| Issue |
ESAIM: Procs
Volume 60, 2017
Journées MAS 2016 de la SMAI – Phénomènes complexes et hétérogènes
|
|
|---|---|---|
| Page(s) | 90 - 103 | |
| DOI | https://doi.org/10.1051/proc/201760090 | |
| Published online | 14 December 2017 | |
Spiking neurons : interacting Hawkes processes, mean field limits and oscillations*
CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. email : eva.loecherbach@u-cergy.fr
This paper gives a short survey of some aspects of the study of Hawkes processes in high dimensions, in view of modeling large systems of interacting neurons. We first present a perfect simulation result allowing for a graphical construction of the process in infinite dimension. Then we turn to the study of mean field limits and show how in some cases the delays present in the Hawkes intensities give rise to oscillatory behavior of the limit process.
Résumé
Cet article résume brièvement certains aspects de l’étude de processus de Hawkes en grande dimension, en vue d’une modélisation de systèmes de neurones en interactions. Nous présentons d’abord un résultat de simulation parfaite qui donne lieu à une construction graphique du processus en dimension infinie. Ensuite nous étudions des limites de champ moyen de processus de Hawkes et montrons comment les délais apparaissant dans le processus d’intensité engendrent des oscillations du processus limite.
© EDP Sciences, SMAI 2017
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