Neural Fields and Noise-Induced Patterns in Neurons on Large Disordered Networks

TL;DR

This paper develops a rigorous mean-field framework for large disordered neural networks with noise, revealing noise-induced pattern formation such as Turing bifurcations and spiral waves, and analyzing finite-size effects.

Contribution

It introduces a mean-field limit for complex neural networks with stochastic forcing and demonstrates noise-induced patterns and deviations, advancing understanding of neural dynamics.

Findings

Mean-field equations resemble Wilson-Cowan models

Identification of Turing-like bifurcations due to noise

Numerical evidence of noise-induced spiral waves

Abstract

We study pattern formation in class of a large-dimensional neural networks posed on random graphs and subject to spatio-temporal stochastic forcing. Under generic conditions on coupling and nodal dynamics, we prove that the network admits a rigorous mean-field limit, resembling a Wilson-Cowan neural field equation. The state variables of the limiting systems are the mean and variance of neuronal activity. We select networks whose mean-field equations are tractable and we perform a bifurcation analysis using as control parameter the diffusivity strength of the afferent white noise on each neuron. We find conditions for Turing-like bifurcations in a system where the cortex is modelled as a ring, and we produce numerical evidence of noise-induced spiral waves in models with a two-dimensional cortex. We provide numerical evidence that solutions of the finite-size network converge weakly to solutions of the mean-field model. Finally, we prove a Large Deviation Principle, which provides a means of assessing the likelihood of deviations from the mean-field equations induced by finite-size effects.