Paper Info
Title
A Probabilistic Analysis of Volume Transmission in the Brain.
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 concentration is approximately constant across the extracellular space. Second, by computing an explicit formula for this constant, we find that it depends on very few details in the problem. In particular, this constant does not depend on the number or arrangement of nerve varicosities, the geometry or size of the extracellular space, or firing correlations between neurons. The biological implications of these results will be explored in a forthcoming paper.
Year
DOI
Venue
2018
10.1137/16M1102252
SIAM JOURNAL ON APPLIED MATHEMATICS
Keywords
Field
DocType
stochastic switching,volume transmission,neuromodulation,stochastic hybrid system,random PDE,piecewise deterministic Markov process,randomly switching boundary
Nucleus,Mathematical analysis,Brain Nucleus,Piecewise-deterministic Markov process,Probabilistic analysis of algorithms,Brownian motion,Partial differential equation,Diffusion equation,Mathematics,Bounded function
Journal
Volume
Issue
ISSN
78
2
0036-1399
Citations 
PageRank 
References 
0
0.34
0
Authors
1
Name
Order
Citations
PageRank
sean d lawley1116.68