Mean-field limit of generalized Hawkes processes

TL;DR

This paper extends multivariate Hawkes processes to include age dependence, analyzes their mean-field limit as the number of processes grows, and links microscopic models to macroscopic PDE descriptions of neurons.

Contribution

It introduces a generalized age-dependent Hawkes process model and establishes its mean-field limit, connecting microscopic and macroscopic neuronal models.

Findings

System approximated by i.i.d. age-dependent processes

Generalizes previous mean-field results

Links microscopic Hawkes models to PDE neuron models

Abstract

We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point. Within this class, we investigate the limit behaviour, when n goes to infinity, of a system of n mean-field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by Delattre, Fournier and Hoffmann (2015). In continuity with the work of Chevallier et al. (2015), the second goal of this paper is to give a proper link between these generalized Hawkes processes as microscopic models of individual neurons and the age-structured system of partial differential equations introduced by Pakdaman, Perthame and Salort (2010) as macroscopic model of neurons.