Predictable Mean-Field Chaos in Random Recurrent Networks

Abstract

Dynamical mean-field theory recasts deterministic chaos in random recurrent networks as an effective stochastic process. We show that for analytic nonlinearities with sufficiently fast Fourier decay, this stochasticity is only apparent: the continuous past of a realized mean-field trajectory uniquely determines its future. The mean-field theory is therefore not merely an ensemble description, but a conditional prediction theory for individual trajectories. Unfolding the power spectrum into a Krylov state space exposes how this latent determinism is organized across an infinite hierarchy of temporal modes. The associated Krylov growth rate sets the complexity of finite-resolution prediction and upper-bounds the largest Lyapunov exponent in this class of networks. Thus, microscopic sensitivity and predictive complexity are distinct aspects of mean-field chaos. Our results extend Krylov growth ideas developed for Hamiltonian chaotic dynamics to classical dissipative systems.