中科院数学与系统科学研究院
数学研究所
学术报告
调和分析和偏微分方程研讨班
报告人：Zhennan Zhou (Peking University)
题 目：Fokker-Planck equations of neuron ensembles: rigorous justification and exploring its blowup behavior
时 间：2022.08.23（星期二）17:00-18:00
地 点：N913
摘 要：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 new generalized solution based on reformulating the PDE model with a specific change of variable in time. A firing rate dependent timescale is introduced, in which the transformed equation can be shown to be globally well-posed for any connectivity parameter even in the event of the blow-up. The generalized solution is then defined via the backward change of timescale, and it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness in the original timescale.
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