Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels

J Chevallier,A Duarte,E Löcherbach,G Ost

doi:10.1016/j.spa.2018.02.007

Abstract

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in Rd and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis.

Highlights

The aim of this paper is to present a microscopic model describing a large network of spatially structured interacting neurons, and to study its large population limits
We provide a rigorous derivation of the neural field equation based on a thorough mean field analysis
A spike of a pre-synaptic neuron leads to a change in the membrane potential of the post-synaptic neuron, possibly after some delay

Summary

Introduction

The aim of this paper is to present a microscopic model describing a large network of spatially structured interacting neurons, and to study its large population limits. This convergence is expressed in terms of the empirical measure of the spike counting processes associated with the neurons as well as the empirical measure correspondent to their position. The heuristic relies on the following argument: at the limit, we expect that the firing rate at time t of the neurons near location y should be approximately equal to λ(t, y) Taking this into account, Equation (3.12) is the limit version of the interaction structure of our system described in Definition 1.

Collecting all the estimates we then conclude that

If d

Yet lim sup E lim sup