  In the mean field integrate-and-fire model, the dynamics of a typical neuron
within a large network is modeled as a diffusion-jump stochastic process whose
jump takes place once the voltage reaches a threshold. In this work, the main
goal is to establish the convergence relationship between the regularized
process and the original one where in the regularized process, the jump
mechanism is replaced by a Poisson dynamic, and jump intensity within the
classically forbidden domain goes to infinity as the regularization parameter
vanishes. On the macroscopic level, the Fokker-Planck equation for the process
with random discharges (i.e. Poisson jumps) are defined on the whole space,
while the equation for the limit process is on the half space. However, with
the iteration scheme, the difficulty due to the domain differences has been
greatly mitigated and the convergence for the stochastic process and the firing
rates can be established. Moreover, we find a polynomial-order convergence for
the distribution by a re-normalization argument in probability theory. Finally,
by numerical experiments, we quantitatively explore the rate and the asymptotic
behavior of the convergence for both linear and nonlinear models.
