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 ${\beta }_i$ of the neuron $i\in I$ is bounded, we construct an algorithm based on a priori realizations of the Poisson process $(M^i,i\in I)$. We show that there exists a critical value ${\delta }_c$ such that if $\underline{\delta }>{\delta }_c$ (where $\underline{\delta }={\inf }_i{\delta }_i$ with ${\delta }_i=\frac{{\beta }_{i\ast }}{{\beta }_i^{\ast }-{\beta }_{i\ast }}$ ) the process is ergodic.