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

## TL;DR

This paper rigorously derives neural field equations as mean field limits of large spatially extended systems of nonlinear Hawkes processes, providing a mathematical foundation for neural modeling with exponential memory kernels.

## Contribution

It introduces a rigorous derivation of neural field equations from large systems of nonlinear Hawkes processes with spatial extension and exponential memory kernels.

## Key findings

- Proves convergence of neuron dynamics to a neural field equation as the number of neurons grows.
- Establishes a mean field limit for spatially extended nonlinear Hawkes processes.
- Provides a mathematical foundation for neural modeling using mean field theory.

## 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.