Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains

摘要

A system {X(t)} = {(X_1(t),X_2(t),...,X_N(t))} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X_i(t)} (i = 1,2,...,N) is partitioned into two aggregates. Interaction between the processes {X_1(t)}, {X_2(t)},..., {X_N(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {X(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {X_i(t)} corresponds to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.