On a toy network of neurons interacting through their dendrites
| Autor: | Nicolas Fournier, Etienne Tanré, Romain Veltz |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour les Neurosciences (MATHNEURO), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Biological neural networks Ulam’s problem Dendrite Longest increasing subsequence 01 natural sciences Combinatorics 010104 statistics & probability FOS: Mathematics medicine nonlinear stochastic differential equations Limit (mathematics) Uniqueness Ulam's problem 0101 mathematics 60K35 60J75 92C20 Mathematics Propagation of chaos Quantitative Biology::Neurons and Cognition Probability (math.PR) 010102 general mathematics Ode Front (oceanography) [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] medicine.anatomical_structure 60K35 FOS: Biological sciences Quantitative Biology - Neurons and Cognition 92C20 Mean-field limit Neurons and Cognition (q-bio.NC) Soma Electric potential 60J75 Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2020, 56 (2), pp.1041-1071. ⟨10.1214/19-AIHP993⟩ Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2020, 56 (2), pp.1041-1071. ⟨10.1214/19-AIHP993⟩ Ann. Inst. H. Poincaré Probab. Statist. 56, no. 2 (2020), 1041-1071 Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2020, 56 (2), pp.1041-1071 |
| ISSN: | 0246-0203 1778-7017 |
| DOI: | 10.1214/19-aihp993 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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