Fluctuating Landscapes and Heavy Tails in Animal Behavior

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Fluctuating Landscapes and Heavy Tails in Animal Behavior

Antonio Carlos Costa, Gautam Sridhar, Claire Wyart, and Massimo Vergassola

PRX Life 2, 023001 – Published 2 April 2024

Abstract

Animal behavior is shaped by a myriad of mechanisms acting on a wide range of scales, which hampers quantitative reasoning and the identification of general principles. Here, we combine data analysis and theory to investigate the relationship between behavioral plasticity and heavy-tailed statistics often observed in animal behavior. Specifically, we first leverage high-resolution recordings of Caenorhabditis elegans locomotion to show that stochastic transitions among long-lived behaviors exhibit heavy-tailed first-passage-time distributions and correlation functions. Such heavy tails can be explained by slow adaptation of behavior over time. This particular result motivates our second step of introducing a general model where we separate fast dynamics on a quasistationary multiwell potential, from nonergodic, slowly varying modes. We then show that heavy tails generically emerge in such a model, and we provide a theoretical derivation of the resulting functional form, which can become a power law with exponents that depend on the strength of the fluctuations. Finally, we provide direct support for the generality of our findings by testing them in a C. elegans mutant where adaptation is suppressed and heavy tails thus disappear, and recordings of larval zebrafish swimming behavior where heavy tails are again prevalent.

Received 23 January 2023
Accepted 6 March 2024
Corrected 15 April 2024

DOI:https://doi.org/10.1103/PRXLife.2.023001

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Animal behavior, social interactions & emergent phenomena Biological movement Emergent biological functions from complex interactions Locomotion

Brownian dynamics First passage problems Fokker–Planck equation Langevin equation Markovian processes Non-Markovian processes Stochastic analysis

Physics of Living SystemsStatistical Physics & ThermodynamicsNonlinear Dynamics

Corrections

15 April 2024

Correction: Support information in the Acknowledgment section was incomplete and has been fixed.

Authors & Affiliations

Antonio Carlos Costa ^1,*, Gautam Sridhar ², Claire Wyart ², and Massimo Vergassola ¹

¹Laboratoire de Physique de l'Ecole normale supérieure (ENS), Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
²Sorbonne University, Paris Brain Institute (ICM), Inserm U1127, CNRS UMR 7225, Paris, France

^*Present address: Sorbonne University, Paris Brain Institute (ICM), Inserm U1127, CNRS UMR 7225, Paris, France.

Article Text

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Issue

Vol. 2, Iss. 2 — April - June 2024

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Images

Figure 1
Power-law distributions observed across species. (a) Distribution of step lengths $ℓ$ for an individual basking shark (Cetorhinus maximus) (adapted from Ref. [19]). The inset shows the probability density $f (ℓ)$ in a log-log scale, and the power-law fit $ℓ^{- μ}$ with $μ = 2.3$ . Assuming a constant speed, the time spent in a step would also be distributed as $f (t) \approx t^{- μ}$ . (b) Distribution of the time between decisions in a choice task by Sprague Dawley rats (adapted from Ref. [20]). The inset shows the same curve in log-log scale, and the power-law fit $t^{- μ}$ with $μ = 1.97$ . (c) Distribution of the duration of clockwise (gray) and counterclockwise (black) rotations of a single Escherichia coli motor (adapted from Ref. [21]). The inset shows the complementary cumulative distribution function (black) with a superposed power law $t^{- 1}$ (gray), corresponding to a probability density $\sim t^{- 2}$ .
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Figure 2
A reduced-order model of C. elegans foraging dynamics. (a) From video imaging data, we measure local tangent angles along the body to obtain the body posture vector $θ_{t}$ as in Ref. [28]. A series of $K^{*}$ such vectors are then stacked to yield the variable $X_{K^{*}} (t)$ defined in the text, which captures short-term memory. (b) A high-fidelity Markov model of the dynamics is obtained using methods in Refs. [10, 16]. The first nontrivial eigenvector $ϕ_{2}$ of the inferred Markov chain captures the long-time dynamics of the system, as discussed in the body of the paper. We represent the high-dimensional state space $X_{K^{*}}$ through a two-dimensional (2D) UMAP embedding as in Ref. [16] (left), and color-code each point by its projection along $ϕ_{2}$ . An example 10-min-long centroid trajectory color-coded by $ϕ_{2}$ is shown on the right. The example showcases how negative and positive values of $ϕ_{2}$ correspond to forward “runs” and combinations of reversals, ventral and dorsal turns during “pirouettes.” (c) Example time series of $ϕ_{2}$ illustrating the stochastic hopping between runs and pirouettes. (d) Left: Distribution function of observing a run or a pirouette with a duration longer than $τ_{beh}, 1 - P (τ_{beh} \leq t)$ , estimated from the experimental data (black), simulations of Eq. (2), i.e., the dynamics projected onto $ϕ_{2}$ (blue), and simulations of Eq. (1), i.e., of the full unprojected model (gray). While simulations capture the sum of exponential functions (gray dashed line) that approximates the bulk of the distribution, heavy tails observed in the data are not well captured. Right: Connected autocorrelation function $C_{ϕ_{2}} (τ)$ for the data (black) and simulations of the projected (unprojected) model [blue (gray)]. Simulations fail again in predicting the long-range correlations exhibited by the data. Note that the projected and the full model yield similar results, illustrating the efficiency of our projection method. Error bars represent 95% confidence intervals bootstrapped across worms.
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Figure 3
A time-varying potential landscape captures heavy tails in C. elegans behavior. (a) The time-dependent potential landscape for the eigenmode $ϕ_{2}$ discussed in the text. As time goes on (blue to yellow), the barrier remains close to $ϕ_{2} = 0$ (black dot on the $ϕ_{2}$ axis) while the “run” well becomes deeper over time. That witnesses adaptation towards increasingly performing runs. (b) The probability of observing a run or a “pirouette” with a duration $> τ_{beh}, 1 - P (τ_{beh} \leq t)$ (left), and the connected autocorrelation function, $C_{ϕ_{2}} (τ)$ (right), as obtained from data (black), simulations of the static model in Eq. (2) [blue, same as Fig. 1], and simulations of the time-dependent model in Eq. (4) (orange). Note that the last captures heavy tails and long-range correlations observed in the data. Error bars represent 95% confidence intervals bootstrapped across worms.
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Figure 4
Emergence of heavy tails in the first-passage-time distribution of a slowly driven double-well potential. (a) Schematic of the variation of the double-well potential with $s$ (colored from blue to red; the black line represents $s = μ_{s}$ ). (b) Probability density function (PDF) of times spent in a potential well (or equivalently, first passage time distribution) from numerical simulations of Eq. (12) for different values of $τ_{s}$ and $T_{s} = T_{x} / 2$ (see Appendix pp1). When $τ_{s} \to 0$ , the potential landscape relaxes to its mean value faster than the time to escape the well, resulting in an exponential with a hopping rate corresponding to $μ_{s}$ (black line). As $τ_{s}$ approaches $T_{exp}$ , we observe a transition from exponential to power law, and in the limit of large $τ_{s}$ we obtain the power law Eq. (13) (black dashed line). (c) PDF of times spent in a potential well from numerical simulations of Eq. (12) for large $τ_{s} = 10^{3} T_{expt}$ and different values of $T_{s}$ (see Appendix pp1). As predicted, the tail of the distribution behaves as $f (t) \sim t^{- 2 - \frac{T_{x}}{2 T_{s}}}$ (colored lines) with an exponent that approaches $- 2$ as $T_{s} \to \infty$ (black dashed line).
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Figure 5
Power-law correlations and finite-size corrections in a slowly driven double-well potential. (a) Estimated nonconnected correlation function ${\tilde{C}}_{x} (τ) = 〈 x (t) x (t + τ) 〉$ (see Appendix pp1) for the position $x$ of a particle in a double-well potential driven on a timescale $τ_{s} = 10^{2} T_{expt}$ . Our prediction $C_{x} (τ) \sim τ^{- \frac{T_{x}}{2 T_{s}}}$ is validated (solid lines). Error bars represent 95% confidence intervals across 50 000 simulations. (b) Connected autocorrelation function, $C_{x} (τ)$ , directly estimated through time averages (see Appendix pp1), for a double-well potential driven on a timescale $τ_{s} = 10^{2} T_{expt}$ . Due to the existence of timescales $\sim T_{expt}$ , finite-size corrections $C_{c}$ are present and generate long-range anticorrelations in $C_{x} (τ)$ , as predicted in Appendix pp6 (solid lines). For both $C_{x} (τ)$ and $C_{c} (τ)$ we normalize the correlation functions by dividing by their value at $τ = 1 lag = 5 \times 10^{- 4} T_{expt}$ . Error bars represent 95% confidence intervals across 50 000 simulations. (c) Finite-size correction $C_{c} (τ)$ vs $T_{s}$ (see Appendix pp6). As we increase $T_{s}$ , the range of observed $ω$ grows, and so do finite-size corrections, which result in stronger anticorrelations (blue). Conversely, for small $T_{s}$ finite-size effects are negligible as the longest sampled $ω^{- 1} ≪ T_{expt}$ .
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Figure 6
Testing our theoretical predictions: loss-of-function mutation in the npr-1 gene of C. elegans ablates heavy-tailed statistics, and larval zebrafish exhibit heavy tails in their long-lived behavior. (a) Probability of observing “run” and “pirouette” states as a function of time for npr-1 mutants. Unlike wild-type N2 worms, the distribution of these states is constant over time. We estimate the fraction of time spent either performing a run or a pirouette in 3-min-long windows, and obtained bootstrapped averages across all seven worms. (b) Left: Complementary cumulative distribution function of observing a run or a pirouette with a duration $> τ_{beh}, 1 - P (τ_{beh} \leq t)$ [same as Fig. 2, left]. The distribution is well approximated by a sum of exponential functions: a longer one that corresponds to the runs and a shorter one that corresponds to the pirouettes. Unlike N2 worms, the first-passage-time distribution in npr-1 mutants is not heavy tailed. Right: Autocorrelation function $C_{ϕ_{2}}$ of the slow mode $ϕ_{2}$ inferred from the npr-1 mutant data. Unlike wild-type N2 worms, nontrivial long-range correlations are absent and instead correlations decay within a minute. (c) We collected data from Ref. [63] in which larval zebrafish are exposed to a chasing dot stimulus for 5 s every 2 min for at least 1 hour. We proceed as for C. elegans (see Appendix pp1) and find that the longest-lived dynamics $ϕ_{2}$ corresponds to a “wandering-cruising” axis, in which the fish either engages in bout sequences with large orientation changes (“wandering”), or performs sequences of smoother forward bouts (“cruising”), Fig. S7(b) [36]. Over time, we observe that the probability of the “cruising” state slowly increases over time, becoming more prevalent than the “wandering” state. We estimate the probability of being in the cruising and wandering states in 5-min windows, and show the average probability bootstrapped across the 11 fish. (d) Left: Complementary cumulative distribution function of observing cruising or wandering with a duration $> τ_{beh}, 1 - P (τ_{beh} \leq t)$ [same as Fig. 2, left]. We plot the data in black, as well as the distribution obtained from simulations with a stationary model ( $V (ϕ_{2})$ , blue) and a time-dependent model ( $V (ϕ_{2}, t)$ , orange) (see Appendix pp1 for details). As for C. elegans N2 worms foraging, we find that time-dependent parameters are required to accurately predict the tail of the distribution. Right: Autocorrelation function $C_{ϕ_{2}}$ of the slow mode dynamics obtained from the larval zebrafish data (black) as well as simulations from the stationary model (blue) and the time-dependent model (orange). Notably, we find that larval zebrafish exhibit heavy-tailed statistics due to the explicit time dependency of the behavioral dynamics.
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