The Mathematical Institute, University of Oxford, Eprints Archive

Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Hinch, R. and Chapman, S. J. (2005) Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks. European Journal of Applied Mathematics, 16 (4). pp. 427-446. ISSN 0956-7925

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Abstract

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations.

Item Type:Article
Subjects:A - C > Biology and other natural sciences
D - G > Difference and functional equations
O - Z > Probability theory and stochastic processes
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:294
Deposited By:Jon Chapman
Deposited On:03 Oct 2006
Last Modified:20 Jul 2009 14:20

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