

Preprint 62/2009
Stochastic partial differential equations in Neurobiology
Henry Tuckwell
Contact the author: Please use for correspondence this email.
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