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Empirical clustering of bursts of openings in markov and semi-markov models of single channel gating incorporating time interval omission

Part of: Markov processes Physiological, cellular and medical topics Special processes Basic linear algebra

Published online by Cambridge University Press:  01 July 2016

Frank Ball
Frank Ball*
Affiliation:
University of Nottingham
*
* Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than t 0 . Let N(t) be the number of bursts commencing in (0, t]. Then are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.

The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.

Keywords

CONTINUOUS-TIME MARKOV CHAINMARKOV RENEWAL PROCESSAGGREGATED PROCESSSINGLE ION CHANNEL MODELLINGFUNDAMENTAL MATRIXSPECTRAL REPRESENTATIONDIFFERENTIAL-DIFFERENCE EQUATIONPERTURBATION THEORY

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K15: Markov renewal processes, semi-Markov processes 92C05: Biophysics 15A99: Miscellaneous topics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 909 - 946
Copyright
Copyright © Probability Trust 1997 

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