Generalized Glauber Dynamics for Inference in Biology

Large interacting systems in biology often exhibit emergent dynamics, such as coexistence of multiple timescales, manifested by fat tails in the distribution of waiting times. While existing tools in statistical inference, such as maximum entropy models, reproduce the empirical steady-state distributions, it remains challenging to learn dynamical models. We present a novel inference method, called generalized Glauber dynamics. Constructed through a non-Markovian fluctuation dissipation theorem, generalized Glauber dynamics tunes the dynamics of an interacting system, while keeping the steady-state distribution fixed. We motivate the need for the method on real data from Eco-HAB, an automated habitat for testing behavior in groups of mice under seminaturalistic conditions, and present it on simple Ising spin systems. We show its applicability for experimental data by inferring dynamical models of social interactions in a group of mice that reproduce both its collective behavior and the long tails observed in individual dynamics.

Popular Summary

Biological systems, from cellular tissues to animal groups, often exhibit collective behavior that emerges from dynamical interactions between their individual components. Recent improvements in data acquisition methods and carefully controlled experiments allow for the discovery of the rules governing the emergence of this complex behavior. However, most of these analyses explore the static properties of these inherently dynamical systems. Existing approaches for learning dynamical models often do not correctly reproduce steady-state properties. Here, inspired by the behavior of a horde of mice living in a controlled ecological habitat, we propose a method that learns the dynamical interaction rules in a way that automatically reproduces the static properties.

Specifically, we propose an inference method, termed “generalized Glauber dynamics,” that tunes the dynamics of the system while keeping the statistics of the static distribution fixed. In practice, this allows for the inference to be separated into two parts: first, inference of the steady-state distribution using maximum entropy models, and second, tuning the dynamics to match the data. We validate this approach on data from an experiment on social interactions among male mice cohabitating in an artificial environment. The inference method reproduces their collective behavior and individual dynamics.

This work provides an essential step to bridging the steady-state distribution of living systems with their collective dynamics. Our model, while general and applicable to a broad class of systems, does not solve the complete inference problem for all dynamical systems but fills a niche of interpretable dynamical models that link to the steady-state landscape.

Figure 1

Collective dynamics among social mice. (a) The Eco-HAB experimental setup (top view) with C57BL6/J male mice ($N=15$) placed in four interconnected chambers. The two chambers with food are labeled by letter $F$. The location of each mouse is recorded using mouse-embedded RFID and antennas at the edge of the tubes (indicated by black bars). (b) Example trace for the colocalization patterns of a group of 15 mice over 10 full days, consisting of alternating dark and light cycles (12 h). The red vertical lines indicate the beginning of each dark cycle. (c) Pairwise connected correlation function of mice colocalization $C_{ij}$ (with error bar computed by bootstrap). Cyclically shuffled data show no correlation, while data shuffled within the same day show a strongly reduced correlation. (d) Autocorrelation function for the number of mice in a given box $C_n(t)$ as a function of time difference, computed from the mice position between $13:00$ and $19:00$ each day, a period of the intensified activity chosen for the presented analysis. Error bars are the standard error from the mean across 10 days. (e) Mean waiting time for each mouse, defined to be the time a mouse spent staying in a given box before exits. Each dot indicates a different day. (f) Distribution of waiting times normalized by their mean for each mice. Distributions collapse across all mice, and decay slower than exponentially, indicating the existence of long timescales.

Figure 2

Glauber dynamics based on the inferred steady-state distribution, Eq. (1), fails to predict the longtime dynamics of mice. (a) The normalized mean transition rate is well reproduced by Glauber dynamics with the parameters of the steady-state model, but not by Metropolis-Hasting dynamics. (b) However, Glauber dynamics with the parameters of the steady-state model do not reproduce the long tails of the waiting time distribution. The dynamics are also not reproduced by a nonparametric estimation of the transition rates (see SM, Fig. S3 [53]).

Figure 3

Toy models with generalized Glauber dynamics. (a) Schematics of the GGD. In addition to classical couplings $J_{ij}$, by considering coupling to an oscillator bath, the observed degrees of freedom are coupled to their own memory [through kernel ${\mathrm{\Gamma }}_i(t)$], to that of their neighbors [through kernels $G_{ij}(t)$], as well as to noise sources ${\xi }_i(t)$ correlated across time and variables. Panel (b) shows for a single Ising spin that adding an exponentially decaying memory kernel can create long tails in the waiting distribution. (c) A single four-state variable with memory can create a bias in the tendency to continue transitioning in the same direction as in the previous transition, versus going back to the previous state. (d) In a multiple spin system, the dynamics can depend on the history of other spins, illustrated by 10 Ising spins arranged in a loop.

Figure 4

Example of GGD inference with an expectation-maximization (EM) algorithm on a single Ising spin under an oscillating field $h(t)=\sin (0.15t)$. The memory kernel is $\mathrm{\Gamma }(t)=A\exp (-t/\tau )$, with true parameters $A=0.8$, $\tau =19.5\mathrm{s}$, and the baseline transition rate $\mu =4{\mathrm{s}}^{-1}$. The simulation was conducted for a duration of $T=300\mathrm{s}$, using a time step of $\mathrm{\Delta }t=0.1\mathrm{s}$. (a) The combination of $h(t)$, the noise $\xi$, and the effective local field due to history of spin flips, $h_{\mathrm{mem}}$ [the second and third terms of Eq. (2)] generates the sample spin trajectory $s_t$. (b) Given the spin trajectory $\sigma (t)$ and $h(t)$, the EM algorithm recovers an ensemble of possible realizations of the hidden noise $\xi (t)$. With parameters inferred using the EM algorithm, the simulated trajectories recover the waiting time distribution (c) and the autocorrelation decay (d) as the data. The envelope of the curve is the standard deviation.

Figure 5

The generalized Glauber dynamics on top of the static maximum entropy model on Eco-HAB data reproduces the long tail of both the shape (a) and the mean (b) of the waiting time distributions. The Pearson correlation coefficient $\rho$ is given in the plot.