On a finite-size neuronal population equation

Population equations for infinitely large networks of spiking neurons have a long tradition in theoretical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear stochastic integral equation. We prove that, in the case of leaky integrate-and-fire (LIF) neurons with escape noise and granting a simplification, the equation is well-posed and stable in the sense of Bremaud-Massoulie. The proof combines methods from Markov processes and nonlinear Hawkes processes.