Authors: Quentin Cormier ¹

• 1 Inria, CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris

We investigate a stochastic network composed of Integrate-and-Fire spiking neurons, focusing on its mean-field asymptotics. We consider an invariant probability measure of the McKean-Vlasov equation and establish an explicit sufficient condition to ensure the local stability of this invariant distribution. Furthermore, we prove a conjecture proposed initially by J. Touboul and P. Robert regarding the bistable nature of a specific instance of this neuronal model.

• 37A30 - Ergodic theorems, spectral theory, Markov operators
• 45D05 - Volterra integral equations
• 60H10 - Stochastic ordinary differential equations (aspects of stochastic analysis)
• 60K35 - Interacting random processes; statistical mechanics type models; percolation theory