Long time behavior of a mean-field model of interacting neurons

TL;DR

This paper analyzes the long-term behavior of a McKean-Vlasov stochastic differential equation modeling neuron membrane potentials, demonstrating convergence to an invariant measure under small interaction parameters and external current perturbations.

Contribution

It provides a rigorous analysis of the convergence to equilibrium for a neuron model driven by Poisson processes, extending results to general external currents and the nonlinear case.

Findings

Solutions converge to a unique invariant measure for small interaction parameters.

Global bounds on jump rates are established and used to analyze convergence.

The results are extended to nonlinear McKean-Vlasov equations.

Abstract

We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.