Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T09:13:04.889Z Has data issue: false hasContentIssue false

Aggregated semi-Markov processes incorporating time interval omission

Published online by Cambridge University Press:  01 July 2016

Frank Ball*
Affiliation:
University of Nottingham
Robin K. Milne*
Affiliation:
University of Western Australia
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
Postal address: Department of Mathematics, University of Nottingham, Nottingham, NG7 2RD, UK.
∗∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
∗∗∗Postal address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.

Abstract

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, F. G. (1990) Aggregated Markov processes with negative exponential time interval omission. Adv. Appl. Prob. 22, 802830.CrossRefGoogle Scholar
Ball, F. G. and Rice, J. A. (1989) A note on single-channel autocorrelation functions. Math. Biosci. 97, 1726.CrossRefGoogle ScholarPubMed
Ball, F. G. and Sansom, M. S. P. (1987) Temporal clustering of ion channel openings incorporating time interval omission. IMA J. Math. Med. Biol. 4, 333361.CrossRefGoogle ScholarPubMed
Ball, F. G. and Sansom, M. S. P. (1988a) Aggregated Markov processes incorporating time interval omission. Adv. Appl. Prob. 20, 546572.CrossRefGoogle Scholar
Ball, F. G. and Sansom, M. S. P. (1988b) Single-channel autocorrelation functions: the effects of time interval omission. Biophys. J. 53, 819832.CrossRefGoogle ScholarPubMed
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. London 236, 385416.Google ScholarPubMed
Ball, F. G., Kerry, C. J., Ramsey, R. L., Sansom, M. S. P. and Usherwood, P. N. R. (1988) The use of dwell time cross-correlation functions to study single-ion channel gating kinetics. Biophys. J. 54, 309320.CrossRefGoogle ScholarPubMed
Ball, F. G., Yeo, G. F., Milne, R. K., Edeson, R. O., Madsen, B. W. and Sansom, M. S. P. (1991) Single ion channel models incorporating aggregation and time interval omission. Submitted.Google Scholar
Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
Blatz, A. L. and Magleby, K. L. (1986) Correcting single channel data for missed events. Biophys. J. 49, 967980.CrossRefGoogle ScholarPubMed
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123187.CrossRefGoogle Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1977) Relaxation and fluctuations of membrane currents that flow through drug-operated channels. Proc. R. Soc. London 199, 231262.Google 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. London 300, 159.Google ScholarPubMed
Colquhoun, D. and Hawkes, A. G. (1987) A note on correlations in single ion channel records. Proc. R. Soc. London 230, 1552.Google ScholarPubMed
Dabrowski, A. R., Mcdonald, D. and Rösler, U. (1990) Renewal properties of ion channels. Ann. Statist. 18, 10911115.CrossRefGoogle Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks. A Markov Renewal Approach. John Hopkins University Press, Baltimore MD.Google Scholar
Edeson, R. O., Yeo, G. F., Milne, R. K. and Madsen, B. W. (1990) Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling. Math: Biosci. 102, 75104.Google ScholarPubMed
Feigin, P. D., Tweedie, R. L. and Belyea, C. (1983) Weighted area techniques for explicit parameter estimation. Austral. J. Statist. 25, 116.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II, 2nd edn Wiley, New York.Google Scholar
Fredkin, D. and Rice, J. A. (1986) On aggregated Markov processes. J. Appl. Prob. 23, 208214.CrossRefGoogle Scholar
Fredkin, D. R., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. In Proc. Conf. Honor of Jerzy Neyman and Jack Keifer , ed. Le Cam, Lucien and Olshen, Richard, Vol. 1, pp. 269289. Wadsworth, Belmont, CA.Google Scholar
Horn, R. and Lange, K. (1983) Estimating kinetic constants from single channel data. Biophys. J. 43, 207223.CrossRefGoogle ScholarPubMed
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Kemeny, J. and Snell, L. (1969) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
Kienker, P. (1989) Equivalence of aggregated Markov models of ion channel gating. Proc. R. Soc. London 236, 269309.Google ScholarPubMed
Kijima, S. and Kijima, H. (1987) Statistical analysis of channel current from a membrane patch II. A stochastic theory of a multi-channel system in the steady-state. J. Theoret. Biol. 128, 435455.CrossRefGoogle Scholar
Laurence, A. F. and Morgan, B. J. T. (1987) Selection of the transformation variable in the Laplace transform method of estimation. Austral. J. Statist. 29, 113127.CrossRefGoogle Scholar
Liebovitch, L. S., Fischbarg, J. and Koniarek, J. P. (1987) Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes. Math. Biosci. 84, 3768.CrossRefGoogle Scholar
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979) Multivariate Analysis. Academic Press, London.Google Scholar
Milne, R. K., Yeo, G. F., Edeson, R. O. and Madsen, B. W. (1988) Stochastic modelling of a single ion channel: an alternating renewal approach with application to limited time resolution. Proc. R. Soc. London 233, 247292.Google ScholarPubMed
Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 12311242.CrossRefGoogle Scholar
Roux, B. and Sauve, R. (1985) A general solution to the time interval omission problem applied to single channel analysis. Biophys. J. 48, 149158.CrossRefGoogle Scholar
Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744756.CrossRefGoogle Scholar
Sakmann, B. and Neher, E. (1983) Single-Channel Recording. Plenum Press, New York.Google Scholar
Seneta, E. (1973) Non-negative Matrices. An Introduction to Theory and Applications. Allen and Unwin, London.Google Scholar
Sumita, U. and Rieders, M. (1988) First passage times and lumpability of semi-Markov processes. J. Appl. Prob. 25, 675687.CrossRefGoogle Scholar
Sumita, U. and Rieders, M. (1989) Lumpability and time-reversibility in the aggregation-disaggregation method for large Markov chains. Commun. Statist.Stoch. Models 5, 6381.CrossRefGoogle Scholar
Yang, G. L. and Swenberg, C. E. (1990) Estimation of open dwell time and problems of identifìability in channel experiments. In Stoch. Methods Biological Intelligence. To appear.Google Scholar
Yeo, G. F., Milne, R. K., Edeson, R. O. and Madsen, B. W. (1988) Statistical inference from single channel records: two-state Markov model with limited time resolution. Proc. Roy. Soc. London. 235, 6394.Google ScholarPubMed
Yeo, G. F., Edeson, R. O., Milne, R. K. and Madsen, B. W. (1989) Superposition properties of independent ion channels. Proc. Roy. Soc. London 238, 155170.Google ScholarPubMed