  Volume transmission is an important neural communication pathway in which
neurons in one brain region influence the neurotransmitter concentration in the
extracellular space of a distant brain region. In this paper, we apply
asymptotic analysis to a stochastic partial differential equation model of
volume transmission to calculate the neurotransmitter concentration in the
extracellular space. Our model involves the diffusion equation in a
three-dimensional domain with interior holes that randomly switch between being
either sources or sinks. These holes model nerve varicosities that alternate
between releasing and absorbing neurotransmitter, according to when they fire
action potentials. In the case that the holes are small, we compute
analytically the first two nonzero terms in an asymptotic expansion of the
average neurotransmitter concentration. The first term shows that the
concentration is spatially constant to leading order and that this constant is
independent of many details in the problem. Specifically, this constant first
term is independent of the number and location of nerve varicosities, neural
firing correlations, and the size and geometry of the extracellular space. The
second term shows how these factors affect the concentration at second order.
Interestingly, the second term is also spatially constant under some mild
assumptions. We verify our asymptotic results by high-order numerical
simulation using radial basis function-generated finite differences.
