Stability and mean-field limits of age dependent Hawkes processes

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 behavior 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. du réseau est supprimée ou au moins modifiée. Nous améliorons les résultats de stabilité classiques pour les processus de Hawkes dans ce cadre. En particulier, nous n’avons ni besoin de supposer que les intensités sont bornées, ni d’imposer une condition aux normes Lipschitz des fonctions taux de saut. Lorsque les interactions entre les neurones sont du type champ moyen, nous étudions les limites en grande population et nous démontrons la propriété de propagation du chaos du système.