Age Dependent Hawkes Process

TL;DR

This paper introduces the Age Dependent Hawkes process, enhancing neural network models by incorporating post-jump behavior, which improves stability analysis and allows for large network limit studies.

Contribution

It extends classical Hawkes processes by including age dependence, enabling modeling of refractory periods and improving stability without strict bounds.

Findings

Improved stability results for age-dependent Hawkes processes.

Established propagation of chaos in mean field neural networks.

Demonstrated modeling of neural refractory periods.

Abstract

In the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process, the Age Dependent Hawkes process, which incorporates individual post-jump behaviour into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean field type, we study large network limits and establish the propagation of chaos property of the system.