Hierarchy of chaotic dynamics in random modular networks

TL;DR

The paper investigates how hierarchical modular connectivity shapes chaotic dynamics in randomly connected neural populations. Using dynamical mean-field theory (DMFT) and simulations, it derives coupled order-parameter equations for level-wise variances $q_j(t)$ and analyzes the maximal Lyapunov exponent $\lambda_{max}$ across macroscopic and microscopic regimes, revealing high-dimensional (microscopic) chaos, low-dimensional (macroscopic) chaos, and a multiscale crossover where different chaos diagnostics diverge. It shows that chaos can be attenuated by adding noise to strongly modular connectivity or by introducing modular structure into predominantly random networks, and that a loosely balanced multilevel hierarchy drives the system toward the edge of chaos. The multilevel generalization suggests a general mechanism by which hierarchical organization enhances the robustness of critical-like dynamics, with implications for information flow across brain hierarchies. An adaptation algorithm balancing activity across levels demonstrates that maintaining near-equal level contributions naturally positions the system near criticality across a range of settings.

Abstract

We introduce a model of randomly connected neural populations and study its dynamics by means of the dynamical mean-field theory and simulations. Our analysis uncovers a rich phase diagram, featuring high- and low-dimensional chaotic phases, separated by a crossover region characterized by low values of the maximal Lyapunov exponent and participation ratio dimension, but with high values of the Lyapunov dimension that change significantly across the region. Counterintuitively, chaos can be attenuated by either adding noise to strongly modular connectivity or by introducing modularity into random connectivity. Extending the model to include a multilevel, hierarchical connectivity reveals that a loose balance between activities across levels drives the system towards the edge of chaos.

Figures (8)

• Figure 1: Visualizations of the weight matrix (top row), its eigenvalues (second row), and steady-state activities of five sample neurons from a shared population (third row) and five sample populations (bottom row) in four phases predicted by the mean-field theory: quiescent (${\sigma=\sigma_{\mu}=0.5}$), $\mu$ (microscopic chaos, ${\sigma=4, \sigma_{\mu}=0.5}$), $\mu+M$ (multiscale chaos, ${\sigma=4, \sigma_{\mu}=6}$), and $M$ (macroscopic chaos, ${\sigma=1, \sigma_{\mu}=5}$). Red dashed lines in the eigenvalue spectra represent the unit circle in the complex plane.
• Figure 2: The mean-field phase diagram. Lines represent transition lines: $\sigma^*_{\mu}$ (solid) and $\sigma^{EoHC}_{\mu}$ (dashed). Phases are denoted by symbols $M$ (macroscopic chaos) and $\mu$ (microscopic chaos). Heat maps represent the values of $\lambda_{coherent}$(left), $\lambda_{random}$(center), and $\lambda_{max}$(right).
• Figure 3: Statistics of the typical autonomous dynamics as functions of $\sigma_{\mu}$ for $\sigma=3.5$(left) and as functions of $\sigma$ for $\sigma_{\mu}=8$(right). Dots and dashed lines correspond to results of computer simulations and predictions of the mean-field theory, respectively. Vertical gray lines represent transition lines: $\lambda_{coherent}=0$ (solid) and $\lambda_{random}=0$ (dashed). Top: Total ($q$), macroscopic ($q_m$), and microscopic ($q-q_m$) variances. Center: Maximal Lyapunov exponents computed either directly from simulations or as within-subspace theoretical predictions. Bottom: Normalized dimensionality of the steady-state neural activity manifold as measured by the covariance matrix (PR dimension) or Lyapunov exponents (KY dimension).
• Figure 4: Evolution of order (top) and control (bottom) parameters during the adaptation process in networks with two levels ($P_1=P_2=100$). Five columns correspond to different seeds (i.e., independent realizations of the weights and initial conditions). Dashed lines denote the desired activity levels (top) and the corresponding control parameters, as predicted by the mean-field equations (bottom). Other parameters: $\eta=0.2$, $\hat{q}_L=0.8$.
• Figure 5: Same as Fig. \ref{['fig:SM_adaptation_L2']} but in networks with three levels ($P_1=P_2=P_3=22$).