Kinetic description and numerical study of a network of noisy resonate and fire neurons - long version

Networks of coupled resonate and fire neurons are studied in their mean-field limit through the formal derivation of a nonlinear kinetic Fokker-Planck equation in a half-plane with a threshold-reset mechanism. After structural considerations on the model and on the associated partial differential equation, an original finite-difference scheme is introduced and studied, with non-negativity and mass-preservation properties proved rigorously. A consistency analysis of the scheme is carried out, and numerical experiments indicate first-order convergence in space and time. Numerical simulations are then presented, illustrating the accuracy of the mean-field description, the efficiency of the implementation, and several dynamical regimes of the model. These experiments also lead to a number of conjectures concerning long-time behavior, the emergence of population-level self-sustained oscillations, and possible bifurcation phenomena.