Ergodicity Breaking and High-Dimensional Chaos in Random Recurrent Networks

TL;DR

The paper extends the Sompolinsky–Crisanti–Sommers framework by introducing a nonzero mean coupling $J_0$ to break balance in a random recurrent network, revealing a richer phase diagram that includes a symmetry-broken synchronous-chaos (SC) phase alongside quiescent, asynchronous-chaos (AC), and persistent-activity (PA) phases. Using dynamical mean-field theory (DMF) and simulations, it derives self-consistent equations for the mean $M$ and autocorrelation, maps the phase boundaries to SK spin-glass analogies, and develops a potential-landscape formulation to describe non-fixed-point dynamics. A key finding is ergodicity breaking in SC, quantified by an excess long-time autocorrelation $\Delta > 0$ and a decomposition of inputs into fast, ergodic fluctuations plus quenched, disorder-induced biases that create a continuum of disorder-dependent chaotic attractors. The work connects high-dimensional chaos in recurrent networks to spin-glass physics, showing striking parallels to SK but with crucial dynamical differences, and discusses implications for computation in disordered networks and broader criticality concepts in neural systems.

Abstract

The neural model introduced by Sompolinsky, Crisanti, and Sommers (SCS) nearly four decades ago has become a paradigmatic framework for studying complex dynamics in random recurrent networks. In its original formulation, with balanced positive and negative couplings, the model exhibits two phases: a quiescent regime, where all activity ceases, and a regime of ongoing irregular collective activity, termed asynchronous chaos (AC), in which state variables fluctuate strongly in time and across units but average to zero across the network. Building on recent work, we analyze an extension of the SCS model that breaks this coupling balance, yielding a richer phase diagram. In addition to the classical quiescent and AC phases, two additional regimes emerge, marked by spontaneous symmetry breaking. In the persistent-activity (PA) phase, each unit settles into a distinct, stable activation state. In the synchronous-chaotic (SC) phase, dynamics remain irregular and chaotic but fluctuate around a nonzero mean, generating sustained long-time autocorrelations. Using analytical techniques based on dynamical mean-field theory, complemented by extensive numerical simulations, we show how structural disorder gives rise to symmetry and ergodicity breaking. Remarkably, the resulting phase diagram closely parallels that of the Sherrington-Kirkpatrick spin-glass model, with the onset of the SC phase coinciding with the transition associated with replica-symmetry breaking. All key features of spin glasses, including ergodicity breaking, have clear counterparts in this recurrent network context, albeit with crucial idiosyncratic differences, highlighting a unified perspective on complexity in disordered systems.

Figures (6)

• Figure 1: Phase diagram and representative trajectories of the random recurrent network model. Panel A: Phase diagram in the $(J_0 / J, 1/gJ)$ plane, from DMF analyses and confirmed by simulations. Four dynamical regimes are shown with their order parameters: quiescent (Q) $M = 0$, $q = 0$; persistent activity (PA) $M \neq 0$, $q = 0$; asynchronous chaos (AC) $M = 0$, $q \neq 0$, with vanishing long-lag correlations; and synchronous chaos (SC) $M \neq 0$, $q \neq 0$, with persistent correlations. Q–PA and Q–AC transitions follow Eq. \ref{['Eq. fp condition']}, while non-quiescent transitions are obtained via steady-state analysis (Sec. \ref{['sec:nonstationary']}). Markers indicate the parameter choices for panels (B)–(D). Panels B/C/D: Representative states from simulations of the microscopic dynamics, Eq.\ref{['eq:scs_dynamics']}, with $N = 2000$ neurons. For each state, we show: the eigenvalue spectrum of the synaptic matrix, which in lowermost case includes an outlier; the time evolution of individual unit dynamics $x_i(t)$ along with the population mean $\hat{M}$ (blue line); and the autocorrelation function $\hat{C}(\tau)$ as a function of lag time $\tau$ along with $\hat{M}^2$ (dashed, blue line).
• Figure 2: Effective potential $V(C; C_0, M)$ as a function of the autocorrelation $C$, shown for different initial conditions $C_0$ (beige, pink, and blue-gray curves) and for representative parameter sets $(J_0/J, 1/gJ)$ corresponding to the AC (panel A), SC (panel B), and PA (panel C) states. In each panel, the order parameter $M$ is fixed to the value dynamically selected by the system in the corresponding state, as determined by Eqs. (\ref{["Eq. V'"]}--\ref{['Eq. V"']}). The three choices of $C_0$ illustrate key dynamical scenarios: the beige curve corresponds to possible oscillatory (unstable) solutions, the pink curve corresponds to the dynamically selected $C_0$ (in panels A and B) giving a stable solution, and the blue-gray curve shows the fixed-point (at $C=q=0$) value determined by Eq. (\ref{['Eq. q']}).
• Figure 3: Self–consistent evaluation of $\Xi(C;C_0^{\text{sel}},M)$ for the dynamically selected values of $C_0^{\text{sel}}$ and $M$ in each phase: asynchronous chaos (AC, panel A), synchronous chaos (SC, panel B), and persistent activity (PA, panel C). Each panel shows $\Xi(C;C_0^{\text{sel}},M)$ (black) together with the diagonal $C$ (grey), restricted to $C\in[0,1]$. The red circle marks the self–consistent intersection $C_\infty$ satisfying Eq. (\ref{["Eq. V'"]}); this point is a local maximum of the potential. Insets display the corresponding effective potential $V(C;C_0^{\text{sel}},M)$ computed at the same parameters; the red triangle indicates the local maximum at $C_\infty$.
• Figure 4: Ergodicity breaking in the broken-symmetry (SC and PA) phases. Panel A shows disorder-averaged statistics of the excess-correlation parameter $\hat{\Delta} = \mathrm{Var}[\hat{m}]$ for $1/gJ = 0.25$ (red circles) together with the analytical prediction $\Delta$ (solid black line) obtained by numerically solving Eq. (\ref{['eq:Delta_SC_self']}). Panels B–C display representative trajectories for a single disorder realization at $J_0/J = 1.20$ (SC) and $J_0/J = 1.60$ (PA), showing the activity $x_i(t)$ of two units and their inner means $\hat{m}_i$ (dashed lines). The corresponding autocorrelations $\hat{C}_{ii}(\tau)$ converge to $\hat{m}_i^2$, forming sample-dependent plateaus consistent with $\Delta > 0$. Simulations used $N = 2000$ neurons, $S = 100$ disorder realizations, and trajectory length $T = 2000$ time units.
• Figure 5: Symmetry breaking across dynamical phases. Panel A: Heat map of the empirical density of $\hat{\mu}_i/(gJ_0)$ versus $J_0/J$, with black lines showing $\pm \hat{M}$ and the shaded region marking the SC phase. The distribution evolves from unimodal in AC to bimodal in PA, with intermediate broadening in SC. Panels B–D: Representative histograms at $J_0/J = 0.60, 1.35,$ and $2.45$ (AC, SC, PA). Violet histograms show the Gaussian distribution of the sample-dependent mean for a single network $\mathbf{W}$ realization, centered —in the broken-symmetry phases-- at $-\hat{M}$, while light-green histograms show the distribution across multiple realizations, such that the positive and negative branches overlap. The variance $\mathrm{Var}[\hat{\mu}]$ is shown as a function of $N$, illustrating that in the SC phase the width vanishes in the thermodynamic limit, while in the AC and PA phases the distribution remains broad, reflecting persistent heterogeneity across units. Simulations used $N = 5000$, $S = 200$, and $T = 2000$.