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.

Zhennan Zhou graduated from the School of Mathematics at Jilin University in 2009 with a Bachelor's degree. In 2014, he obtained a Ph.D degree from the Department of Mathematics at the University of Wisconsin-Madison in the United States. From 2014 to 2017, he served as an Assistant Research Professor at Duke University in the United States. Between 2017 and 2023, he held the position of Assistant Professor at the Beijing International Center for Mathematical Research, Peking University. He joined the Westlake University in March 2024. Zhennan Zhou's primary research interests lie in the applied analysis of differential equations and stochastic models, as well as the design and analysis of numerical methods for scientific problems. These problems stem from diverse fields such as quantum mechanics, theoretical chemistry, and biology, and often involve multiscale phenomena. He particularly focuses on strategies related to model reduction and efficient simulation techniques.