On a toy network of neurons interacting through their dendrites

TL;DR

This paper models a large network of neurons with dendritic interactions, proving a mean-field limit as the network size grows, and explores a variant with explicit solutions.

Contribution

It introduces a novel stochastic neuron model with dendritic front interactions and establishes a mean-field limit for large networks.

Findings

Proved existence and uniqueness of the mean-field limit.

Derived explicit formulas in a special case.

Connected neuron dynamics to longest increasing subsequence results.

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 $v_{min}$, 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 \cite{dz} 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 $v_{max}$, and find an explicit formula for the (heuristic) mean-field limit.