Interacting Hawkes processes with multiplicative inhibition

TL;DR

This paper introduces a new class of mean-field nonlinear Hawkes processes modeling reciprocal interactions between excitatory and inhibitory neuronal populations, analyzing their well-posedness, dynamics, and long-term behavior, including potential limit cycles.

Contribution

It presents a novel mathematical model incorporating multiplicative inhibition and additive retroaction, with rigorous analysis of system dynamics and long-term behavior.

Findings

Well-posedness of the interacting Hawkes system established

Analysis of large population dynamics conducted

Numerical evidence suggests inhibition and retroaction can lead to limit cycles

Abstract

In the present work, we introduce a general class of mean-field interacting nonlinear Hawkes processes modelling the reciprocal interactions between two neuronal populations, one excitatory and one inhibitory. The model incorporates two features: inhibition, which acts as a multiplicative factor onto the intensity of the excitatory population and additive retroaction from the excitatory neurons onto the inhibitory ones. We give first a detailed analysis of the well-posedness of this interacting system as well as its dynamics in large population. The second aim of the paper is to give a rigorous analysis of the longtime behavior of the mean-field limit process. We provide also numerical evidence that inhibition and retroaction may be responsible for the emergence of limit cycles in such system.