Fast-slow asymptotics for a Markov chain model of fast sodium current

Modern models of excitability of cardiac cells include description of ionic channels in terms of deterministic Markov chains. The transition rates in these chains often vary by several orders of magnitude, which makes numerical simulations more difficult and necessitates development of specialized numerical approaches. We follow the usual wisdom that the small parameters in a mathematical model can be turned from an impediment into an advantage by development of asymptotic description exploiting those small parameters. The usual problem with experiment-derived (rather than postulated) models is that small parameters in them are not defined a priori but need to be identified, and sometimes this can be done equally plausibly in more than one way. In this paper, we show how the standard fast-slow asymptotic theory can be applied to the Markov chain models of ionic channels using one important model of this class as an example. We explore three selected ways of identifying small parameters in this model, and investigate the factors on which the utility of resulting asymptotics depends.