In this talk, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. In the first part of the talk, we present an alternative way to derive such Fokker-Planck equations from the microscopic model based on a novel iterative expansion. With this formulation, we prove that the probability density function of the “leaky integrate-and-fire” type stochastic process is a classical solution to the Fokker-Planck equation. Secondly, we propose a conservative and positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models. 

Zhennan Zhou is an assistant professor at BICMR, Peking University since 2017. He was a William W. Elliott Assistant Research Professor at Duke University, from 2014 to 2017. He obtained his PhD degree from University of Wisconsin-Madison in 2014. His primary research interests are in the applied analysis of PDE and stochastic models, and numerical approximation of scientific problems arising from quantum mechanics, theoretical chemistry, solid state physics, materials science, biology, etc.