Asymptotic and numerical analysis of a stochastic PDE model of volume transmission
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 threedimensional 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 highorder numerical simulation using radial basis functiongenerated finite differences.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.11680
 Bibcode:
 2018arXiv181211680L
 Keywords:

 Mathematics  Probability;
 Mathematics  Numerical Analysis;
 Quantitative Biology  Neurons and Cognition
 EPrint:
 29 pages, 4 figures. Accepted to SIAM Multiscale Modeling and Simulation