Interacting Hawkes processes with multiplicative inhibition

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.

Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity

“…An example is the neural Hawkes process, presented in Mei and Eisner (2017); Zuo et al (2020), which combines a multivariate Hawkes process and a recurrent neural network architecture. In Duval et al (2021), a multiplicative model considers two sets of neuronal populations, one exciting and another inhibiting, and each intensity function is the product of two non-linear functions (one for each group). Another model is presented in Olinde and Short (2020) and called self-limiting Hawkes process.…”