Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-27T22:56:20.069Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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.
Get access Rights & Permissions [Opens in a new window]

Abstract

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brémaud, P. and Massoulié, L. (1996).Stability of nonlinear Hawkes processes.Ann. Prob. 24,1563–1588.CrossRefGoogle Scholar
[2] Chevallier, J. (2015).Mean-field limit of generalized Hawkes processes.Preprint. Available at https://arxiv.org/abs/1510.05620v1.Google Scholar
[3] Comets, F.,Fernández, R. and Ferrari, P. A. (2002).Processes with long memory: regenerative construction and perfect simulation.Ann. Appl. Prob. 12,921–943.CrossRefGoogle Scholar
[4] Daley, D. J. and Vere-Jones, D. (2003).An Introduction to the Theory of Point Processes: Elementary Theory and Methods,Vol. I,2nd edn.Springer,New York.Google Scholar
[5] Delattre, S.,Fournier, N. and Hoffmann, M. (2016).Hawkes processes on large networks.Ann. App. Prob. 26,216–261.Google Scholar
[6] Duarte, A. (2015).Stochastic models in neurobiology: from a multiunitary regime to EEG data.Doctoral Thesis, University of Sao Paulo.Google Scholar
[7] Duarte, A. and Ost, G. (2016).A model for neural activity in the absence of external stimuli.Markov Process. Relat. Fields 22,37–52.Google Scholar
[8] Ferrari, P. A.,Maass, A.,Martínez, S. and Ney, P. (2000).Cesàro mean distribution of group automata starting from measures with summable decay.Ergodic Theory Dynam. Systems 20,1657–1670.CrossRefGoogle Scholar
[9] Fournier, N. and Löcherbach, E. (2016).On a toy model for interacting neurons.Ann. Inst. H. Poincaré Prob. Statist. 52,1844–1876.CrossRefGoogle Scholar
[10] Galves, A. and Löcherbach, E. (2013).Infinite systems of interacting chains with memory of variable length–a stochastic model for biological neural nets.J. Statist. Phys. 151,896–921.CrossRefGoogle Scholar
[11] Hansen, N. R.,Reynaud-Bouret, P. and Rivoirard, V. (2015).Lasso and probabilistic inequalities for multivariate point processes.Bernoulli 21,83–143.Google Scholar
[12] Hawkes, A. G. (1971).Spectra of some self-exciting and mutually exciting point processes.Biometrika 58,83–90.Google Scholar
[13] Hawkes, A. G. and Oakes, D. (1974).A cluster process representation of a self-exciting process.J. Appl. Prob. 11,93–503.CrossRefGoogle Scholar
[14] Jaisson, T. and Rosenbaum, M. (2015).Limit theorems for nearly unstable Hawkes processes.Ann. Appl. Prob. 25,600–631.Google Scholar
[15] Liggett, T. M. (1985).Interacting Particle Systems.Springer,Berlin.CrossRefGoogle Scholar
[16] Massoulié, L. (1998).Stability results for a general class of interacting point processes, and applications.Stoch. Process. Appl. 75,1–30.Google Scholar
[17] Møller, J. and Rasmussen, J. G. (2005).Perfect simulation of Hawkes processes.Adv. Appl. Prob. 37,629–646.Google Scholar
[18] Reynaud-Bouret, P. and Schbath, S. (2010).Adaptive estimation for Hawkes processes: application to genome analysis.Ann. Statist. 38,2781–2822.Google Scholar