# Asymptotic and numerical analysis of a stochastic PDE model of volume transmission

## TL;DR

This paper develops an asymptotic and numerical analysis of a stochastic PDE model for volume transmission, revealing that neurotransmitter concentration is primarily spatially uniform and independent of many details, with second-order effects influenced by various factors.

## Contribution

The paper provides the first analytical asymptotic expansion for a stochastic PDE model of volume transmission, highlighting the spatial uniformity of neurotransmitter concentration and its dependence on second-order factors.

## Key findings

- Concentration is spatially constant to leading order.
- Leading order concentration is independent of nerve varicosities and geometry.
- Second order effects depend on firing correlations and space geometry.

## Abstract

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.