Interacting Hawkes processes with multiplicative inhibition

Céline Duval (1, 2) , Eric Luçon (1, 2) , Christophe Pouzat (3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 FP2M - Fédération Parisienne de Modélisation Mathématique
3 IRMA - Institut de Recherche Mathématique Avancée

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.

Domaines: Probabilités [math.PR]

Céline Duval, Eric Luçon, Christophe Pouzat. Interacting Hawkes processes with multiplicative inhibition. Stochastic Processes and their Applications, 2022, 148, pp.180-226. ⟨10.1016/j.spa.2022.02.008⟩. ⟨hal-03233710⟩