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Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling
R O Edeson1, G F Yeo, R K Milne
1Department of Anaesthesia, Sir Charles Gairdner Hospital, Nedlands, Western Australia.
Mathematical Biosciences
|November 1, 1990
Summary
This study introduces a graph-based method to calculate sojourn time distributions in classes of states for stochastic processes. This approach simplifies analysis, particularly for complex systems like ion-channel kinetics where individual states are often indistinguishable.
Area of Science:
- Stochastic processes
- Applied mathematics
- Biophysics
Background:
- Sojourn time distributions are crucial in analyzing stochastic processes, especially in fields like ion-channel kinetics.
- Individual states in complex systems are often experimentally indistinguishable, necessitating analysis of class sojourn times.
Purpose of the Study:
- To develop a novel graph-based methodology for deriving sojourn time distributions within classes of states in stochastic processes.
- To provide a framework applicable to systems where individual states are not easily differentiated.
Main Methods:
- Representing the stochastic process and its transitions using graph theory.
- Modifying the graph through state composition to define a new Markov chain.
- Expressing class sojourn times as random sums within the new Markov chain framework.
- Utilizing symmetry properties to simplify derivations.
Main Results:
- Derivation of class sojourn time distributions through graph composition.
- Demonstration of the method's applicability using ion-channel kinetics models.
- Discussion of parameter estimation for Markov processes with exponentially distributed sojourn times.
Conclusions:
- The graph-based composition method offers an algorithmic and potentially simplified approach to calculating class sojourn time distributions.
- This methodology is particularly valuable for analyzing systems with indistinguishable states, such as in biophysical modeling.
- The study provides explicit estimating equations for sequential models relevant to nicotinic receptor kinetics.