Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On a toy network of neurons interacting through their dendrites

Nicolas Fournier, Etienne Tanré, and Romain Veltz

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Abstract

Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{\mathrm{min}}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_{n}$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{\min },\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N\to \infty $ with $w_{n}\simeq N^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni (Ann. Probab. 23 (1995) 852–878) concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value $v_{\mathrm{max}}$, and find an explicit formula for the (heuristic) mean-field limit.

Résumé

Considérons un grand nombre $n$ de neurones, chacun étant connecté à environ $N$ autres, choisis au hasard. Quand un neurone décharge, ce qui se produit au hasard à un certain taux en fonction de son potentiel électrique, son potentiel est remis à une valeur minimale $v_{\min }$, ce qui déclenche, après un petit délai, deux fronts sur les dendrites (linéaires) de tous les neurones auxquels il est connecté. Les fronts se déplacent à vitesse constante. Lorsque deux fronts (sur la dendrite du même neurone) entrent en collision, ils s’annihilent. Lorsqu’un front touche le soma d’un neurone, son potentiel est augmenté d’une petite valeur $w_{n}$. Entre les sauts, les potentiels des neurones évoluent dans $[v_{\min },\infty )$, suivant une EDO. Nous prouvons l’existence et l’unicité d’une limite champ moyen du système lorsque $n,N\to \infty $ avec $w_{n}\simeq N^{-1/2}$ obtenue de manière heuristique. Nous utilisons certaines versions récentes des résultats de Deuschel et Zeitouni (Ann. Probab. 23 (1995) 852–878) concernant la taille de la sous-suite croissante la plus longue d’une suite i.i.d. de points du plan. Nous étudions également, dans un cas très particulier, un modèle légèrement différent où les neurones déchargent quand leur potentiel atteint une valeur maximale $v_{\mathrm{max}}$. Nous trouvons heuristiquement une expression explicite pour la limite de champ moyen.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1041-1071.

Dates
Received: 17 May 2018
Revised: 22 March 2019
Accepted: 4 April 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345629

Digital Object Identifier
doi:10.1214/19-AIHP993

Mathematical Reviews number (MathSciNet)
MR4076775

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J75: Jump processes 92C20: Neural biology

Keywords
Mean-field limit Propagation of chaos Nonlinear stochastic differential equations Ulam’s problem Longest increasing subsequence Biological neural networks

Citation

Fournier, Nicolas; Tanré, Etienne; Veltz, Romain. On a toy network of neurons interacting through their dendrites. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1041--1071. doi:10.1214/19-AIHP993. https://projecteuclid.org/euclid.aihp/1584345629


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