Multi-class oscillating systems of interacting neurons

TL;DR

This paper analyzes large multi-class neuron systems modeled by nonlinear Hawkes processes, deriving mean field limits, oscillatory behaviors, and diffusion approximations to understand their complex dynamics.

Contribution

It introduces a rigorous derivation of mean field limits for multi-class interacting neurons and explores oscillatory behaviors and diffusion approximations in these systems.

Findings

Mean field limit described by nonlinear differential equations

Oscillatory behaviors identified in the limit system

Central limit theorems established for fluctuations

Abstract

We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations.