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Mathematical Medicine and Biology 2000 17(3):263-293; doi:10.1093/imammb/17.3.263
© 2000 by Institute of Mathematics and its Applications
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Stochastic models for systems of interacting ion channels

FRANK G. BALL{dagger}, ROBIN K. MILNE{ddagger} and GEOFFREY F. YEO§

School of Mathematical Sciences, The University of Nottingham Nottingham, NG7 2RD, UK
Department of Mathematics, The University of Western Australia Nedlands, 6907, Australia
Mathematics and Statistics, DSE, Murdoch University Murdoch, 6150, Australia

{dagger}Email: fgb{at}maths.nott.ac.uk

{ddagger}Email: milne{at}maths.uwa.edu.au

§Email: yeo{at}prodigal.murdoch.edu.au

We consider a variety of Markov based models for systems of ion channels exhibiting dependence between channels. It is shown how many useful properties which may be calculated for an aggregated single-channel model, or a system of independent channels, can be extended to various types of interacting channel systems. Key structure and results from the theory of aggregated Markov processes are summarized in a convenient matrix form. These are then applied to the superposition of independent and dependent channels, including a patch of channels in a random environment, and a system of channels with spatial interactions. Calculations based on the resultant matrix expressions and intensity arguments can be implemented straightforwardly in a matrix-oriented package such as Matlab. The role of reversibility is also studied. A number of examples illustrate the strengths of the methods and enable numerical comparisons between the different types of systems.

Keywords: aggregated Markov chain; dwell-times; intensities; matrix methods; random environment; receptors; reversibility; spatial dependence; superposition


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