DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS SERIES B Volume 9, Number 1, January 2008 pp. 145–162 PHASE-LOCKING AND ARNOLD CODING IN PROTOTYPICAL NETWORK TOPOLOGIES Stefan Martignoli and Ruedi Stoop Institute for Neuroinformatics UZH / ETHZ Winterthurerstrasse 190 8057 Z¨urich, Switzerland Abstract. Phase- and frequency-locking phenomena among coupled biologi- cal oscillators are a topic of current interest, in particular to neuroscience. In the case of mono-directionally pulse-coupled oscillators, phase-locking is well understood, where the phenomenon is globally described by Arnold tongues. Here, we develop the tools that allow corresponding investigations to be made for more general pulse-coupled networks. For two bi-directionally coupled oscil- lators, we prove the existence of three-dimensional Arnold tongues that mediate from the mono- to the bi-directional coupling topology. Under this transforma- tion, the coupling strength at which the onset of chaos is observed is invariant. The developed framework also allows us to compare information transfer in feedforward versus recurrent networks. We find that distinct laws govern the propagation of phase-locked spike-time information, indicating a qualitative difference between classical artificial vs. biological computation. 1. Introduction. Stable limit-cycle oscillations arise as the solutions of a vari- ety of autonomous nonlinear differential equations. Primarily, they are generated when systems pass either through saddle-node or through Hopf bifurcations. Corre- spondingly, stable limit-cycles are widely observed in physics, chemistry and biology. As an example, the most salient neuron models, such as the Hodgkin-Huxley, the Fitzhugh-Nagumo, the Morris-Lecar, or the spatially detailed compartmental cable models of neurons yield stable limit-cycle solutions in biophysically relevant param- eter regimes. Whereas for low-dimensional model systems the limit cycle property can be verified from the equations, in the case of the high-dimensional cable model or for biological neurons, the limit-cycle property is exhibited by the phenomenon of phase- and frequency locking. In biology, the limit-cycles interact by means of sharp voltage pulses, the spikes. For mono-directionally periodically pulse-perturbed limit-cycle oscillators, it is sim- ple to derive one-dimensional maps that describe this interaction. These maps are mathematically well understood: The emerging phenomena of phase- and frequency- locking are globally organized along Arnold tongues. For more general networks, a corresponding theory is still missing. Here, we develop the tools that are applicable, in principle, to arbitrary network topologies. Using this approach, we first generalize mono-directional phase-locking to bi-directional coupling, where a smooth change 2000 Mathematics Subject Classification. Primary: 37E10; Secondary: 37E45. Key words and phrases. Arnold tongues, phase-locked oscillators, recurrent topology. This research was supported by the SNF-grant 65293. 145 146 STEFAN MARTIGNOLI AND RUEDI STOOP of the involved interaction strengths, characterizes the transition from the feedfor- ward to the recurrent topology. Our first finding will be that during this transition, the Arnold tongues undergo a continuous transformation as well. The developed description also allows to investigate and compare general feedforward versus re- current networks. Here our main finding is that the properties of how spike-time information is processed in these networks, differ in a salient way, which may explain the observed increased efficacy of biological neural networks, which predominantly are of recurrent type. We start with a review of phase-locking phenomena and their relation to the circle-map universality class. Our present study is motivated by recent neurobi- ological experiments, where experimentally measured response functions provide insights into the dynamics of coupled neurons, and point out a potential role of phase-locking as a mechanism of information encoding. In Section 2, we develop a novel discrete phase-response map that can be used to describe arbitrary networks of pulse-coupled limit-cycle oscillators. Using this tool, we investigate in Sections 3-4 bi-directional pair coupling, where we prove the existence of three-dimensional Arnold tongues that connect the feedforward and recurrent topologies of coupled pairs of neurons in a continuous way. Using coupled neurons as elements, we will construct and analyze more general networks in Sections 5. In particular, the ef- ficacy of information processing in feedforward and in recurrent networks will be compared. 1.1. Circle-map universality. Locking was first described in 1665 by Christiaan Huygens [1], who discovered that two pendulum clocks, attached on opposite sides of a wall, tend to synchronize their frequencies in anti-phase. In order to describe the physics behind this phenomenon, he developed a theory, which today still provides the most salient insights into the phenomenon. More recently, in a different context, Arnold [2] provided a refined mathematical description of locking. He studied the return map of a periodically forced oscillator by means of the equation K x = f (x )= x +Ω sin(2πx ) mod1, (1) n+1 Ω,K n n − 2π n the so-called sine-, or Arnold circle-map, where Ω = TS/T0 is the frequency ratio of the two oscillators, and K denotes the mono-directional perturbation strength. The connection between limit-cycle oscillators and the sine-map can be derived from the so-called kicked rotator model [3]. For this system, the equations of motion can be analytically reduced to the one-dimensional discrete sine-map, in the limit of very strong damping. One essential insight added by Arnold is that the sine-map is characteristic for the universality class of circle-maps of the form fΩ,K : [0, 1] [0, 1], → x = f (x )= x +Ω g(K, x ) mod1, (2) n+1 Ω,K n n − n where the phase response function (PRF) g(K, xn), defined by g(K, x) := T (K, x)/T0, g(K, 0) = g(K, 1)=1, g(0, x)=1, (3) measures the effect T (K, x) of a perturbation of strength K, arriving at phase x of an oscillator of intrinsic period T0. This discrete map formulation can be seen as the Poincar´esection of a phase model of the motion along the stable limit-cycle [4]. In the latter picture, a perturbed limit-cycle oscillator is represented by a differential PHASE-LOCKING IN PROTOTYPICAL NETWORK TOPOLOGIES 147 equation of the form N φ˙ = ω + K P (φj )R(φ), (4) j=1 X where φ(t) is the phase, ω the intrinsic constant angular velocity, R(φ) the sensitiv- ity function and K the coupling strength. P (φj ) is called the influence function and can be interpreted as a stimulating waveform, originating from a source of pertur- bation labelled by j. For weak coupling, the sensitivity function is equivalent to an ’infinitesimal’ PRF 1 g(K = ǫ, φ). We will use the discrete-time model introduced by Arnold. − As an example, the PRF of a sinusoidal limit-cycle perturbed by spikes is given by K g(K, x)=1 sin(2πx), (5) ± 2π · where the plus sign refers to in-phase, and the minus sign to anti-phase 1/1- synchronization. For specific physical or biological oscillators, the PRF can be determined by simple experiments [5, 6, 7]. Eq. (5) may serve as a guiding line for many experimental pulse-perturbed limit-cycles in the following sense. An experi- mentally obtained return map fΩ,K belongs to the circle-map universality class if the following requirements are met [8]: a) fΩ,K (x) is a diffeomorphism for K <KC , where KC is called the critical in- teraction strength −1 b) At K = KC , fΩ,K(x) has a cubic inflection point and fΩ,K(x) is no longer dif- ferentiable c) For K>KC, fΩ,K(x) is non-invertible. The salient properties of the circle-map universality class (Eq. (2)) in the Ω,K - parameter space are the following [9, 8] { } Arnold tongues Tp/q of stable phase-locking emerge at every rational winding • number ω = p/q from the Ω-axis. The intervals of locking expand with in- creasing interaction strength K. At a critical interaction strength KC [2], the tongues start to overlap. For subcritical interaction strength, the widths of the Arnold tongues are • described by sequences of rational Farey-tree numbers Q [0, 1] (global scaling law) [10]. ∈ As a function of Ω, at K = KC , the winding numbers ω form a devil’s stair- • case [2], where the union of locked intervals has full measure [11]. Local scaling laws govern the transition from quasiperiodicity to chaos [12]. • 1.2. Phase-locking in neurobiology. Experimentally measured PRFs g(K, x) can be used to study phase-locking in neurobiology. Provided that a modeled or real neuron has a sufficiently stable limit-cycle behavior (which is generally the case), experimental results of spiking neurons perturbed by strong spikes can be modeled by circle-maps (Eq. (2)). In the last decade, PRFs have been measured for a variety of biological and model neurons [13, 14, 6, 7, 15, 16, 17, 18, 19]. Stoop et al. [14] measured the PRFs of both inhibitory and excitatory stimulated pyramidal neurons in rat neocortex and determined experimentally the corresponding Arnold tongues (see Fig. 1). 148 STEFAN MARTIGNOLI AND RUEDI STOOP In particular, it was proven that only strong inhibitory stimulations could lead to chaotic response, on a small, though nonzero set of the Ω,K -parameter space [14]. More recently, Netoff et al. [16] induced 1/1-locking{ in hybrid} circuits, in which neurons from the hippocampal formation of rats were coupled to virtual neurons, by taking advantage of previously measured PRFs. Similar dynamic-clamp techniques were used by Pervouchine et al. [17] in three-cell networks of the entorhinal cortex, resulting in the claim that the measured PRFs offer a possible explanation for the generation of β-frequency oscillations.