Dynamic Mean-Field Theory for Continuous Random Networks

Abstract

This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. The position is an element of a measurable space. Such a network corresponds to a statistical field theory (STF) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks (NNs). Each of these discrete networks corresponds to a lattice SFT, where the action contains a finite number of neurons and two scalar fields for each neuron. In this article, we develop mathematically rigorous, continuous versions of the mean-field theory approximation and the double-copy system that allow us to derive a condition for the criticality of continuous stochastic networks via the largest Lyapunov exponent. We use two basic architectures; in the first one, the space of neurons is the real line, and then the neurons are organized in one layer; in the second one, the space of neurons is the p-adic line, and then the neurons are organized in an infinite, fractal, tree-like structure. We also studied a toy model of a continuous Gaussian network with a continuous phase transition. This behavior matches the critical brain hypothesis, which states that certain biological neuronal networks work near phase transitions.