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

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

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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