Neurotransmitter concentrations in the presence of neural switching in one dimension

Sean D.  Lawley; Janet A.  Best; Michael C.  Reed

doi:10.3934/dcdsb.2016046

Article Contents

2016, Volume 21, Issue 7: 2255-2273. Doi: 10.3934/dcdsb.2016046

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Neurotransmitter concentrations in the presence of neural switching in one dimension

1.
Department of Mathematics, University of Utah, Salt Lake City, UT 84112
2.
Department of Mathematics, The Ohio State University, Columbus, OH 43210
3.
Department of Mathematics, Duke University, Durham, NC 27708

Received: August 31, 2015

Revised: February 29, 2016

Published: August 2016

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Abstract

In volume transmission, neurons in one brain nucleus send their axons to a second nucleus where neurotransmitter is released into the extracellular space. One would like methods to calculate the average amount of neurotransmitter at different parts of the extracellular space, depending on neural properties and the geometry of the projections and the extracellular space. This question is interesting mathematically because the neuron terminals are both the sources (when they are firing) and the sinks (when they are quiescent) of neurotransmitter. We show how to formulate the questions as boundary value problems for the heat equation with stochastically switching boundary conditions. In one space dimension, we derive explicit formulas for the average concentration in terms of the parameters of the problems in two simple prototype examples and then explain how the same methods can be used to solve the general problem. Applications of the mathematical results to the neuroscience context are discussed.

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Mathematics Subject Classification: Primary: 35R60, 60H15; Secondary: 92C20.

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