Perfect simulation for interacting Hawkes processes with variable length memory

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 $\delta_i = \frac{\beta_{i* }}{\beta^*_i-\beta_{i*}}$ ) the process is ergodic.