Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On a toy model of interacting neurons

Nicolas Fournier and Eva Löcherbach

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We continue the study of a stochastic system of interacting neurons introduced in De Masi, Galves, Löcherbach and Presutti (J. Stat. Phys. 158 (2015) 866–902). The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to $0$ and all other neurons receive an additional amount $1/N$ of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its average potential. We prove propagation of chaos of the system, as $N\to\infty $, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of (J. Stat. Phys. 158 (2015) 866–902), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/\sqrt{N}$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique nontrivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.


Cet article continue l’étude du système stochastique de neurones en interaction introduit par De Masi, Galves, Löcherbach et Presutti (J. Stat. Phys. 158 (2015) 866–902). Le système est composé de $N$ neurones. Chaque neurone décharge un potentiel d’action à des instants aléatoires, à un taux qui dépend de son potentiel de membrane. Ce potentiel est alors remis à $0$, et tous les autres neurones reçoivent une charge supplémentaire de $1/N$. De plus, des synapses électriques induisent une dérive déterministe qui attire le système vers sa valeur moyenne. Nous établissons la propriété de propagation du chaos lorsque $N\to\infty$, vers la solution d’une équation différentielle stochastique non-linéaire à sauts. Nous améliorons les résultats obtenus dans (J. Stat. Phys. 158 (2015) 866–902) puisque (i) nous levons la condition de support compact imposée aux données initiales, (ii) nous obtenons une vitesse de convergence en $1/\sqrt{N}$. Enfin, nous proposons une étude de l’équation limite : nous décrivons la forme de ses lois marginales (en temps), nous démontrons l’existence d’une unique loi invariante non-triviale et montrons que la mesure invariante triviale n’est pas attractive. Enfin, nous obtenons la convergence vers l’équilibre dans un cas particulier.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1844-1876.

Received: 13 October 2014
Revised: 3 May 2015
Accepted: 22 July 2015
First available in Project Euclid: 17 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes 60F17: Functional limit theorems; invariance principles

Piecewise deterministic Markov processes Mean-field interaction Biological neural nets Interacting particle systems Nonlinear stochastic differential equations


Fournier, Nicolas; Löcherbach, Eva. On a toy model of interacting neurons. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1844--1876. doi:10.1214/15-AIHP701.

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