Propagation of chaos in neural fields

TL;DR

This paper proves the propagation of chaos in neural field models with infinite neuronal types and space-dependent delays, deriving new infinite-dimensional delayed integro-differential equations for neural activity.

Contribution

It introduces a novel mesoscopic limit for neural fields with multiple neuronal types and delays, establishing propagation of chaos and deriving new mean-field equations.

Findings

Propagation of chaos is proven under mild assumptions.

Derived new infinite-dimensional delayed integro-differential equations.

Applied equations to a specific model showing deterministic nonlinear dynamics.

Abstract

We consider the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields. The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve the quite fine structure of the neuron's activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We also show how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations. These results have several theoretical implications in neuroscience we review in the discussion.