Scale invariance in the dynamics of spontaneous behavior Alex Proekta,b,1, Jayanth R. Banavarc, Amos Maritand,1, and Donald W. Pfaffb aDepartment of Anesthesiology, Weill Cornell Medical College, New York, NY 10065; bLaboratory for Neurobiology and Behavior, The Rockefeller University, New York, NY 10065; cDepartment of Physics, University of Maryland, College Park, MD 20742; and dDepartment of Physics, University of Padova, Istituto Nazionale di Fisica Nucleare and Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, 35131 Padova, Italy Contributed by Donald W. Pfaff, April 24, 2012 (sent for review January 6, 2012) Typically one expects that the intervals between consecutive invariance in the behavior of animals and the lack of universality of occurrences of a particular behavior will have a characteristic time the scaling exponents. scale around which most observations are centered. Surprisingly, Behavior is often conceived as serving a particular purpose or as the timing of many diverse behaviors from human communication a response to a specific stimulus. However, even in the relative to animal foraging form complex self-similar temporal patterns re- absence of these, all animals including humans readily exhibit produced on multiple time scales. We present a general frame- spontaneous behavior. Spontaneous activation of behavior is the work for understanding how such scale invariance may arise in simplest case of animal behavior because it avoids the complexities nonequilibrium systems, including those that regulate mammalian added by specific behavioral tasks (e.g., refs. 6 and 8), interactions behaviors. We then demonstrate that the predictions of this frame- among individuals (2, 3), and the specifics of the structure of the work are in agreement with detailed analysis of spontaneous mouse environment (11–13). Understanding the dynamics of spontaneous behavior observed in a simple unchanging environment. Neural behavior therefore is a prerequisite for understanding behavioral systems operate on a broad range of time scales, from milliseconds dynamics in more complex settings and is the focus of our analysis. to hours. We analytically show that such a separation between time scales could lead to scale-invariant dynamics without any fine Results tuning of parameters or other model-specific constraints. Our ana- There is preliminary evidence that the dynamics of spontaneous lyses reveal that the specifics of the distribution of resources or behavior may exhibit scale invariance (e.g., refs. 14, 16, and 17). NEUROSCIENCE competition among several tasks are not essential for the expres- Here, we carry out detailed analysis of spatial and temporal sion of scale-free dynamics. Rather, we show that scale invariance distributions of spontaneous mouse activity in a simple and un- observed in the dynamics of behavior can arise from the dynamics changing environment (Materials and Methods). A representative intrinsic to the brain. recording of distance traveled by a mouse in consecutive 1-s intervals is shown in Fig. 1A. Although at the largest time scale, power law | criticality periodicity in the amount of locomotion related to the light:dark cycle is seen, at all finer time scales the patterns of locomotion PHYSICS special class of phenomena exists in the vicinity of a contin- and rest are irregular. Regardless of time scale, the pattern Auous phase transition where processes at the microscopic, appears unchanged over several orders of magnitude pointing to macroscopic, and, indeed, all intermediate scales are essentially a scale invariance of rest/activity fluctuations. Here, we develop a similar except for a change in scale. Remarkably, the timing of general framework to account for this scale invariance. many animal and human behaviors also exhibits this scale in- variance, evidenced by the fact that the temporal pattern remains Scale-Invariant Dynamics Are Generically Observed for a Broad Class unchanged regardless of the time scale on which the data are of Processes. We begin by observing that the autocorrelation of plotted (1–17) (see, e.g., Fig. 1 A and B and Table 1). More for- the rest intervals rapidly approaches zero (Fig. 1C). Thus, each individual rest interval can be considered independently (i.e., the mally, a function f(x) is said to be scale-invariant, if on multiplying system is memoryless). This observation is confirmed by the fact the argument of the function by some constant scaling factor (λ), − β that the return maps constructed from a sequence of consecutive λ λ ( +1) f x — one obtains f( x)= ( ) the same shape is retained but rest intervals are similar to those constructed from a dataset in with a different scale. It is straightforward to show that a function which the order of the rest intervals was randomized (data not fi ∼ −(β+1) that satis es this property is a power law p(x) x , where shown). Let t=0be the time at which the mouse starts resting. β > ( +1) is the scaling exponent. Let p (t) represent the probability that the mouse has not moved > In physical systems, one observes scale invariance near a critical in the time interval between 0 and t. Note that p (t) is the sur- point. It has been suggested that the presence of power laws in vivor function and is equal to (1 − cumulative probability dis- diverse living systems might imply that biological systems are tribution of rest intervals). The probability that the mouse is still > > poised in the vicinity of a continuous phase transition (e.g., ref. at rest at time t+dtis therefore p (t + dt)={1− rdt}p (t), 18). There are, however, fundamental differences between scale where rdt is the hazard function, or the probability that the invariance exhibited by biological and physical systems. Criticality mouse moves in the interval dt given that it was still at rest at > > is confined to a small region in parameter space, and it is not clear time t. Thus, dp (t)/dt = −rp (t). This simple equation has been how diverse biological systems are fine-tuned to exhibit criticality. studied extensively in many different contexts (e.g., ref. 20). Critical systems can be categorized in terms of a small number of universality classes and, depending on fundamental properties such as dimensionality and the symmetry of ordering (19), only a Author contributions: A.P. and D.W.P. designed experiments; A.P. performed experi- few sets of scaling exponents are observed. However, the dynam- ments; J.R.B. and A.M. performed theoretical work; A.P., J.R.B., and A.M. analyzed data; and A.P., J.R.B., A.M., and D.W.P. wrote the paper. ics of behavior exhibit considerable variation in the values of the The authors declare no conflict of interest. scaling exponents (Table 1). Finally, critical systems are at equi- 1To whom correspondence may be addressed. E-mail: [email protected] or amos.maritan@ librium, whereas most processes occurring in living systems in- pd.infn.it. cluding animal behavior are nonequilibrium. Thus, a fundamentally This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. different picture is needed to explain the ubiquity of scale 1073/pnas.1206894109/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1206894109 PNAS Early Edition | 1of6 Downloaded by guest on October 2, 2021 In the stationary case, by definition r does not depend on time (i.e., is a constant), which we denote as 1/T. In this case one readily obtains p > ðtÞ¼e − t=T , a distribution with a well-defined temporal scale. Thus, to exhibit scale invariance, the processes that activate behavior must necessarily be nonstationary, and, > therefore, nonequilibrium. The time dependence of p is quite generally described by the following equation: R t > − rðt′Þdt′ > p ðtÞ¼e τ p ðτÞ: [1] The lower limit of the integral, τ,isa“microscopic” time scale below which one cannot make meaningful observations of whether the mouse is moving or at rest. This time scale does not play any role in what follows except providing a measure of the unit time scale and the right units for proportionality constants (see below). Clearly, an infinite resting time is not consistent with a living mouse. We therefore postulate the existence of an emergent characteristic time scale (T), which limits the maximum time that an animal can stay at rest. In the case of spontaneous behavior without any additional constraints, this time may be determined by metabolic needs—eventually, the mouse has to move to obtain food or water. The specifics of the processes underlying the emergence of the characteristic time scale are likely distinct for different systems. Nevertheless, the key point is that regardless of specific mechanisms, the consequences of the existence of such a time scale are generically valid and are the focus of our analysis. To understand the origins of scale invariance, we now examine time dependence of r. Because subsequent rest intervals are uncorrelated, and there are no other relevant variables related to the environmental stimuli, r can be a function of only τ, t, and T, r=G(τ,t,T), where G is a yet unspecified function. We note that r has units of inverse time, whereas all arguments of G have units of time. Thus, if all arguments of G were expressed in units of seconds instead of minutes for instance (i.e., multiplied by a factor of 60), then r would be scaled by a factor of 1/60. More generally, this dimensionality argument demands that G(τ,t,T) must satisfy the following scaling form ðλτ; λ ; λ Þ¼1 ðτ; ; Þ; λ: G t T λ G t T for any À Á Setting λ ¼ 1=t, one finds G τ=t; 1; T=t ¼ t × Gðτ; t; TÞ¼rt: The most general form of r is therefore: 1 ^ ^ r ¼ Gðτ=t; t=TÞ where Gðτ=t; t=TÞ¼Gðτ=t; 1; T=tÞ: [2] t The key observation is that there is considerable simplification when the separation between the microscopic and macroscopic time scales is large τ << t < T.