On a Finite-Size Neuronal Population Equation

Population equations for infinitely large networks of spiking neurons have a long tradition in theoret-ical 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 neurons with escape noise and for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Bre'\maud and Massoulie'. 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.