Beijing Institute of Mathematical Sciences and Applications

Yi Zhu

Zhennan Zhou

Thursday, May 12, 2022 10:00 AM - 11:30 AM

Tencent 836 6547 4971 ()

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 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. Secondly, we are concerned with a kinetic model for neuron networks. Individual neurons are characterized by their voltage and conductance, the dynamics of the voltage is influenced by the conductance and when the voltage is reaching a threshold, it is immediately reset to a lower value. By exploring a series of toy models, we aim to identify the cause of the emergence of time-periodic solutions in such Fokker-Planck equations.

周珍楠，北京大学北京国际数学研究中心研究员、博士生导师。2014 年在美国威斯康辛大学麦迪逊分校获得博士学位，2014-2017 年在美国杜克大学担任助理研究教授，2017 年加入北京大学北京国际数学研究中心，任研究员、博士生导师。主要研究领域为微分方程的应用分析，微分方程数值解，应用随机分析，随机模拟等，特别是关注来源于自然科学的应用数学问题。入选中组部第十四批“千人计划”青年人才项目（2018），入选北京市科协（2020-2022 年）青年人才托举工程项目。