Scale Invariance in the Dynamics of Spontaneous Behavior

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 invariance, 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 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. In this case, the dependence on the microscopic time scale τ drops out and Eq. 2 can be reduced ^ ¼ Gð0; t=TÞ ¼ fðt=TÞ to r t t . 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 p > ðtÞ¼e−t=T . More generally, f(z) has the following small z expansion (valid when t < T):

Fig. 1. Dynamics of spontaneous fluctuations between activity and rest are which the mouse stayed at the same location (SI Materials and Methods). scale-invariant. (A) Distance traveled by a single C57 adult male mouse Each mouse was recorded continuously for ∼22 d (15,027,257 dwell times maintained in a constant environment measured every second. White and total, 1,669,695 ± 432,903 dwell times per mouse ± SD). Distributions are black rectangles show light and dark phases (12 h:12 h), respectively. Each linear on a log-log scale and have similar slopes. The dashed line is a guide to subsequent lower trace shows progressive zooming in on the time axis in the the eye. (C) Lag autocorrelation function of the dwell times shown in B. region shown by the red dots (number at the right of the x axis shows the Different colors show different mice. Inset shows zooming in on the x axis. range plotted). Regardless of scale, the pattern of fluctuations between The autocorrelation function decays rapidly and does not exhibit significant locomotion and rest remains unchanged. (B) Cumulative distributions of oscillatory behavior. Thus, subsequent dwell times are to a large degree dwell times for nine male mice. Dwell time was defined as the time during independent.

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1206894109 Proekt et al. Downloaded by guest on October 2, 2021 Table 1. Timing of many diverse behaviors is scale-invariant Examples of scale-free dynamics Scaling exponent (β +1)

Time intervals between e-mail communications (2) ∼1 Time it took Einstein and Darwin to reply to letters (3) ∼3/2 Times that a human stays within a small area based on currency dispersal (4) ∼1.6 Times that a human stays within a small area based on phone records (5) ∼1.8 Times between movie ratings on Netflix (6) ∼1.5–2.7 Times between car movements in Florence (7) ∼0.97 Times between print job submissions (8) ∼1.76 Times between hospitalizations for exacerbations of schizophrenia (9) ∼1.21 Times that a healthy human stays at rest (10) ∼1.9 Times that a depressed human stays at rest (10) ∼1.7 Waiting times during foraging of spider monkeys (11) ∼1.7 Lengths of step sizes (or times between changes in direction) during foraging of marine predators (12, 13) 1.4–3 Times between turns of a fruitfly in a featureless environment (14) 1–2 Times between turns of a fruitfly tracking an odor (15) ∼1.3 Times that a mouse rests (16) 1.97 Times that a rat rests (17) 1.7

Scale-invariant distribution of time intervals characterizing the dynamics of many behaviors follow a power law—the probability of occurrence of a time − β interval (t)isp(t) ∼ t ( +1).

point to the differences in the species-specific processes that give fðzÞ¼β þ γzδ þ ... β; γ ≥ δ > : with 0 and 0 rise to the macroscopic time scale. One possible explanation of the fi This equation leads to species-speci c macroscopic time scale are differences in their metabolic rates. The basal metabolic rate and the total energy > −β −β ð−γ=δð = Þδþ...Þ NEUROSCIENCE p ðtÞ¼t Fðt=TÞ ∝ t e t T ; [3] stored in an organism limit the total amount of time that an animal can go without food and water. Note that because the metabolic where F is a function of the dimensionless ratio t/T. Only in the rate is known to scale as approximately the 3/4 power of body mass special case when β =0, Eq. 3 yields a distribution with char- and the total energy stored would scale as body mass, character- acteristic time scale T: pure exponential if δ = 1 or a stretched istic macroscopic times ought to scale approximately as organism exponential if δ < 1. In the more general case, however, when β > mass to the quarter power. > fi 0, p (t) is a power law modi ed by an exponential or stretched PHYSICS exponential. Thus, scale-invariant dynamics are generically ob- Dynamics of Activation of Behavior Are Not Affected by Location served for a broad class of nonstationary Markov processes given Preference and the Familiarity of the Environment. We observe that, only the separation between the microscopic and macroscopic even given a fairly uniform environment, animals display robust time scales without the need for additional model-specific con- location preferences, readily seen both in terms of the number of straints or fine-tuning of parameters. visits and the fraction of total time spent in different locations Furthermore, the scaling function f(t/T )= rt,asdefined above, (Fig. 2 A and B). One might naively expect that, at each location, is not expected to have a universal form but rather depends on there is a distribution of resting times each with its own well-de- the specifics of the dynamical processes that give rise to it. Thus, fined time scale reflecting location preference—the mouse stays in contrast to equilibrium critical phenomena, the scaling expo- longer at preferred locations. This intuition, however, is not borne β ¼ ð Þ¼ nent f 0 limt=T→0 rt is not universal. out in the data (Fig. 2D). Despite significant scatter, it can be seen that many of the distributions are linear on a log-log plot and, Dynamics of Activation of Spontaneous Behavior Exhibit Scale thus, lack a characteristic scale. The linear regimes extend up to a Invariance. We studied the distribution of resting times by using fi different cutoff for each location. rigorous statistical techniques (21) and con rmed that the distri- If the spontaneous activation of behavior is probing the dy- bution is consistent with power law up to a cutoff that occurs on the namics intrinsic to the nervous system, we would expect these scale of 1,000 s in all mice studied (SI Materials and Methods, dynamics to be independent of location. Then the dwell times at a Statistical Analysis of the Distribution: Maximum Likelihood location visited N times should correspond to N extractions from Estimation). Fig. 1B shows the cumulative distribution of dwell a power law distribution of dwell times (Fig. 1). We hypothesized times for nine wild-type male mice. The distributions are linear on therefore that the apparent differences between distributions a log-log scale and have similar slopes. The dynamics of activation fl fi of spontaneous behavior are thus scale-invariant even in the ab- shown in Fig. 2D re ect nite sample size effects. One prediction of this hypothesis is that the total time spent at a particular lo- sence of any priorities and goals (2, 3) or complex distribution of 1/β resources (11–13), just as predicted by our theory. We note that cation should scale as N . Testing this prediction, however, is not the resting time intervals in mice maintained in a simple and practical because the distribution of sums of numbers drawn from unchanging environment and in humans operating in a totally a power law distribution is itself a power law (SI Materials and unconstrained environment (5) scale similarly. In both cases, the Methods, Dependence of Total Time Spent on a Number of Visits). scaling exponent β is approximately the same (∼0.7) (Table S1), Thus, large differences between estimates of total time spent at ð Þ¼ two locations visited the same number of times are expected. resulting> in the resting time distribution, p(t), scaling as p t − dp ðtÞ ∝ −ð1þβÞ ∝ −1:7 To demonstrate that the differences among distributions shown dt t t . It is also noteworthy, that the dynamics of foraging behaviors in diverse species observed in their natural in Fig. 2D can be accounted for by finite size effect, it is useful to environments have also been shown to scale similarly (e.g., refs. define a characteristic time scale (tN) determined by the number of 12 and 13). The cutoff times reflecting the macroscopic time scale visits for each location and use this time scale to rescale the loca- are quite different depending on the species. This observation may tion-specific distributions. Note that the mean dwell time diverges

Proekt et al. PNAS Early Edition | 3of6 Downloaded by guest on October 2, 2021 Fig. 2. Dynamics of behavior are invariant with respect to location and novelty of the environment. Data in A–E were all collected from the same mouse. Fraction of time spent (A), and number of visits (B) to different locations inside the cage for a single mouse recorded continuously for 22 d at steady state ﬁ ¼ ht2 i beginning at least 2 wk after rst being introduced into the cage. Peaks indicate clear location preferences. (C) Estimate of a characteristic time tN hti for different locations. (D) Cumulative distributions of dwell times at each individual location. Distributions from locations visited more than 10,000 times are shown in red; distributions from all other locations are shown in blue. Linear regimes, indicating power law distributions, are observed extending upto

different cutoffs depending on the location. (E and F) Location-speciﬁc distributions rescaled by tN. Only locations visited more than 10,000 times are shown because relatively small sample sizes lead to unreliable estimates of tN. β for this particular mouse was estimated to be 0.69. E shows data collected at steady state. The data in E is replotted in F to illustrate the collapse of the dynamics observed during the initial habituation phase (green) lasting 3 d starting at the initial introduction into the cage onto the steady-state behavior (red). The collapse indicates that the dynamics operate in a largely location-independent manner and are not affected by the relative novelty or familiarity of the environment.

in the large N limit and, therefore, is not a suitable characteristic The data in Fig. 2 A–E were recorded at “steady state” after time scale (see below). the initial habituation period lasting at least 2 wk. Upon initial We begin by determining the expected value for the largest exposure to a new environment, animals display higher levels of dwell time observed after N extractions from a cumulative power activity and, through the process of exploration, establish their law distribution characterized by the scaling exponent β by re- location preferences (22). Distinction between the initial habit- quiring that there is at least one dwell time greater than tN. uation phase and the subsequent steady-state behavior is com-

1 monly made in behavioral analysis. However, Fig. 2F shows that > β Np ðtN Þ ∼ ⇒ tN ∼ τN after appropriate rescaling, location-specific distributions from the initial habituation phase collapse onto those observed at This equation suggests that the empirical cumulative distribution fi steady state. Therefore, the dynamics of spontaneous activity are is given by the standard nite-size scaling form largely independent of location and relative novelty or familiarity fl τβ τ β of the environment and, thus, likely re ect the underlying in- > ð j Þ¼ ð = Þ¼ ^ð = Þ τ; trinsic dynamics of the nervous system. pempir t N Fsamp t tN F t tN for t t tN Note that although rest intervals can be objectively defined based on the data, the definition of activity intervals is not straightfor- where F ðzÞ¼zβF^ðzÞ approaches 1 when z approaches 0 and samp ward. The mouse can be said to be at rest if it has been observed to decreases rapidly at large z. Note that F is simply related to samp be at the same location for a time interval greater than some sampling in contrast to F(t/T) in Eq. 3. The k-th moment, < k> threshold Tth. The distribution of activity intervals can therefore be denoted as t , of the empirical probability distribution (for fi > β de ned as the distribution of times between two consecutive resting k ) is given by ≥ episodes Tth. Clearly, the duration of activity intervals will depend Z ∞ > dp ðtjNÞ t2 on the particular choice of Tth. We predict that the cumulative k empir ¼ k ∝ k−β ⇒ τ ≡ ∝ : t dt t tN charac h i tN distributions of activity intervals should collapse after rescaling the 0 dt t time axis by Tth . Unlike in Fig. 2, however, only the time but not the Note that this equation implies that all moments k > β diverge. cumulative distribution is scaled by a function of Tth, and the scaling β Because β < 1 for all mice studied, the mean and all higher collapse is independent of (SI Materials and Methods, Waiting moments are divergent. Thus, the appropriate characteristic Time Distributions). As predicted, the raw waiting time distributions time, τcharac, at each site is given by measuring the ratio of the (Fig. 3A) collapse onto a universal curve after rescaling (Fig. 3B). ht2i The same applies to the dataset taken as a whole and to that at moments for the dwell times at that site (Fig. 2C). If the hti a single location (Fig. 3C). Thus, much like the distribution of rest, dynamics are indeed location-invariant, we predict that plots of the distribution of activity durations exhibits scaling collapse. Note τβ > ð j Þ : =τ charac pempir t N vs t charac should collapse. We demonstrate that, remarkably, our derivation is equally valid for spontaneous that this prediction is in good accord with the data in Fig. 2E. mouse behavior and for the distribution of time intervals of after-

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.1206894109 Proekt et al. Downloaded by guest on October 2, 2021 Fig. 3. Activity intervals exhibit scaling collapse. (A) Cumulative waiting time (t) distributions between rest intervals t ≥ Tth (between 30 and 1,000 s) for a single wild-type mouse observed continuously for ∼22 d at steady-state conditions. (B) Waiting time distributions rescaled by Tth.(C) Cumulative waiting time distribution between events t ≥ Tth observed at a single location (most visited location was used to improve sampling). The distributions collapse for thresholds between 30 and 1,000 s. Note that the x axes in plots B and C are on different scales, because plot C considers events occurring at a single location. Thus, the same dynamics operate at the level of the whole environment and at the level of a single location.

shocks greater than a certain threshold magnitude after a major separation of timescales is abolished). Thus, it is easily seen that quake, as observed by Bak et al. (23). The essential requirement the same intrinsic dynamics can give rise to both types of animal again is the emergence of a characteristic macroscopic time scale foraging. For the dynamics of human communication (2, 3), the arising from relaxational processes in the earth. macroscopic time scale is imposed by the emails or letters being deleted or being no longer readily accessible after a certain time. Dependence of the Dynamics on the Phase of the Light:Dark Cycle. Indeed, very few of us reply to every letter we receive (e.g., ref. 3). Behavior of most animals, including humans, is profoundly in- ﬂ Motivated by work on equilibrium critical phenomena, differ- uenced by the circadian cycle manifested as consolidated periods ences in the scaling exponents between seemingly diverse behav- of sleep and wakefulness allocated preferentially to the speciﬁc iors such as human communication (2, 3) and animal foraging (e.g. phases of the light:dark cycle. Mice are nocturnal and have con-

ref. 12) were taken to imply fundamental differences in the un- NEUROSCIENCE solidated periods of sleep during the light phase (24) expressed both in the increase in the total sleep time and durations of in- derlying processes. However, here we show that unlike in equi- dividual sleep episodes (25). This observation suggests that, during librium critical phenomena, the scaling exponent is free to assume the light phase, the dynamics of rest/activity ﬂuctuations should be a broad range of values, allowing one to unify seemingly disparate dominated by the dynamics of sleep. Thus, we predict that the observations within a single framework. power law should be disrupted during the light phase (SI Materials We note that although our framework allows for an arbitrary scaling exponent (β + 1), most of the scaling exponents in Table

and Methods, Generalization to Three States: Effects of the Light: PHYSICS Dark Cycle). Data in Fig. 4 confirm this prediction. During the 1 lie in the range between 1 and 3. This observation raises the light phase, there are more long rest durations than would be question whether there is some evolutionary advantage to expected for power-law distributed data. a particular value of the exponent (e.g., ref. 28). This notion is consistent with our observations that the scaling exponent is ap- Discussion proximately the same for different individual mice. Alternatively, Our framework has a straightforward physical and biologically a much simpler explanation for the observed range of scaling plausible interpretation. When β > 0, the probability of transition exponents needs to be entertained. The lower cutoff of the range is out of the resting state, ∼β/t, decreases as a function of time. If imposed by the fact that distributions where β +1< 1 are non- this tendency acted unopposed, the longer the system was at rest, normalizable. However, if β +1> 3, then the power law behavior the less likely it will be to activate behavior. The existence of a is characterized by a steep decay and, thence, is difficult to es- macroscopic characteristic time T, however, provides an opposing tablish over a significant range of time. 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 aggre- gation of motor acts, regardless of their specifics, is governed by scale-free dynamics suggests the existence of an elementary un- differentiated 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 Fig. 4. Scale invariance is disrupted during the light phase of the light:dark foraging resembles a Lévy flight (i.e., a random walk with indi- cycle. Cumulative distribution of rest intervals analyzed separately for the vidual steps sampled from a power law distribution) when food is dark (red) and light (blue) phases of the light:dark cycle. The rest interval was defined as the number of consecutive seconds without locomotion. For scarce, but when food is abundant, foraging resembles a con- the light phase, power law behavior was consistent with the data for only ventional random walk (12). In the latter case, the animal is likely one mouse, whereas, for the dark phase, a power law was consistent with to stumble upon food before it explores the dynamic range be- the data for eight mice according to the methodology outlined in SI Mate- tween the microscopic and macroscopic time scales (i.e., the rials and Methods.

Proekt et al. PNAS Early Edition | 5of6 Downloaded by guest on October 2, 2021 One may speculate that the dynamics of behavior evolved to was placed either in the middle or one of the sides of the cage. The data confer a selective survival advantage. Scale-free dynamics offer were recorded onto a personal computer (Dell) by using VersaMax analyzer several fundamental advantages. The ability to balance conflicting software version 3.41. demands such as the need for rest along with the ability to readily The total distance traveled (TOTDIST parameter in VersaDAT) was sampled respond to salient stimuli (29) over a range of physiologically every second. Rest interval was defined as the number of consecutive seconds relevant time scales and the ability to produce the greatest rep- during which the distance traveled was zero. ertoire of responses with efficient coding of information (30–32) We also studied the entire behavioral sequence consisting of a set of ... are all simultaneously fulfilled by scale-free dynamics. Interestingly, observations {(l1,d1) (ln,dn)} where l denotes location inside the cage and d denotes the dwell time at that location. Because the position of the mouse scale-free behavior arises quite naturally without specifically in- was recorded by using a grid, the mouse is seen as moving on a lattice where voking prioritization of tasks, complex interactions among indi- each subsequent location (l ) is usually in the immediate vicinity of the viduals in social dynamics, or complex environmental structure in n+1 preceding location (ln). The time spent in motion between two immediately foraging. adjacent locations is negligible. To construct this dataset, we sampled beam It is known that systems consisting of many interacting non- breaks at 20-ms intervals and defined the mouse location as a center of a linear elements exhibit emergent behavior that is not present in rectangle formed by the first and last beam broken by the mouse in each any one of the constituent elements (32–34). Our work demon- direction. Dwell time was defined as the time during which the mouse lo- strates that the essential ingredient for scale-invariant behavior is cation stayed constant. the existence of a characteristic macroscopic time scale rather Note, that the TOTDIST parameter reflects only directed locomotion and than the specifics of the mechanisms or models that give rise to it. disregards back and forth motion. Thus, rest intervals correspond to times between consecutive episodes of locomotion. In contrast, dwell times cor- Materials and Methods respond to time intervals between any detectable motion which allows for All animal procedures were approved by The Rockefeller University’s Animal greater temporal resolution. The distributions of rest intervals and dwell Care and Use Committee in accordance with the Animal Welfare Act and the times are statistically identical. Department of Health and Human Services Guide for the Care and Use of As controls, we recorded from empty cages and cages containing an im- Laboratory Animals. mobile object approximately the size of a mouse for 2 d continuously. No Spontaneous activity was measured as described (35). Adult C57 mice were activity at all was recorded in either one of these cases. housed individually inside a VersaMax monitor (Accuscan Instruments) All analysis was performed off line by using custom-written programs in × × consisting of an acrylic cage (18 cm 29 cm 13 cm) equipped with hori- Mathematica (Wolfram Research). The details of the statistical analysis and zontal infrared beams and sensors spaced 2.54 cm (1 inch) apart in the mathematical derivations can be found in SI Materials and Methods. horizontal plane. Each VersaMax monitor was placed inside a larger wooden chamber with its dedicated light and ventilation systems used to minimize ACKNOWLEDGMENTS. We thank Ana C. Ribeiro for providing the initial the potential for transmission of sounds and other signals between animals. dataset that motivated this study and Cori Bargmann and Eric Siggia for Animals were maintained on the 12 h:12 h light:dark cycle at constant their insightful comments. This work was supported by a Foundation for temperature of ∼22 °C. Food and water were available ad libitum. Food Anesthesia Education and Research grant (to A.P.) and a grant from the pellets were randomly scattered on top of the bedding. The water bottle Cassa di Risparmio di Padova e Rovigo foundation (to A.M.).

1. Barabasi AL (2010) Bursts: The Hidden Pattern Behind Everything We Do (Penguin, 19. Stanley HE (1999) Scaling, universality, and renormalization: Three pillars of modern New York, NY). critical phenomena. Rev Mod Phys 71:S358–S366. 2. Barabási AL (2005) The origin of bursts and heavy tails in human dynamics. Nature 20. Cox DR (1962) Renewal Theory (Spottiswoode Ballantyne and Co. Ltd., London and – 435:207 211. Colchester). 3. Oliveira JG, Barabási AL (2005) Human dynamics: Darwin and Einstein correspondence 21. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. – patterns. Nature 437:1251 1251. SIAM Rev 51:661–703. 4. Brockmann D, Hufnagel L, Geisel T (2006) The scaling laws of human travel. Nature 22. Dvorkin A, Benjamini Y, Golani I (2008) Mouse cognition-related behavior in the 439:462–465. open-field: Emergence of places of attraction. PLOS Comput Biol 4:e1000027. 5. Song CM, Koren T, Wang P, Barabasi AL (2010) Modelling the scaling properties of 23. Bak P, Christensen K, Danon L, Scanlon T (2002) Unified scaling law for earthquakes. human mobility. Nat Phys 6:818–823. Phys Rev Lett 88:178501. 6. Zhou T, Kiet HAT, Kim BJ, Wang BH, Holme P (2008) Role of activity in human dy- 24. Naylor E, et al. (2000) The circadian clock mutation alters sleep homeostasis in the namics. Epl-. Europhys Lett 82:28002. – 7. Bazzani A, Giorgini B, Rambaldi S, Gallotti R, Giovannini L (2010) Statistical laws in mouse. J Neurosci 20:8138 8143. urban mobility from microscopic GPS data in the area of Florence. J Stat Mech 210: 25. Chemelli RM, et al. (1999) Narcolepsy in orexin knockout mice: Molecular genetics of – 05001. sleep regulation. Cell 98:437 451. 8. Harder U, Paczuski M (2006) Correlated dynamics in human printing behavior. Physica 26. Mullin FJ, Kleitman N (1938) Variations in threshold of auditory stimuli necessary to A 361:329–336. awaken the sleeper. Am J Physiol 123:477–481. 9. Dunki RM, Ambuhl B (1996) Scaling properties in temporal patterns of schizophrenia. 27. Pfaff D, Ribeiro A, Matthews J, Kow LM (2008) Concepts and mechanisms of gener- Physica A 230:544–553. alized central nervous system arousal. Ann N Y Acad Sci 1129:11–25. 10. Nakamura T, et al. (2007) Universal scaling law in human behavioral organization. 28. Pfaff D, Banavar JR (2007) A theoretical framework for CNS arousal. Bioessays 29: Phys Rev Lett 99:138103. 803–810. 11. Ramos-Fernandez G, et al. (2004) Levy walk patterns in the foraging movements of 29. Viswanathan GM, et al. (1999) Optimizing the success of random searches. Nature – spider monkeys (Ateles geoffroyi). Behav Ecol Sociobiol 55:223 230. 401:911–914. 12. Humphries NE, et al. (2010) Environmental context explains Lévy and Brownian 30. Schneidman E, Berry MJ, 2nd, Segev R, Bialek W (2006) Weak pairwise correlations – movement patterns of marine predators. Nature 465:1066 1069. imply strongly correlated network states in a neural population. Nature 440: 13. Sims DW, et al. (2008) Scaling laws of marine predator search behaviour. Nature 451: 1007–1012. 1098–1102. 31. Beggs JM (2008) The criticality hypothesis: How local cortical networks might opti- 14. Maye A, Hsieh CH, Sugihara G, Brembs B (2007) Order in spontaneous behavior. PLoS mize information processing. Philos Transact A Math Phys Eng Sci 366:329–343. ONE 2:e443. 32. Kinouchi O, Copelli M (2006) Optimal dynamical range of excitable networks at 15. Reynolds AM, Frye MA (2007) Free-flight odor tracking in Drosophila is consistent – with an optimal intermittent scale-free search. PLoS ONE 2:e354. criticality. Nat Phys 2:348 352. fi 16. Nakamura T, et al. (2008) Of mice and men—universality and breakdown of behav- 33. Hop eld JJ (1982) Neural networks and physical systems with emergent collective – ioral organization. PLoS ONE 3:e2050. computational abilities. Proc Natl Acad Sci USA 79:2554 2558. 17. Anteneodo C, Chialvo DR (2009) Unraveling the fluctuations of animal motor activity. 34. Bak P (1997) How Nature Works: The Science of Self-Organized Criticality (Copernicus, Chaos 19:033123. New York). 18. Mora T, Bialek W (2011) Are biological systems poised at criticality? J Stat Phys 144: 35. Garey J, et al. (2003) Genetic contributions to generalized arousal of brain and be- 268–302. havior. Proc Natl Acad Sci USA 100:11019–11022.

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