北京雁栖湖应用数学研究院

朱毅

周珍楠

2022年05月12日 10:00 至 11:30

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，2009年毕业于吉林大学数学学院，获学士学位，2014年毕业于美国威斯康辛大学麦迪逊分校数学系，获博士学位。2014年至2017年在美国杜克大学担任助理研究教授。2017年至2023年在北京大学北京国际数学研究中心担任助理教授。2024年3月全职加入西湖大学，担任理论科学研究院特聘研究员。周珍楠博士的主要研究兴趣在于微分方程和随机模型的应用分析，以及科学问题的数值方法设计和算法分析。 这些科学问题源于量子力学、理论化学、生物学等领域，而且普遍存在多尺度的现象，尤其关注相关的模型约化策略以及高效模拟方法。