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Generalised eigenproblems arising in aggregated markov processes allowing for time interval omission

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

Published online by Cambridge University Press:  01 July 2016

Assad Jalali  and
Alan G. Hawkes
Assad Jalali
Affiliation:
University of Wales, Swansea
Alan G. Hawkes
Affiliation:
University of Wales, Swansea
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Abstract

We consider a continuous-time Markov chain in which one cannot observe individual states but only which of two sets of states is occupied at any time. Furthermore, we suppose that the resolution of the recording apparatus is such that small sojourns, of duration less than a constant deadtime, cannot be observed. We obtain some results concerning the poles of the Laplace transform of the probability density function of apparent occupancy times, which correspond to a problem about generalised eigenvalues and eigenvectors. These results provide useful asymptotic approximations to the probability density of occupancy times. A numerical example modelling a calcium-activated potassium channel is given. Some generalisations to the case of random deadtimes complete the paper.

Keywords

EIGENVALUESOCCUPANCY TIMESION CHANNELS

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 15A18: Eigenvalues, singular values, and eigenvectors 92C05: Biophysics 92C30: Physiology (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 302 - 321
Copyright
Copyright © Applied Probability Trust 1992 

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References

Ball, F. G. and Sansom, M. S. P. (1988) Aggregated Markov processes incorporating time interval omission. Adv. Appl. Prob. 20, 546–572.CrossRefGoogle Scholar
Ball, F. G. and Sansom, M. S. P. (1989) Ion channel gating mechanisms: model identification and parameter estimation from single channel recordings. Proc. R. Soc. Lond. B 236, 385–416.Google ScholarPubMed
Ball, F. G. (1990) Aggregated Markov processes with negative exponential time interval omission. Adv. Appl. Prob. 22, 802–830.CrossRefGoogle Scholar
Bellman, R. (1970) Introduction to Matrix Analysis, 2nd edn, McGraw-Hill, New York.Google Scholar
Blatz, A. L. and Magleby, K. L. (1986) Correcting single channel data for missed events. Biophys. J. 49, 967–980.CrossRefGoogle ScholarPubMed
Colquhoun, D. and Hawkes, A. G. (1982) On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Phil. Trans. R. Soc. Lond. B 300, 1–59.Google ScholarPubMed
Colquhoun, D. and Sigworth, F. J. (1983) Fitting and statistical analysis of single channel records. In Single Channel Recording, ed. Sakmann, B. and Neher, E., Plenum, New York, 191–263.CrossRefGoogle Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Crouzy, S. C. and Sigworth, F. J. (1990) Yet another approach to the dwell time omission problem of single channel analysis. Biophys. J. 58, 731–743.CrossRefGoogle Scholar
Fredkin, D. R., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. In Proceedings of the Berkeley Conference in Honour of Jerzy Neyman and Jack Kiefer, ed. LeCam, L. M. and Ohlsen, R. A., Belmont, Wadsworth, 269–289.Google Scholar
Hawkes, A. G., Jalali, A. and Colquhoun, D. (1990) The distributions of the apparent open times and shut times in a single channel record when brief events can not be detected. Phil. Trans R. Soc. Lond. A 332, 511–538.Google Scholar
Hawkes, A. G., Jalali, A. and Colquhoun, D. (1992) Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events. Phil. Trans. R. Soc. London B. To appear.Google Scholar
Hawkes, A. G. and Sykes, A. M. (1990) Equilibrium distributions of finite-state Markov processes. IEEE Trans. Reliability 39, 592–595.CrossRefGoogle Scholar
Jalali, A. and Hawkes, A. G. (1992) The distribution of apparent occupancy times in a two-state Markov process in which brief events can not be detected. Adv. Appl. Prob. 24, 288–301.CrossRefGoogle Scholar
Lancaster, P. (1969) Theory of Matrices. Academic Press, New York.Google Scholar
Roux, B. and Sauve, R. (1985) A general solution of the time interval omission problem applied to single channel analysis. Biophys. J. 48, 149–158.CrossRefGoogle ScholarPubMed
Magleby, K. S. and Pallota, B. L. (1983) Calcium dependence of open and shut interval distributions from calcium-activated potassium channels in cultured rat muscle. J. Physiol. (Lond.) 344, 585–604.CrossRefGoogle ScholarPubMed
Morrison, D. F. (1967) Multivariate Statistical Methods. McGraw-Hill, New York.Google Scholar
Ostrowski, A. M. (1964) Positive matrices and functional analysis. In Recent Advances in Matrix Theory, ed. Schneider, H., University of Wisconsin Press, Madison, 125–138.Google Scholar
Seneta, E. (1973) Non-negative Matrices. George Allen and Unwin, London.Google Scholar
Smith, M. G. (1966) Laplace Transform Theory. Van Nostrand, London.Google Scholar
Taussky, O. (1964) On the variation of the characteristic roots of a finite matrix under various changes of its elements. In Recent Advances in Matrix Theory, ed. Schneider, H., University of Wisconsin Press, Madison, 125–138.Google Scholar