Network of interacting neurons with random synaptic weights

¹ Mathematics Institute, University of Warwick, Coventry, United Kingdom
e-mail: p.grazieschi@bath.ac.uk
² Department of Mathematics, University of Pisa, Pisa, Italy
e-mail: marta.leocata@gmail.com
³ Université Côte d’Azur, CNRS, I3S, France
e-mail: mascart@i3s.unice.fr
⁴ Laboratoire Jean Kuntzmann, Université Grenoble Alpes (UFR IM2AG), Grenoble, France
e-mail: Julien.Chevallier1@univgrenoble-alpes.fr
⁵ Université Côte d’Azur, CNRS, LJAD, France
e-mail: francois.delarue@unice.fr
⁶ Université Côte d’Azur, Inria, France.
e-mail: Etienne.Tanre@inria.fr

Abstract

Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.