Long-term stability of interacting Hawkes processes on random graphs

TL;DR

This paper studies the long-term stability of a large network of interacting Hawkes processes modeling neurons, showing stability of synaptic activity over polynomial time horizons in the subcritical regime with exponential memory kernels.

Contribution

It provides a rigorous analysis of the stability of interacting Hawkes processes on random graphs, extending understanding of neural network dynamics over long times.

Findings

Stability of synaptic current as network size grows

Validity of results up to polynomial time in N

Applicable to subcritical regimes with exponential kernels

Abstract

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the segment $[0,1]$, and the connectivity between neurons is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices through a spatial kernel. The main result of the paper concerns the longtime stability of the synaptic current of the population, as $N\to\infty$, in the subcritical regime in case the synaptic memory kernel is exponential, up to time horizons that are polynomial in $N$.