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 for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Br\'emaud-Massouli\'e. The proof combines methods from Markov processes taking values in the space of positive measures and nonlinear Hawkes processes. For applications, we also provide efficient simulation algorithms.