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

By R. Hinch and S. J. Chapman


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

Topics: Biology and other natural sciences, Difference and functional equations, Probability theory and stochastic processes
Year: 2005
OAI identifier:

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  13. (1996). Calcium sparks and [Ca2+]i waves in cardiac myocytes,
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  15. (1993). Effect of focusing of caustics on exit phenomena in systems lacking detailed balance,
  16. (1981). Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusion fields,
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  18. (1999). Local control models of cardiac excitation-contraction coupling: a possible role for allosteric interactions between ryanodine receptors,
  19. (1995). Metastable internal layer dynamics for the viscous Cahn-Hilliard equation,
  20. (1982). Nonlinear fluctuations: The problem of deterministic limit and reconstruction of stochastic dynamics,
  21. (1997). Numerical simulations of local Ca2+ movements during l-type Ca2+ channel gating in the cardiac diad.,
  22. (1995). On the exponentially slow motion of a viscous shock,
  23. (1991). Perturbation Methods, doi
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  26. (1982). Stochastic pricesses: Time evolution, symmetries and linear response,
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  28. (2005). Synapses and Muscles, Springer-Verlag
  29. (1992). Theory of excitation–coupling in cardiac muscle.,

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