  In this work, the primary goal is to establish rigorous connection between
the Fokker-Planck equation of neural networks with its microscopic model: the
diffusion-jump stochastic process that captures the mean field behavior of
collections of neurons in the integrate-and-fire model. The proof is based on a
novel iteration scheme: with an auxiliary random variable counting the firing
events, both the density function of the stochastic process and the solution of
the PDE problem admit series representations, and thus the difficulty in
verifying the link between the density function and the PDE solution in each
sub problem has been greatly mitigated. The iteration approach provides a
generic frame in integrating the probability approach with PDE techniques, with
which we prove that the density function of the diffusion-jump stochastic
process is indeed the classical solution of the Fokker-Planck equation with a
unique flux-shift structure.
