Invariant Manifolds of Binomial-Like Nonautonomous Master Equations

Abstract

It has recently been shown that certain finite-state jump processes used to model ion channel kinetics give rise to nonautonomous master equations which possess low-dimensional invariant manifolds. In particular, for the model of an ion channel with many identical and independent subunits existing in either an open or closed state, the one-dimensional manifold of binomial distributions, whose parameter is the probability of a single subunit being in an open state, is an invariant manifold of the associated master equation, and it is the only such one-dimensional manifold. In this paper, we make explicit how the invariance of this manifold arises from the structure of the transition rate matrix of the master equation. We also show that, despite not having any apparent underlying structure or symmetry akin to the ion channel example, there exists a two-parameter family of master equations that shares the essential features of this example and therefore possesses unique, one-dimensional invariant manifolds similar to the manifold of binomial distributions. The mere existence of such a family suggests that there may be other nonautonomous master equations that are devoid of any obvious structure but nevertheless possess invariant manifolds of low dimension.