Network of interacting neurons with random synaptic weights

Paolo Grazieschi (1) , Marta Leocata (2) , Cyrille Mascart (3) , Julien Chevallier (4) , François Delarue (5) , Etienne Tanré (6)

1 Department of Mathematics, University of Warwick
2 University of Pisa - Università di Pisa
3 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO
4 SVH - Statistique pour le Vivant et l’Homme
5 LJAD - Laboratoire Jean Alexandre Dieudonné
6 TOSCA-POST - Simuler et calibrer des modèles stochastiques

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, 2, 3, 7, 6, 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 .

Domaines: Probabilités [math.PR]

Paolo Grazieschi, Marta Leocata, Cyrille Mascart, Julien Chevallier, François Delarue, et al.. Network of interacting neurons with random synaptic weights. ESAIM: Proceedings and Surveys, 2019, CEMRACS 2017 - Numerical methods for stochastic models: control, uncertainty quantification, mean-field, 65, pp.445-475. ⟨10.1051/proc/201965445⟩. ⟨hal-01928990⟩