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

1997 ◽  
Vol 29 (4) ◽  
pp. 909-946 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Difference Equation ◽  
Renewal Process ◽  
Markov Models ◽  
Single Channel ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Channel Process ◽  
Gating Mechanism ◽  
Differential Difference Equation ◽  
Image Position

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 t0. 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.

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

1997 ◽  
Vol 29 (04) ◽  
pp. 909-946 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Difference Equation ◽  
Renewal Process ◽  
Markov Models ◽  
Single Channel ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Channel Process ◽  
Gating Mechanism ◽  
Differential Difference Equation ◽  
Image Position

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.

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
Image Position

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

1997 ◽  
Vol 29 (01) ◽  
pp. 92-113 ◽  
Author(s):  
Frank Ball ◽  
Sue Davies
Keyword(s):  
Markov Chain ◽  
Ion Channel ◽  
State Space ◽  
Continuous Time ◽  
Markov Models ◽  
Numerical Study ◽  
Channel Gating ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean[N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

Two Markov Models of Neighborhood Housing Turnover

1972 ◽  
Vol 4 (2) ◽  
pp. 133-146 ◽  
Author(s):  
G Gilbert
Keyword(s):  
Mathematical Models ◽  
Markov Processes ◽  
Renewal Process ◽  
Markov Models ◽  
Transition Probabilities ◽  
Markov Renewal Process ◽  
Turnover Process

This paper develops two mathematical models of housing turnover in a neighborhood. The first of these draws upon the theory of non-homogeneous Markov processes and includes the effects of present neighborhood composition upon future turnover probabilities. The second model considers the turnover process as a Markov renewal process and therefore allows the inclusion of length of occupancy as a determinant of transition probabilities. Example calculations for both models are included, and procedures for using the models are outlined.

Aggregated Markov processes incorporating time interval omission

1988 ◽  
Vol 20 (03) ◽  
pp. 546-572 ◽  
Author(s):  
Frank Ball ◽  
Mark Sansom
Keyword(s):  
State Space ◽  
Renewal Process ◽  
Numerical Study ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Closed Sets ◽  
Finite State ◽  
Dynamic Stochastic ◽  
Stochastic Properties ◽  
Finite State Space

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (04) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

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.

Aggregated Markov processes incorporating time interval omission

1988 ◽  
Vol 20 (3) ◽  
pp. 546-572 ◽  
Author(s):  
Frank Ball ◽  
Mark Sansom
Keyword(s):  
State Space ◽  
Renewal Process ◽  
Numerical Study ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Closed Sets ◽  
Finite State ◽  
Dynamic Stochastic ◽  
Stochastic Properties ◽  
Finite State Space

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (4) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

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.

scholarly journals Analysis of single ion channel data incorporating time-interval omission and sampling

2005 ◽  
Vol 3 (6) ◽  
pp. 87-97 ◽  
Author(s):  
Yu-Kai The ◽  
Jens Timmer
Keyword(s):  
Markov Models ◽  
Dead Time ◽  
Single Channel ◽  
Hidden Markov ◽  
Interval Length ◽  
Time Interval ◽  
Sampled Data ◽  
The Dead ◽  
Single Channel Currents ◽  
Channel Currents

Hidden Markov models are widely used to describe single channel currents from patch-clamp experiments. The inevitable anti-aliasing filter limits the time resolution of the measurements and therefore the standard hidden Markov model is not adequate anymore. The notion of time-interval omission has been introduced where brief events are not detected. The developed, exact solutions to this problem do not take into account that the measured intervals are limited by the sampling time. In this case the dead-time that specifies the minimal detectable interval length is not defined unambiguously. We show that a wrong choice of the dead-time leads to considerably biased estimates and present the appropriate equations to describe sampled data.

scholarly journals Linking Exponential Components to Kinetic States in Markov Models for Single-Channel Gating

2008 ◽  
Vol 132 (2) ◽  
pp. 295-312 ◽  
Author(s):  
Christopher Shelley ◽  
Karl L. Magleby
Keyword(s):  
Rate Constants ◽  
Time Distribution ◽  
Dwell Time ◽  
Markov Models ◽  
Single Channel ◽  
Closed Interval ◽  
Discrete State ◽  
Closed Intervals ◽  
Gating Mechanism ◽  
The Relationship

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1–59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

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