Stability of wandering bumps for Hawkes processes interacting on the circle

TL;DR

This paper analyzes the long-term behavior of a large population of interacting neurons modeled by Hawkes processes on a circle, showing convergence of the system's phase to a Brownian motion over polynomial time scales.

Contribution

It demonstrates the stability of wandering bumps in Hawkes process models with cosine connectivity, revealing their phase dynamics converge to Brownian motion as population size grows.

Findings

The population's voltage phase converges to a Brownian motion over polynomial timescales.

The system admits a stable manifold of stationary solutions.

Long-term proximity of voltage to this manifold is established.

Abstract

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the circle $[-\pi, \pi]$, and the connectivity between neurons is given by a cosine kernel. The firing rate function is a sigmoid. The large population limit admits a locally stable manifold of stationary solutions. The main result of the paper concerns the long-time proximity of the synaptic voltage of the population to this manifold in polynomial times in $N$. We show in particular that the phase of the voltage along this manifold converges towards a Brownian motion on a time scale of order $N$.