Perfect simulation for interacting Hawkes processes with variable length memory

TL;DR

This paper introduces a perfect simulation method for a nonlinear multivariate Hawkes process with variable memory length, modeling neuronal activity, and establishes conditions for ergodicity based on neuron spiking rates.

Contribution

It presents a novel graphical construction and perfect simulation algorithm for the process, along with ergodicity conditions based on neuron spiking rate bounds.

Findings

Existence of a critical threshold elta_c for ergodicity.

Construction of a stationary version of the process.

Algorithm based on Poisson process realizations.

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