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Stochastic Methods in Neuroscience$
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Carlo Laing and Gabriel J Lord

Print publication date: 2009

Print ISBN-13: 9780199235070

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199235070.001.0001

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Markov Chain Models of Ion Channels and Calcium Release Sites

Markov Chain Models of Ion Channels and Calcium Release Sites

(p.29) 2 Markov Chain Models of Ion Channels and Calcium Release Sites
Stochastic Methods in Neuroscience

Jeffrey R. Groff

Hilary DeRemigio

Gregory D. Smith

Oxford University Press

This chapter is an introduction to modelling stochastically gating ion channels using continuous-time discrete-state Markov chains. Analytical and numerical methods are presented for determining steady-state statistics of single channel gating, including the stationary distribution and open and closed dwell times. Model reduction techniques such as fast–slow analysis and state lumping are discussed as well as Gillespie’s method for simulating stochastically gating ion channels. Techniques for the estimation of model parameters and identification of model topology are briefly discussed, as well as the thermodynamic requirements that constrain the selection of rate constants. Approaches for modelling clusters of interacting ion channels using Markov chains are also summarized. Our presentation is restricted to Markov chain models of intracellular calcium release sites where clusters of calcium release channels are coupled via changes in the local calcium concentration and exhibit stochastic calcium excitability reminiscent of calcium puffs and sparks. Representative release site simulations are presented showing how phenomena such as allosteric coupling and calcium-dependent inactivation, in addition to calcium-dependent activation, affect the generation and termination of calcium puffs and sparks. The chapter concludes by considering the state space explosion that occurs as more channels are included in Markov chain models of calcium release sites. Techniques used to mitigate against this state space explosion are discussed, including the use of Kronecker representations and mean-field approximations.

Keywords:   Markov chains, ion channel gating, coupled gating, intracellular calcium release, inositol 1,4,5-trisphosphate receptors, ryanodine receptors, calcium coupling, allosteric coupling, mean-field coupling, puffs, sparks, stochastic automata

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