Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

TL;DR

This paper derives mean-field equations for large stochastic neural networks with delays, revealing how noise influences complex collective behaviors like bifurcations, oscillations, and chaos in neural activity.

Contribution

It provides a rigorous derivation of Gaussian mean-field equations with delays for stochastic neural networks, enabling analysis of noise-induced transitions and bifurcations.

Findings

Noise stabilizes homogeneous solutions

Bifurcations lead to oscillations and chaos

Noise induces diverse neural activity patterns

Abstract

In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.