  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.
