Gammaitoni, L., F. Marchesoni, and S. Santucci, 1995, Phys. Ha¨nggi, P., 1993, in Activated Barrier Crossing, edited by G. R. Rev. Lett. 74, 1052. Fleming and P. Ha¨nggi (World Scientific, London), pp. 268– Gammaitoni, L., M. Martinelli, L. Pardi, and S. Santucci, 1991, 292. Phys. Rev. Lett. 67, 1799. Ha¨nggi, P., and R. Bartussek, 1996, in Nonlinear Physics of Gammaitoni, L., M. Martinelli, L. Pardi, and S. Santucci, 1992, Complex Systems: Current Status and Future Trends, edited Mod. Phys. Lett. B 6, 197. by J. Parisi, S. C. Mu¨ ller, and W. Zimmermann, Lecture Gammaitoni, L., M. Martinelli, L. Pardi, and S. Santucci, 1993, Notes in Physics 476 (Springer, Berlin, New York), p. 294. J. Stat. Phys. 70, 425. Ha¨nggi, P., H. Grabert, G. L. Ingold, and U. Weiss, 1985, Phys. Gammaitoni, L., E. Menichella-Saetta, S. Santucci, and F. Rev. Lett. 55, 761. Marchesoni, 1989, Phys. Lett. A 142, 59. Ha¨nggi, P., and P. Jung, 1995, Adv. Chem. Phys. 89, 239. Gammaitoni, L., E. Menichella-Saetta, S. Santucci, F. Ha¨nggi, P., P. Jung, and F. Marchesoni, 1989, J. Stat. Phys. 54, Marchesoni, and C. Presilla, 1989, Phys. Rev. A 40, 2114. 1367. Gerstein, G. L., and B. Mandelbrot, 1964, Biophys. J. 4, 41. Ha¨nggi, P., P. Jung, C. Zerbe, and F. Moss, 1993, J. Stat. Phys. Gingl, Z., L. B. Kiss, and F. Moss, 1995, Europhys. Lett. 29, 70, 25. 191. Ha¨nggi, P., F. Marchesoni, and P. Grigolini, 1984, Z. Phys. B Gitterman, M., I. B. Khalfin, and B. Ya. Shapiro, 1994, Phys. 56, 333. Lett. A 184, 339. Ha¨nggi, P., F. Marchesoni, and P. Sodano, 1988, Phys. Rev. Gluckman, B. J., T. I. Netoff, E. J. Neel, W. L. Ditto, M. Lett. 60, 2563. Spano, and S. J. Schiff, 1996, Phys. Rev. Lett. 77, 4098. Ha¨nggi, P., P. Talkner, and M. Borkovec, 1990, Rev. Mod. Goel, N. S., and N. Richter-Dyn, 1974, Stochastic Models in Phys. 62, 251. Biology (Academic, New York). Ha¨nggi, P., and H. Thomas, 1982, Phys. Rep. 88, 207. Golding, B., N. M. Zimmermann, and S. N. Coppersmith, Heneghan, C., C. C. Chow, J. J. Collins, T. T. Imhoff, S. B. 1992, Phys. Rev. Lett. 68, 998. Lowen, and M. C. Teich, 1996, Phys. Rev. E 54, R2228. Go´ mez-Ordo´ñez, J., and M. Morillo, 1994, Phys. Rev. E 49, Hibbs, A. D., A. L. Singsaas, E. W. Jacobs, A. R. Bulsara, J. J. 4919. Bekkedahl, and F. Moss, 1995, J. Appl. Phys. 77, 2582. Gong, D., G. Hu, X. Wen, C. Yang, G. Qin, R. Li, and D. Hohmann, W., J. Mu¨ ller, and F. W. Schneider, 1996, J. Chem. Ding, 1992, Phys. Rev. A 46, 3243; Phys. Rev. E 48, 4862 (E). Phys. 100, 5388. Gong, D., G. R. Qin, G. Hu, and X. D. Weng, 1991, Phys. Lett. Hu, G., 1993, Phys. Lett. A 174, 247. A 159, 147. Hu, G., A. Daffertshofer, and H. Haken, 1996, Phys. Rev. Lett. Goychuk, I. A., E. G. Petrov, and V. May, 1996, Chem. Phys. 76, 4874. Lett. 353, 428. Hu, G., T. Ditzinger, C. Z. Ning, and H. Haken, 1993, Phys. Grabert, H., and H. Wipf, 1990, in Festko¨rperprobleme– Rev. Lett. 71, 807. Advances in Solid State Physics, edited by U. Ro¨ ssler Hu, G., H. Haken, and C. Z. Ning, 1992, Phys. Lett. A 172, 21. (Vieweg, Braunschweig), Vol. 30, 1. Hu, G., H. Haken, and C. Z. Ning, 1993, Phys. Rev. E 47, 2321. Graham, R., and A. Schenzle, 1982, Phys. Rev. A 25, 1731. Hu, G., H. Haken, and F. Xie, 1996, Phys. Rev. Lett. 77, 1925. Grebogi, C., E. Ott, and J. A. Yorke, 1987, Physica D 7, 181. Hu, G., G. Nicolis, and C. Nicolis, 1990, Phys. Rev. A 42, 2030. Grifoni, M., and P. Ha¨nggi, 1996a, Phys. Rev. Lett. 76, 1611. Hu, G., G. R. Qing, D. C. Gong, and X. D. Weng, 1991, Phys. Grifoni, M., and P. Ha¨nggi, 1996b, Phys. Rev. E 54, 1390. Rev. A 44, 6424. Grifoni, M., L. Hartmann, S. Berchtold, and P. Ha¨nggi, 1996, I, L., and J.-M. Liu, 1995, Phys. Rev. Lett. 74, 3161. Phys. Rev. E 53, 5890; 56, 6213 (E). Iannelli, J. M., A. Yariv, T. R. Chen, and Y. H. Zhuang, 1994, Grifoni, M., M. Sassetti, P. Ha¨nggi, and U. Weiss, 1995, Phys. Appl. Phys. Lett. 65, 1983. Rev. E 52, 3596. Imbrie, J., A. C. Mix, and D. G. Martinson, 1993, Nature (Lon- Grifoni, M., M. Sassetti, J. Stockburger, and U. Weiss, 1993, don) 363, 531. Phys. Rev. E 48, 3497. Inchiosa, M., and A. Bulsara, 1995a, Phys. Rev. E 52, 327. Grigolini, P., and F. Marchesoni, 1985, Adv. Chem. Phys. 62, Inchiosa, M., and A. Bulsara, 1995b, Phys. Rev. E 53, 327. 29. Inchiosa, M., and A. Bulsara, 1995c, Phys. Lett. A 200, 283. Grigorenko, A. N., V. I. Konov, and P. I. Nikitin, 1990, JETP Inchiosa, M., and A. Bulsara, 1996, Phys. Rev. E 51, 2021. Lett. 52, 592. Ippen, E., J. Lindner, and W. Ditto, 1993, J. Stat. Phys. 70, 148. Grigorenko, A. N., and P. I. Nikitin, 1995, IEEE Trans. Magn. Jost, B., and B. Sahleh, 1996, Opt. Lett. 21, 287. 31, 2491. Ju¨ licher, F., A. Ajdari, and J. Prost, 1997, Rev. Mod. Phys. 69, Grigorenko, A. N., P. I. Nikitin, A. N. Slavin, and P. Y. Zhou, 1269. 1994, J. Appl. Phys. 76, 6335. Jung, P., 1989, Z. Phys. B 76, 521. Grohs, J., S. Apanasevich, P. Jung, H. Issler, D. Burak, and C. Jung, P., 1993, Phys. Rep. 234, 175. Klingshirn, 1994, Phys. Rev. A 49, 2199. Jung, P., 1994, Phys. Rev. E 50, 2513. Grohs, J., H. Issler, and C. Klingshirn, 1991, Opt. Commun. 86, Jung, P., 1995, Phys. Lett. A 207, 93. 183. Jung, P., 1997, in Stochastic Dynamics, edited by L. Grohs, J., A. Schmidt, M. Kunz, C. Weber, A. Daunois, B. Schimansky-Geier and T. Po¨ schel, Lecutre Notes in Physics Schehr, A. Rupp, W. Dotter, F. Werner, and C. Klingshirn, No. 484 (Springer, Berlin), p. 23. 1989, Proc. SPIE 1127, 39. Jung, P., and R. Bartussek, 1996, in Fluctuations and Order: Grossmann, F., T. Dittrich, P. Jung, and P. Ha¨nggi, 1991, Phys. The New Synthesis, edited by M. Millonas (Springer, New Rev. Lett. 67, 516. York, Berlin), pp. 35–52.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 286 Gammaitoni et al.: Stochastic resonance

Jung, P., U. Behn, E. Pantazelou, and F. Moss, 1992, Phys. Mantegna, R. N., and B. Spagnolo, 1994, Phys. Rev. E 49, Rev. A 46, R1709. R1792. Jung, P., G. Gray, R. Roy, and P. Mandel, 1990, Phys. Rev. Mantegna, R. N., and B. Spagnolo, 1995, Nuovo Cimento D Lett. 65, 1873. 17, 873. Jung, P., and P. Ha¨nggi, 1989, Europhys. Lett. 8, 505. Mantegna, R. N., and B. Spagnolo, 1996, Phys. Rev. Lett. 76, Jung, P., and P. Ha¨nggi, 1990, Phys. Rev. A 41, 2977. 563. Jung, P., and P. Ha¨nggi, 1991a, Phys. Rev. A 44, 8032. Marchesoni, F., 1997, Phys. Lett. A 231, 61. Jung, P., and P. Ha¨nggi, 1991b, Ber. Bunsenges. Phys. Chem. Marchesoni, F., L. Gammaitoni, and A. Bulsara, 1996, Phys. 95, 311. Rev. Lett. 76, 2609. Jung, P., and P. Ha¨nggi, 1993, Z. Phys. B 90, 255. Marchesoni, F., and P. Grigolini, 1983, Physica A 121, 269. Jung, P., J. G. Kissner, and P. Ha¨nggi, 1996, Phys. Rev. Lett. Masoliver, J., A. Robinson, and G. H. Weiss, 1995, Phys. Rev. 76, 3436. E 51, 4021. Jung, P., and G. Mayer-Kress, 1995, Phys. Rev. Lett. 74, 2130. Matteucci, G., 1989, Climate Dynamics 3, 179. Jung, P., and P. Talkner, 1995, Phys. Rev. E 51, 2640. Matteucci, G., 1991, Climate Dynamics 6, 67. Jung, P., and K. Wiesenfeld, 1997, Nature (London) 385, 291. McClintock, P. V. E, and F. Moss, 1989, in Noise in Nolinear Kapitaniak, T., 1994, Phys. Rev. E 49, 5855. Dynamical Systems, Vol. 3, edited by F. Moss and P. V. E. Kim, S., and L. E. Reichl, 1996, Phys. Rev. E 53, 3088. McClintock, (Cambridge University, Cambridge), p. 243. Kiss, L. B., Z. Gingl, Z. Marton, J. Kertesz, F. Moss, G. McNamara, B., and K. Wiesenfeld, 1989, Phys. Rev. A 39, Schmera, and A. Bulsara, 1993, J. Stat. Phys. 70, 451. 4854. Kittel, A., R. Richter, M. Hirsch, G. Fla¨tgen, J. Peinke, and J. McNamara, B., K. Wiesenfeld, and R. Roy, 1988, Phys. Rev. Parisi, 1993, Z. Naturforsch. Teil A 48, 633. Lett. 60, 2626. Melnikov, V. I., 1993, Phys. Rev. E 48, 2481. Kramers, H., 1940, Physica (Utrecht) 7, 284. Millman, J., 1983, Microelectronics, McGraw-Hill Series in Krogh, A., and R. Palmer, 1991, Introduction to the Theory of Electrical Engineering, Electrons and Electronic Circuits Neural Computation (Addsion Wesley, New York), 1991. (McGraw-Hill, New York). Kubo, R., 1957, J. Phys. Soc. Jpn. 12, 570. Morillo, M., J. Go´ mez-Ordo`˜nez, and J. M. Casado, 1995, Phys. Kubo, R., 1966, Rep. Prog. Phys. 29, 255. Rev. E 52, 316. Kubo, R., M. Toda, and N. Hashitsume, 1985, Statistical Phys- Moss, F., 1991, Ber. Bunsenges. Phys. Chem. 95, 303. ics II, Springer Series in Solid State Sciences Vol. 31 Moss, F., 1994, in Contemporary Problems in Statistical Phys- (Springer, Berlin, New York). ics, edited by G. H. Weiss (SIAM, Philadelphia), pp. 205–253. Lambsdorff, M., C. Dornfeld, and C. Klingshirn, 1986, Z. Phys. Moss, F., A. Bulsara, and M. F. Shlesinger, 1993, Eds., The B 64, 409. Proceedings of the NATO Advanced Research Workshop: Leggett, A. J., S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, Stochastic Resonance in Physics and Biology, J. Stat. Phys. 70 A. Garg, and W. Zwerger, 1987, Rev. Mod. Phys. 59,1;67, (Plenum, New York), pp. 1–512. 725(E). Moss, F., J. K. Douglass, L. Wilkins, D. Pierson, and E. Pan- Leibler, S., 1994, Nature (London) 370, 412. tazelou, 1993, Ann. (N.Y.) Acad. Sci. 706, 26. Levin, J. E., and J. P. Miller, 1996, Nature (London) 380, 165. Moss, F., D. Pierson, and D. O’Gorman, 1994, Int. J. Bifurca- Lindner, J. F., B. K. Meadows, W. L. Ditto, M. E. Inchiosa, tion Chaos Appl. Sci. Eng. 4, 1383. and A. Bulsara, 1995, Phys. Rev. Lett. 75,3. Moss, F., and K. Wiesenfeld, 1995a, Sci. Am. 273, 50. Lo¨ cher, M., G. A. Johnson, and E. R. Hunt, 1996, Phys. Rev. Moss, F., and K. Wiesenfeld, 1995b, Spektrum der Wissen- Lett. 77, 4698. schaft, 273 (Oktober), 92. Lo¨ fstedt, R., and S. N. Coppersmith, 1994a, Phys. Rev. Lett. Murray, J. D., 1989, Mathematical Biology (Springer, Berlin). 72, 1947. Ne´da, Z., 1995a, Phys. Lett. A 210, 125. Lo¨ fstedt, R., and S. N. Coppersmith, 1994b, Phys. Rev. E 49, Ne´da, Z., 1995b, Phys. Rev. E 51, 5315. 4821. Neiman, A., and L. Schimansky-Geier, 1994, Phys. Rev. Lett. Longtin, A., 1993, J. Stat. Phys. 70, 309. 72, 2988. Longtin, A., A. Bulsara, and F. Moss, 1991, Phys. Rev. Lett. Neiman, A., and L. Schimansky-Geier, 1995, Phys. Lett. A 197, 67, 656. 379. Longtin, A., A. R. Bulsara, D. Pierson, and F. Moss, 1994, Neiman, A., L. Schimansky-Geier, and F. Moss, 1997, Phys. Biol. Cybern. 70, 569. Rev. E 56, R9. Louis, A. A., and J. P. Sethna, 1995, Phys. Rev. Lett. 74, 1363. Neimann, A., B. Shulgin, V. Anishchenko, W. Ebeling, L. Magnasco, M. O., 1993, Phys. Rev. Lett. 71, 1477. Schimansky-Geier, and J. A. Freund, 1996, Phys. Rev. Lett. Magnus, W., and S. Winkler, 1979, Hill’s Equation (Dover, 76, 4299. New York). Neiman, A., and W. Sung, 1996, Phys. Lett. A 223, 341. Mahato, M. C., and S. R. Shenoy, 1994, Phys. Rev. E 50, 2503. Nicolis, C., 1981, Sol. Phys. 74, 473. Makarov, D. E., and N. Makri, 1995, Phys. Rev. E 52, 5863. Nicolis, C., 1982, Tellus 34,1. Makri, N., 1997, J. Chem. Phys. 106, 2286. Nicolis, C., 1993, J. Stat. Phys. 70,3. Mandel, L., R. Roy, and S. Singh, 1981, in Optical Bistability, Nicolis, C., and G. Nicolis, 1981, Tellus 33, 225. edited by C. M. Bowden, M. Ciftan, and H. R. Robl (Plenum, Nicolis, C., G. Nicolis, and G. Hu, 1990, Phys. Lett. A 151, 139. New York), p. 127. Nicolis, G., C. Nicolis, and D. McKernan, 1993, J. Stat. Phys. Mannella, R., A. Fioretti, L. Fronzoni, B. Zambon, E. Ari- 70, 125. mondo, and S. Chillemi, 1995, Phys. Lett. A 197, 25. Oelschla¨gel, B., T. Dittrich, and P. Ha¨nggi, 1993, Acta Phys. Mansour, M., and G. Nicolis, 1975, J. Stat. Phys. 13, 197. Pol. B 24, 845.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998 Gammaitoni et al.: Stochastic resonance 287

Oppenheim, A. V., and R. W. Schaffer, 1975, Digital Signal Schwartz, D. B., B. Sen, C. N. Archie, and J. E. Lukens, 1985, Processing (Prentice Hall, New York). Phys. Rev. Lett. 55, 1547. Papoulis, A., 1965, Probability, Random Variables, and Sto- Shiino, M., 1987, Phys. Rev. A 36, 2393. chastic Processes (McGraw-Hill, New York). Shneidman, V. A., P. Jung, and P. Ha¨nggi, 1994a, Phys. Rev. Pei, X., J. Wilkens, and F. Moss, 1996, J. Neurophysiol. 76, Lett. 72, 2682. 3002. Shneidman, V. A., P. Jung, and P. Ha¨nggi, 1994b, Europhys. Pe´rez-Madrid, A., and J. M. Rubı´, 1995, Phys. Rev. E 51, 4159. Lett. 26, 571. Phillips, J. C., and K. Schulten, 1995, Phys. Rev. E 52, 2473. Shulgin, B., A. Neiman, and V. Anishchenko, 1995, Phys. Rev. Presilla, C., F. Marchesoni, and L. Gammaitoni, 1989, Phys. Lett. 75, 4157. Rev. A 40, 2105. Simon, A., and A. Libchaber, 1992, Phys. Rev. Lett. 68, 3375. Raikher, Y. L., and V. I. Stepanov, 1994, J. Phys.: Condens. Simonotto, E., M. Riani, C. Seife, M. Roberts, J. Twitty, and F. Matter 6, 4137. Moss, 1997, Phys. Rev. Lett. 78, 1186. Rajaraman, R., 1982, Solitons and Instantons (North-Holland, Smoluchowski, M. v., 1912, Phys. Z. 8, 1069; see p. 1078. Amsterdam). Smoluchowski, M. v., 1914, in Vortra¨ge u¨ber die kinetische Rappel, W. J., and S. H. Strogatz, 1994, Phys. Rev. E 50, 3249. Theorie der Materie und der Elektrizita¨t, edited by M. Planck Reibold, E., W. Just, J. Becker, and H. Benner, 1997, Phys. et al. (Teubner, Leipzig), pp. 89–121. Rev. Lett. 78, 3101. Spano, M. L., M. Wun-Fogle, and W. L. Ditto, 1992, Phys. Reimann, P., R. Bartussek, W. Ha¨ussler, and P. Ha¨nggi, 1996, Rev. A 46, R5253. Phys. Lett. A 215, 26. Stocks, N. G., 1995, Nuovo Cimento D 17, 925. Riani, M., and E. Simonotto, 1994, Phys. Rev. Lett. 72, 3120. Stratonovich, R. L., 1963, Topics in the Theory of Random Riani, M., and E. Simonotto, 1995, Nuovo Cimento D 17, 903. Noise, Vol. I (Gordon and Breach, New York), p. 143ff. Rice, S. O., 1944, Bell Syst. Tech. J. 23, 1; reprinted in N. Vax, Teich, M. C., S. M. Khanna, and P. C. Guiney, 1993, J. Stat. 1954, Noise and Stochastic Processes (Dover, New York). Phys. 70, 257. Rice, S. O., 1948, Bell Syst. Tech. J. 27, 109. Thorwart, M., and P. Jung, 1997, Phys. Rev. Lett. 78, 2503. Risken, H., 1984, The Fokker-Planck Equation, Springer Series Valls, O. T., and G. F. Mazenko, 1986, Phys. Rev. B 34, 7941. in Synergetics Vol. 18 (Springer, Berlin, New York). Van den Broeck, C., J. M. R. Parrando, J. Armero, and A. Risken, H., and H. D. Vollmer, 1989, in Noise in Nonlinear Hernandez-Machado, 1994, Phys. Rev. E 49, 2639. Dynamical Systems; Theory, Experiments, Simulation, Vol. I, van Kampen, N. G., 1992, Stochastic Processes in Physics and edited by F. Moss and P. V. E. McClintock (Cambridge Uni- Chemistry, 2nd ed. (North Holland, Amsterdam, New York). versity, Cambridge), p. 191. Vemuri, G., and R. Roy, 1989, Phys. Rev. A 39, 4668. Rose, J. E., J. F. Brugge, D. J. Anderson, and J. E. Hind, 1967, Vilar, J. M. G., and J. M. Rubi, 1997, Phys. Rev. Lett. 78, 2886. J. Neurophysiol. 30, 769. Weiss, U., 1993, Quantum Dissipative Systems, Series in Mod- Rouse, R., S. Han, and J. E. Lukens, 1995, Appl. Phys. Lett. ern Condensed Matter Physics (World Scientiﬁc, Singapore), 66, 108. Vol. 2. Roy, R., P. A. Schulz, and A. Walther, 1987, Opt. Lett. 12, 672. Wiesenfeld, K., and F. Moss, 1995, Nature (London) 373, 33. Sargent III, M., M. O. Scully, and W. E. Lamb Jr., 1974, Laser Wiesenfeld, K., D. Pierson, E. Pantazelou, C. Dames, and F. Physics (Addison-Wesley, Reading, MA). Moss, 1994, Phys. Rev. Lett. 72, 2125. Schenzle, A., and H. R. Brand, 1979, Phys. Rev. A 20, 1628. Winograd, I. J., T. B. Coplen, J. M. Landwehr, A. C. Riggs, K. Schimansky-Geier, L., J. A. Freund, A. B. Neiman, and B. R. Ludwig, B. J. Szabo, P. T. Kolesar, and K. M. Revesz, Shulgin, 1998, ‘‘Noise Induced Order: Stochastic Resonance,’’ 1992, Science 258, 255. Int. J. Bifurcation Chaos Appl. Sci. Eng. (in press). Wio, H. S., 1996, Phys. Rev. E 54, R3075. Schimansky-Geier, L., and U. Siewert, 1997, in Stochastic Dy- Yang, W., M. Ding, and G. Hu, 1995, Phys. Rev. Lett. 74, 3955. namics, edited by L. Schimansky-Geier, and T. Po¨ schel, Lec- Zambon, B., F. De Tomasi, D. Hennequin, and E. Arimondo, ture Notes in Physics No. 484 (Springer, Berlin), p. 245. 1989, Phys. Rev. A 40, 3782. Schimansky-Geier, L., and Ch. Zu¨ licke, 1990, Z. Phys. B 79, Zhou, T., and F. Moss, 1990, Phys. Rev. A 41, 4255. 451. Zhou, T., F. Moss, and P. Jung, 1990, Phys. Rev. A 42, 3161.

Rev. Mod. Phys., Vol. 70, No. 1, January 1998