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 paper, we formulate and analyze a mathematical model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model consists of the diffusion equation in a bounded two- or three-dimensional domain that contains 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 partial differential equation, we show that each realization of its solution can be represented as a certain expected local time of a Brownian particle in a corresponding realization of a random environment. Using this representation, we prove two surprising results. First, the expected neurotransmitter con...
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