Hawkes processes with variable length memory and an infinite number of components

TL;DR

This paper introduces a novel model of biological neural networks using Hawkes processes with variable memory length and infinitely many components, employing graphical construction and perfect simulation techniques.

Contribution

It develops a new framework for infinite-component Hawkes processes with variable memory, including two models with structured interactions relevant to neural systems.

Findings

Constructed a graphical representation of the process

Developed a perfect simulation algorithm for the stationary process

Described models with saturation thresholds and layered structures

Abstract

In this paper, we build a model for biological neural nets where the activity of the network is described by Hawkes processes having a variable length memory. The particularity of this paper is to deal with an infinite number of components. We propose a graphical construction of the process and we build, by means of a perfect simulation algorithm, a stationary version of the process. To carry out this algorithm, we make use of a Kalikow-type decomposition technique. Two models are described in this paper. In the first model, we associate to each edge of the interaction graph a saturation threshold that controls the influence of a neuron on another. In the second model, we impose a structure on the interaction graph leading to a cascade of spike trains. Such structures, where neurons are divided into layers can be found in retina.