Heterogeneity dominates irreversibility in random Markov models

TL;DR

We address how irreversibility and criticality manifest in random discrete-time Markov models by introducing a two-parameter ensemble with heterogeneity $ε$ and log-asymmetry $γ$. Using random-matrix theory and spectral analysis, we identify a critical locus $ε_c(γ,N)$ separating short- and long-memory phases and derive how entropy production and predictive information depend on the parameters; the approach is then applied to human fMRI and EEG data via maximum-likelihood inference, revealing that real brain dynamics lie near the predicted critical locus with notable subject-to-subject variability. The results show that heterogeneity largely controls observable behavior, while nonequilibrium signatures are intricately intertwined with criticality, implying that scalar irreversibility metrics alone are insufficient to infer time-irreversibility. Across fMRI and EEG, the data exhibit a super-universal placement near the critical line, implying low-dimensional constraints on brain dynamics that transcend measurement modality. These findings unify criticality and nonequilibrium measures in a single framework and provide a principled null model for interpreting complex biological time series.

Abstract

We introduce a two-parameter ensemble of random discrete-time Markov models that simultaneously captures critical slowing down and broken detailed balance. Extending a previously studied heterogeneous Markov ensemble, we incorporate correlations between forward and backward transition rates through a single asymmetry parameter $γ$, while heterogeneity is controlled by $ε$. Using results from random matrix theory, we identify a critical locus $ε_c(γ,N)$ at which relaxation times diverge and spectral universality breaks down. We characterize the behavior of entropy production, predictive information, and relaxation dynamics across the ensemble, showing that many observables depend strongly on heterogeneity but only weakly on asymmetry, except near the symmetric limit. Applying maximum-likelihood inference to human fMRI and EEG data, we find that both modalities operate near the predicted critical locus and occupy a similar region of the $ε-γ$ plane, supporting a super-universality of human brain dynamics. While ensemble averages are well captured by the null model, empirical data exhibit substantially enhanced variability, indicating subject-specific structure beyond random expectations. Our results unify criticality and nonequilibrium measures within a single framework and clarify their intertwined role in the analysis of complex biological dynamics.

Figures (10)

• Figure 1: Predicted critical locus at which long relaxation times emerge, at indicated $N$.
• Figure 2: Transition rate spectra at varying $\gamma$ along the y-axis ($\gamma = +0.9, 0, -0.9$ from top to bottom) and $\epsilon$ along the x-axis ($\epsilon = 10^{-3}, 10^{-1}, 10^{1}$ from left to right). Here $N=32$.
• Figure 3: Expected value of relaxation time $\tau = -1/\log(|\lambda|)$ as a function of $\epsilon$, at various $\gamma$ (see Fig.\ref{['figentropyprod']} for labels), and $N=64$.
• Figure 4: Expected value of entropy production rate $\dot \Sigma$ in units of the discrete time step as a function of $\epsilon/\epsilon_c(\gamma)$, at indicated $\gamma$ and $N=64$ (solid). Theoretical curves valid for $\epsilon \gg \epsilon_c$ are indicated (dashed).
• Figure 5: Predictive information as a function of $\epsilon/\epsilon_c(\gamma)$, at various $\gamma$ from $-0.9$ (light blue) to $+0.9$ (pink); symbols as in Fig.\ref{['figentropyprod']}, and $N=64$.