Gregory Handy, University of Chicago
Stochastic fluctuations drive biological processes from particle diffusion to neuronal spike times. The goal of this talk is to use and extend a variety of mathematical frameworks to understand such fluctuations and derive insight into the corresponding applications. We start by considering n diffusing particles that may leave a bounded domain by either ‘escaping’ through an absorbing boundary or being ‘captured’ by traps that must recharge between captures. We prove that the average number of captured particles grows on the order of log n because of this recharge time, which is drastically different than the linear growth observed for instantaneous recharging. We then examine this process in the limit of large n to investigate the celebrated formula of Berg and Purcell with a modeling framework that uses boundary homogenization to link the diffusion equation to boundary conditions described by nonlinear ordinary differential equations. We end by exploring how the brain leverages interneuron diversity and noisy recurrent connections to assist with cortical computations. Specifically, we utilize linear response theory and mean-field approximations to show how interneurons modulate the level of synchrony in visually induced gamma rhythms.