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

Part of: Stochastic processes Special processes Operations research and management science

Published online by Cambridge University Press:  17 March 2017

Pierre Hodara  and
Eva Löcherbach
Pierre Hodara*
Affiliation:
Université de Cergy-Pontoise
Eva Löcherbach*
Affiliation:
Université de Cergy-Pontoise
*
* Postal address: CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.
* Postal address: CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.
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Abstract

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In this paper we propose a model for biological neural nets where the activity of the network is described by Hawkes processes having a variable length memory. The particularity in this paper is that we deal with an infinite number of components. We propose a graphical construction of the process and build, by means of a perfect simulation algorithm, a stationary version of the process. To implement 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 the retina.

Keywords

Point processmultivariate Hawkes processbiological neural netKalikow-type decompositionperfect simulation

MSC classification

Primary: 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 90B15: Network models, stochastic
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 84 - 107
Copyright
Copyright © Applied Probability Trust 2017 

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