Decomposing Non-Markovian History Dependence

TL;DR

The paper tackles the challenge of quantifying non-Markovian history dependence in biological systems by introducing an information-theoretic decomposition of history dependence: the total dynamical information $I_{tot}$ equals the sum of nonnegative contributions from each history order, $I_{tot} = I_1 + I_2 + \cdots$, with $I_k = h_{k-1} - h_k$. This framework is validated in minimal non-Markovian models and then applied to prolonged fruit-fly behavior recordings, revealing that non-Markovian dependencies scale invariantly across timescales from fractions of a second to minutes, yet the overall strength of non-Markovianity peaks at an intermediate, approximately $7.4$-second timescale. The results demonstrate a principled method to disentangle and quantify multi-order historical dependencies independent of autocorrelation structure, with implications for understanding memory and coarse-graining phenomena in living systems. The approach provides a path toward bounding and approximating long-range dependencies in real data and offers a general, model-agnostic tool for analyzing history dependence across scales. $I_{tot}$, $I_k$, and related quantities are computed from data with finite-data corrections and robust to coarse-graining, enabling applications to other biological contexts beyond fly behavior.

Abstract

Non-Markovian stochastic processes are ubiquitous in biology. Nevertheless, we lack a general framework for quantifying historical dependencies. In this Letter, we propose an information-theoretic approach to decompose history dependence in systems with non-Markovian dynamics, quantifying the information encoded in dependencies of each order. In minimal models of non-Markovian dynamics, we show that this framework correctly captures the underlying historical dependencies, even when autocorrelations do not. In prolonged recordings of fly behavior, we find that the scaling of non-Markovian dependencies is invariant across timescales from fractions of a second to minutes. Despite this invariance, the overall amount of non-Markovian information is non-monotonic, suggesting a unique timescale on which historical dependencies are strongest.

Figures (6)

• Figure 1: Decomposing history dependence in simple stochastic processes. (a) Non-Markovian dynamics are defined by the conditional distribution $p(x_t|x_{t-1},...)$. (b) Increasing knowledge of the past leads to a hierarchy of entropies $h_k$. The total dynamical information $I_\text{tot}$ therefore decomposes into a sum of non-negative contributions $I_k$ from different orders $k$. (c) Schematic of minimal dynamics in which the binary state $x_t$ performs a logical function on $x_{t-1}$ and $x_{t-2}$ with probability of success $\alpha$. (d-e) Dynamical information $I_k$ normalized by the total information $I_\text{tot}$ (d) and autocorrelations $C(k)$ (e) for different logical functions at steady-state. We set $\alpha=0.9$.
• Figure 2: Autocorrelations and dynamical information in a non-Markovian copying process. (a) At each step $t$, the state $k$ steps back is selected with probability $\rho(k)$ and then copied with probability $\alpha$leighton2025companion. (b) Different history dependence distributions $\rho(k)$, including binomial [$\rho(k)={N-1\choose k-1}q^{k-1}(1-q)^{N-k}$ with $N=10,q=0.5$], exponential [$\rho(k)\propto (1-q)^{k-1}$ with $q=0.7$], and power-law [$\rho(k)\propto k^{-s}$ with $s=4,1.5$]. (c-d) Autocorrelations $C(k)$ (c) and dynamical information $I_k$ (d) for copying processes with different history dependencies $\rho(k)$. In all panels, insets display log-log scales and dynamics are defined with $\alpha=0.9$.
• Figure 3: Non-Markovian dependencies in fly behavior. (a) Autocorrelations $C(\tau)$ as a function of time lag $\tau = k\Delta t$. Solid line: mean across all flies. Shaded region: one standard deviation. Dashed line: power-law fit. (b) Dynamical information $I_k$ versus time lag $\tau$ for different levels of temporal coarse-graining $M$. (c) As a function of the dimensionless lag $k$, dynamical information collapses onto a single curve. Dashed line: power-law fit. (d) Markovian ($I_1$) and non-Markovian ($I_{>1} = \sum_{k>1}I_k$, here summed to $k=10$) information versus coarse-graining factor $M$. Inset: For each order $k$, we compute the timescale $\tau^*$ at which $I_k$ is maximized (Fig. \ref{['fig:SI_Fig_OrdervsCG']}). Gold shaded region: maximum and minimum possible $\tau^*$ values. Horizontal dashed line: mean value $\tau^* = 7.4$s.
• Figure 4: Autocorrelation $C(k)$ versus lag $k$ for different levels of temporal coarse-graining. Dashed line: power-law fit for the fine-grained autocorrelations ($M = 1$).
• Figure 5: Dynamical information $I_k$ versus time lag $\tau$ for different levels of temporal coarse-graining $M$. Dashed lines: power-law fits for $k\geq3$. Inset: power-law exponents $\gamma$ versus coarse-graining factor $M$ with error bars reflecting standard errors and dashed line indicating the mean of the five largest exponents.