abstract: Volume transmission is a fundamental neural communication mechanism in which neurons in one brain nucleus modulate the neurotransmitter concentration in the extracellular space of a second nucleus. In this talk, we will describe a mathematical model of volume transmission involving the diffusion equation in a bounded three-dimensional domain with a set of interior holes that randomly switch between being either sources or sinks. The interior holes represent nerve varicosities that are sources of neurotransmitter when firing an action potential and are sinks otherwise. To analyze this random PDE, we will show that its solution can be represented as a certain local time of a Brownian particle in a random environment, and that this representation can be used to prove surprising properties of the solution.
More broadly, we will explain how this probabilistic perspective on Brownian functionals relates to recent results on escape problems involving mean first passage times of diffusion and asymptotic analysis of PDEs.