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# Scale invariance in the dynamics of spontaneous behavior

Contributed by Donald W. Pfaff, April 24, 2012 (sent for review January 6, 2012)

## Abstract

Typically one expects that the intervals between consecutive occurrences of a particular behavior will have a characteristic time scale around which most observations are centered. Surprisingly, the timing of many diverse behaviors from human communication to animal foraging form complex self-similar temporal patterns reproduced on multiple time scales. We present a general framework for understanding how such scale invariance may arise in nonequilibrium systems, including those that regulate mammalian behaviors. We then demonstrate that the predictions of this framework are in agreement with detailed analysis of spontaneous mouse behavior observed in a simple unchanging environment. Neural systems operate on a broad range of time scales, from milliseconds to hours. We analytically show that such a separation between time scales could lead to scale-invariant dynamics without any fine tuning of parameters or other model-specific constraints. Our analyses reveal that the specifics of the distribution of resources or competition among several tasks are not essential for the expression of scale-free dynamics. Rather, we show that scale invariance observed in the dynamics of behavior can arise from the dynamics intrinsic to the brain.

A special class of phenomena exists in the vicinity of a continuous phase transition where processes at the microscopic, macroscopic, and, indeed, all intermediate scales are essentially similar except for a change in scale. Remarkably, the timing of many animal and human behaviors also exhibits this scale invariance, evidenced by the fact that the temporal pattern remains unchanged regardless of the time scale on which the data are plotted (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–17) (see, e.g., Fig. 1 *A* and *B* and Table 1). More formally, a function *f*(*x*) is said to be scale-invariant, if on multiplying the argument of the function by some constant scaling factor (λ), one obtains *f*(λ*x*)=λ^{−(β+1)} *f*(*x*)—the same shape is retained but with a different scale. It is straightforward to show that a function that satisfies this property is a power law *p*(*x*)∼*x*^{−(}^{β+1)}, where (β+1) is the scaling exponent.

In physical systems, one observes scale invariance near a critical point. It has been suggested that the presence of power laws in diverse living systems might imply that biological systems are poised in the vicinity of a continuous phase transition (e.g., ref. 18). There are, however, fundamental differences between scale invariance exhibited by biological and physical systems. Criticality is confined to a small region in parameter space, and it is not clear how diverse biological systems are fine-tuned to exhibit criticality. 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 few sets of scaling exponents are observed. However, the dynamics of behavior exhibit considerable variation in the values of the scaling exponents (Table 1). Finally, critical systems are at equilibrium, whereas most processes occurring in living systems including animal behavior are nonequilibrium. Thus, a fundamentally different picture is needed to explain the ubiquity of scale invariance in the behavior of animals and the lack of universality of the scaling exponents.

Behavior is often conceived as serving a particular purpose or as a response to a specific stimulus. However, even in the relative absence of these, all animals including humans readily exhibit spontaneous behavior. Spontaneous activation of behavior is the simplest case of animal behavior because it avoids the complexities added by specific behavioral tasks (e.g., refs. 6 and 8), interactions among individuals (2, 3), and the specifics of the structure of the environment (11⇓–13). Understanding the dynamics of spontaneous behavior therefore is a prerequisite for understanding behavioral dynamics in more complex settings and is the focus of our analysis.

## Results

There is preliminary evidence that the dynamics of spontaneous behavior may exhibit scale invariance (e.g., refs. 14, 16, and 17). Here, we carry out detailed analysis of spatial and temporal distributions of spontaneous mouse activity in a simple and unchanging environment (*Materials and Methods*). A representative recording of distance traveled by a mouse in consecutive 1-s intervals is shown in Fig. 1*A*. Although at the largest time scale, periodicity in the amount of locomotion related to the light:dark cycle is seen, at all finer time scales the patterns of locomotion and rest are irregular. Regardless of time scale, the pattern appears unchanged over several orders of magnitude pointing to a scale invariance of rest/activity fluctuations. Here, we develop a general framework to account for this scale invariance.

### Scale-Invariant Dynamics Are Generically Observed for a Broad Class of Processes.

We begin by observing that the autocorrelation of the rest intervals rapidly approaches zero (Fig. 1*C*). Thus, each individual rest interval can be considered independently (i.e., the system is memoryless). This observation is confirmed by the fact that the return maps constructed from a sequence of consecutive rest intervals are similar to those constructed from a dataset in which the order of the rest intervals was randomized (data not shown). Let *t = 0* be the time at which the mouse starts resting. Let *p*^{>}(*t*) represent the probability that the mouse has not moved in the time interval between 0 and *t*. Note that *p*^{>}(*t*) is the survivor function and is equal to (1 − cumulative probability distribution of rest intervals). The probability that the mouse is still at rest at time *t + dt* is therefore *p*^{>}(*t* + *dt*) = {1 − *rdt*}*p*^{>}(*t*), where *rdt* is the hazard function, or the probability that the mouse moves in the interval *dt* given that it was still at rest at time *t*. Thus, *dp*^{>}(*t*)/*dt* = −*rp*^{>}(*t*). This simple equation has been studied extensively in many different contexts (e.g., ref. 20).

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 , 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:

The lower limit of the integral, τ, is a “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

Setting , one finds The most general form of *r* is therefore:

The key observation is that there is considerable simplification when the separation between the microscopic and macroscopic time scales is large τ *<< t < T.* In this case, the dependence on the microscopic time scale τ drops out and Eq. **2** can be reduced to .

Thus, the dynamics of activation of behavior are governed by *f—*a function of the dimensionless ratio of *t/T*. For example, *f*(*z*) = *z* leads to . More generally, *f*(*z*) has the following small z expansion (valid when *t < T*):

This equation leads to

where *F* is a function of the dimensionless ratio *t/T*. Only in the special case when β *= 0*, Eq. **3** yields a distribution with characteristic time scale *T*: pure exponential if δ = 1 or a stretched exponential if δ < 1. In the more general case, however, when β *>* 0, *p ^{>}*(

*t*) is a power law modified by an exponential or stretched exponential. Thus, scale-invariant dynamics are generically observed for a broad class of nonstationary Markov processes given only the separation between the microscopic and macroscopic time scales without the need for additional model-specific constraints or fine-tuning of parameters.

Furthermore, the scaling function *f*(*t/T*)=*rt*, as defined above, is not expected to have a universal form but rather depends on the specifics of the dynamical processes that give rise to it. Thus, in contrast to equilibrium critical phenomena, the scaling exponent is not universal.

### Dynamics of Activation of Spontaneous Behavior Exhibit Scale Invariance.

We studied the distribution of resting times by using rigorous statistical techniques (21) and confirmed that the distribution is consistent with power law up to a cutoff that occurs on the scale of 1,000 s in all mice studied (*SI Materials and Methods, Statistical Analysis of the Distribution: Maximum Likelihood Estimation*). Fig. 1*B* shows the cumulative distribution of dwell times for nine wild-type male mice. The distributions are linear on a log-log scale and have similar slopes. The dynamics of activation of spontaneous behavior are thus scale-invariant even in the absence of any priorities and goals (2, 3) or complex distribution of resources (11⇓–13), just as predicted by our theory. We note that the resting time intervals in mice maintained in a simple and unchanging environment and in humans operating in a totally unconstrained environment (5) scale similarly. In both cases, the scaling exponent β is approximately the same (∼0.7) (Table S1), resulting in the resting time distribution, *p(t),* scaling as . It is also noteworthy, that the dynamics of foraging behaviors in diverse species observed in their natural environments have also been shown to scale similarly (e.g., refs. 12 and 13). The cutoff times reflecting the macroscopic time scale are quite different depending on the species. This observation may point to the differences in the species-specific processes that give rise to the macroscopic time scale. One possible explanation of the species-specific macroscopic time scale are differences in their metabolic rates. The basal metabolic rate and the total energy stored in an organism limit the total amount of time that an animal can go without food and water. Note that because the metabolic rate is known to scale as approximately the 3/4 power of body mass and the total energy stored would scale as body mass, characteristic macroscopic times ought to scale approximately as organism mass to the quarter power.

### Dynamics of Activation of Behavior Are Not Affected by Location Preference and the Familiarity of the Environment.

We observe that, even given a fairly uniform environment, animals display robust location preferences, readily seen both in terms of the number of visits and the fraction of total time spent in different locations (Fig. 2 *A* and *B*). One might naively expect that, at each location, there is a distribution of resting times each with its own well-defined time scale reflecting location preference—the mouse stays longer at preferred locations. This intuition, however, is not borne out in the data (Fig. 2*D*). Despite significant scatter, it can be seen that many of the distributions are linear on a log-log plot and, thus, lack a characteristic scale. The linear regimes extend up to a different cutoff for each location.

If the spontaneous activation of behavior is probing the dynamics intrinsic to the nervous system, we would expect these dynamics to be independent of location. Then the dwell times at a location visited *N* times should correspond to *N* extractions from a power law distribution of dwell times (Fig. 1). We hypothesized therefore that the apparent differences between distributions shown in Fig. 2*D* reflect finite sample size effects. One prediction of this hypothesis is that the total time spent at a particular location should scale as *N ^{1/}*

^{β}

*.*Testing this prediction, however, is not practical because the distribution of sums of numbers drawn from a power law distribution is itself a power law (

*SI Materials and Methods, Dependence of Total Time Spent on a Number of Visits*). Thus, large differences between estimates of total time spent at two locations visited the same number of times are expected.

To demonstrate that the differences among distributions shown in Fig. 2*D* can be accounted for by finite size effect, it is useful to define a characteristic time scale (*t _{N})* determined by the number of visits for each location and use this time scale to rescale the location-specific distributions. Note that the mean dwell time diverges in the large

*N*limit and, therefore, is not a suitable characteristic time scale (see below).

We begin by determining the expected value for the largest dwell time observed after *N* extractions from a cumulative power law distribution characterized by the scaling exponent β by requiring that there is at least one dwell time greater than *t _{N}*.

This equation suggests that the empirical cumulative distribution is given by the standard finite-size scaling form

where approaches 1 when *z* approaches 0 and decreases rapidly at large *z*. Note that *F _{samp}* is simply related to sampling in contrast to

*F*(

*t/T*) in Eq.

**3**. The

*k*-th moment, denoted as

*<t*of the empirical probability distribution (for

^{k}>,*k >*β) is given by

Note that this equation implies that all moments *k >* β diverge. Because β < 1 for all mice studied, the mean and all higher moments are divergent. Thus, the appropriate characteristic time, τ_{charac}, at each site is given by measuring the ratio of the moments for the dwell times at that site (Fig. 2*C*). If the dynamics are indeed location-invariant, we predict that plots of should collapse. We demonstrate that this prediction is in good accord with the data in Fig. 2*E*.

The data in Fig. 2 *A*–*E* were recorded at “steady state” after the initial habituation period lasting at least 2 wk. Upon initial exposure to a new environment, animals display higher levels of activity and, through the process of exploration, establish their location preferences (22). Distinction between the initial habituation phase and the subsequent steady-state behavior is commonly made in behavioral analysis. However, Fig. 2*F* shows that after appropriate rescaling, location-specific distributions from the initial habituation phase collapse onto those observed at steady state. Therefore, the dynamics of spontaneous activity are largely independent of location and relative novelty or familiarity of the environment and, thus, likely reflect the underlying intrinsic dynamics of the nervous system.

Note that although rest intervals can be objectively defined based on the data, the definition of activity intervals is not straightforward. The mouse can be said to be at rest if it has been observed to be at the same location for a time interval greater than some threshold *T _{th}*. The distribution of activity intervals can therefore be defined as the distribution of times between two consecutive resting episodes ≥

*T*Clearly, the duration of activity intervals will depend on the particular choice of

_{th}.*T*We predict that the cumulative distributions of activity intervals should collapse after rescaling the time axis by

_{th}.*T*

_{th}_{.}Unlike in Fig. 2, however, only the time but not the cumulative distribution is scaled by a function of

*T*

_{th}_{,}and the scaling collapse is independent of β (

*SI Materials and Methods, Waiting Time Distributions*). As predicted, the raw waiting time distributions (Fig. 3

*A*) collapse onto a universal curve after rescaling (Fig. 3

*B*). The same applies to the dataset taken as a whole and to that at a single location (Fig. 3

*C*). Thus, much like the distribution of rest, the distribution of activity durations exhibits scaling collapse. Note that, remarkably, our derivation is equally valid for spontaneous mouse behavior and for the distribution of time intervals of aftershocks greater than a certain threshold magnitude after a major quake, as observed by Bak et al. (23). The essential requirement again is the emergence of a characteristic macroscopic time scale arising from relaxational processes in the earth.

### Dependence of the Dynamics on the Phase of the Light:Dark Cycle.

Behavior of most animals, including humans, is profoundly influenced by the circadian cycle manifested as consolidated periods of sleep and wakefulness allocated preferentially to the specific phases of the light:dark cycle. Mice are nocturnal and have consolidated periods of sleep during the light phase (24) expressed both in the increase in the total sleep time and durations of individual sleep episodes (25). This observation suggests that, during the light phase, the dynamics of rest/activity fluctuations should be dominated by the dynamics of sleep. Thus, we predict that the power law should be disrupted during the light phase (*SI Materials and Methods, Generalization to Three States: Effects of the Light:Dark Cycle*). Data in Fig. 4 confirm this prediction. During the light phase, there are more long rest durations than would be expected for power-law distributed data.

## Discussion

Our framework has a straightforward physical and biologically plausible interpretation. When β *>* 0, the probability of transition out of the resting state, *∼*β*/t*, decreases as a function of time. If this tendency acted unopposed, the longer the system was at rest, the less likely it will be to activate behavior. The existence of a macroscopic characteristic time *T*, however, provides an opposing tendency. As the system nears this macroscopic time scale, the probability of motion increases and the distribution of resting times deviates from power law. The strength of the threshold stimulus necessary to awaken a human sleeper (26), for instance, follows a pattern consistent with that described above. Note, that in our analysis we do not distinguish between purposeful and motivated behaviors such as eating or drinking and apparently “purposeless” movements. The fact that activation of the aggregation of motor acts, regardless of their specifics, is governed by scale-free dynamics suggests the existence of an elementary undifferentiated process in the nervous system that governs activation of all behaviors (27).

One remarkable conclusion of our work is that in many respects the dynamics of spontaneous behavior in a limited unchanging environment are essentially identical to those observed in complex settings such as foraging. It has been suggested that animal foraging resembles a Lévy flight (i.e., a random walk with individual steps sampled from a power law distribution) when food is scarce, but when food is abundant, foraging resembles a conventional random walk (12). In the latter case, the animal is likely to stumble upon food before it explores the dynamic range between the microscopic and macroscopic time scales (i.e., the separation of timescales is abolished). Thus, it is easily seen that the same intrinsic dynamics can give rise to both types of animal foraging. For the dynamics of human communication (2, 3), the macroscopic time scale is imposed by the emails or letters being deleted or being no longer readily accessible after a certain time. Indeed, very few of us reply to every letter we receive (e.g., ref. 3).

Motivated by work on equilibrium critical phenomena, differences in the scaling exponents between seemingly diverse behaviors such as human communication (2, 3) and animal foraging (e.g. ref. 12) were taken to imply fundamental differences in the underlying processes. However, here we show that unlike in equilibrium critical phenomena, the scaling exponent is free to assume a broad range of values, allowing one to unify seemingly disparate observations within a single framework.

We note that although our framework allows for an arbitrary scaling exponent (β + 1), most of the scaling exponents in Table 1 lie in the range between 1 and 3. This observation raises the question whether there is some evolutionary advantage to a particular value of the exponent (e.g., ref. 28). This notion is consistent with our observations that the scaling exponent is approximately the same for different individual mice. Alternatively, a much simpler explanation for the observed range of scaling exponents needs to be entertained. The lower cutoff of the range is imposed by the fact that distributions where β + 1 < 1 are nonnormalizable. However, if β + 1 > 3, then the power law behavior is characterized by a steep decay and, thence, is difficult to establish over a significant range of time.

One may speculate that the dynamics of behavior evolved to confer a selective survival advantage. Scale-free dynamics offer several fundamental advantages. The ability to balance conflicting demands such as the need for rest along with the ability to readily respond to salient stimuli (29) over a range of physiologically relevant time scales and the ability to produce the greatest repertoire of responses with efficient coding of information (30⇓–32) are all simultaneously fulfilled by scale-free dynamics. Interestingly, scale-free behavior arises quite naturally without specifically invoking prioritization of tasks, complex interactions among individuals in social dynamics, or complex environmental structure in foraging.

It is known that systems consisting of many interacting nonlinear elements exhibit emergent behavior that is not present in any one of the constituent elements (32⇓–34). Our work demonstrates that the essential ingredient for scale-invariant behavior is the existence of a characteristic macroscopic time scale rather than the specifics of the mechanisms or models that give rise to it.

## Materials and Methods

All animal procedures were approved by The Rockefeller University's Animal Care and Use Committee in accordance with the Animal Welfare Act and the Department of Health and Human Services Guide for the Care and Use of Laboratory Animals.

Spontaneous activity was measured as described (35). Adult C57 mice were housed individually inside a VersaMax monitor (Accuscan Instruments) consisting of an acrylic cage (18 cm × 29 cm × 13 cm) equipped with horizontal infrared beams and sensors spaced 2.54 cm (1 inch) apart in the horizontal plane. Each VersaMax monitor was placed inside a larger wooden chamber with its dedicated light and ventilation systems used to minimize the potential for transmission of sounds and other signals between animals. Animals were maintained on the 12 h:12 h light:dark cycle at constant temperature of ∼22 °C. Food and water were available ad libitum. Food pellets were randomly scattered on top of the bedding. The water bottle was placed either in the middle or one of the sides of the cage. The data were recorded onto a personal computer (Dell) by using VersaMax analyzer software version 3.41.

The total distance traveled (TOTDIST parameter in VersaDAT) was sampled every second. Rest interval was defined as the number of consecutive seconds during which the distance traveled was zero.

We also studied the entire behavioral sequence consisting of a set of observations {(*l*_{1},*d*_{1})…(*l*_{n},*d*_{n})} where *l* denotes location inside the cage and *d* denotes the dwell time at that location. Because the position of the mouse was recorded by using a grid, the mouse is seen as moving on a lattice where each subsequent location (*l*_{n+1}) is usually in the immediate vicinity of the preceding location (*l*_{n}). The time spent in motion between two immediately adjacent locations is negligible. To construct this dataset, we sampled beam breaks at 20-ms intervals and defined the mouse location as a center of a rectangle formed by the first and last beam broken by the mouse in each direction. Dwell time was defined as the time during which the mouse location stayed constant.

Note, that the TOTDIST parameter reflects only directed locomotion and disregards back and forth motion. Thus, rest intervals correspond to times between consecutive episodes of locomotion. In contrast, dwell times correspond to time intervals between any detectable motion which allows for greater temporal resolution. The distributions of rest intervals and dwell times are statistically identical.

As controls, we recorded from empty cages and cages containing an immobile object approximately the size of a mouse for 2 d continuously. No activity at all was recorded in either one of these cases.

All analysis was performed off line by using custom-written programs in Mathematica (Wolfram Research). The details of the statistical analysis and mathematical derivations can be found in *SI Materials and Methods*.

## Acknowledgments

We thank Ana C. Ribeiro for providing the initial dataset that motivated this study and Cori Bargmann and Eric Siggia for their insightful comments. This work was supported by a Foundation for Anesthesia Education and Research grant (to A.P.) and a grant from the Cassa di Risparmio di Padova e Rovigo foundation (to A.M.).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. E-mail: proekt{at}gmail.com or amos.maritan{at}pd.infn.it.

Author contributions: A.P. and D.W.P. designed experiments; A.P. performed experiments; 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.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1206894109/-/DCSupplemental.

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