Perfect simulation for interacting Hawkes processes with variable length memory

Abstract

We consider a nonlinear multivariate Hawkes process having a variable length memory which allows to describe the activity of a neuronal network by its membrane potential. We propose a graphical construction of the process and we construct, by means of a perfect simulation algorithm, a stationary version of the process. By making the hypothesis that the spiking rate β i of the neuron i ∈ I is bounded, we construct an algorithm based on a priori realizations of the Poisson process (M i , i ∈ I). We show that there exists a critical value δc such that if δ > δc (where δ = inf i δ i with δ i =) the process is ergodic.