Stochastic partial differential equations in Neurobiology

Henry Tuckwell

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Submission date: 24. Oct. 2009
published in: Stochastic biomathematical models : with applications to neuronal modeling / M. Bachar ... (eds.)
Berlin : Springer, 2013. - P. 149 - 173
(Lecture notes in mathematics ; 2058) 
DOI number (of the published article): 10.1007/978-3-642-32157-3_6
Bibtex
with the following different title: Stochastic partial differential equations in neurobiology : linear and nonlinear models for spiking neurons

Abstract:
Stochastic differential equation models of nerve cells for the most part neglect the spatial dimension. Including the latter leads to stochastic partial differential equations (SPDEs) which allow for the inclusion of important variations in the densities of ion channels. We briefly consider representations of neuronal anatomy in the context of linear SPDE models on line segments with one and two components and present solutions as series of Ornstein-Uhlenbeck processes. The ion currents underlying neuronal spiking are discussed and a general nonlinear SPDE model is presented. In the spatial Hodgkin-Huxley model, excitation is applied over a small region and the spiking activity observed as a function of mean stimulus strength with a view to finding the critical values for repetitive firing. During spiking near those critical values, noise of increasing amplitudes is applied over the whole neuron and over restricted regions. Minima are found in the spike counts which parallel results for the point model and which have been termed inverse stochastic resonance. A stochastic Fitzhugh-Nagumo system is also described and results given for the probability of transmission along a neuron in the presence of noise. The wave phenomenon of cortical spreading depression, which has similarities to neuronal spiking, is modeled as a stochastic reaction-diffusion system with 3-parameter Poisson process sources of potassium ions representing extrusions due to the random firings of neurons. Assuming that in a restricted small area the sources have greater strength than background, the probability of an SD wave is found as a function of the patch size. Also, the probability of elicitation of SD through the occurrence of a patch with compromised metabolic activity, as may occur by virtue of an infarct after stroke, is analyzed in terms of the effect of relative decreases in the strength of the sodium-potassium exchange pump