Fluctuations for Spatially Extended Hawkes Processes

TL;DR

This paper investigates the fluctuations around the mean-field limit of spatially extended Hawkes processes, showing they converge to a Gaussian-driven stochastic differential equation and can be approximated by a stochastic neural field equation.

Contribution

It introduces a central limit theorem for these processes and demonstrates that their fluctuations can be approximated by a stochastic neural field equation, a novel result in the literature.

Findings

Fluctuations converge to a Gaussian-driven stochastic differential equation.

The stochastic differential equation can be approximated by a stochastic neural field equation.

Provides a new understanding of variability in spatially extended neural networks.

Abstract

In a previous paper, it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution $u(t,x)$ of a neural field equation (NFE). The value $u(t,x)$ represents the membrane potential at time $t$ of a typical neuron located in position $x$, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean-field limit $u(t,x)$. Our first main result is a central limit theorem stating that the spatial distribution associated with these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result, we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by $u(t,x)$. To the best of our knowledge, this result appears to be new in the literature.