Comparison of asymptotics of heart and nerve excitability.

Abstract

We analyze the asymptotic structure of two classical models of mathematical biology, the models of electrical action by Hodgkin-Huxley (1952) for a giant squid axon and by Noble (1962) for mammalian Purkinje fibres. We use the procedure of parametric embedding to formally introduce small parameters in these experiment-based models. Although one of the models was designed as a modification of the other, their structure with respect to the small parameters appears to be entirely different: the Hodgkin-Huxley model has two slow and two fast variables, while Noble's model has one slow variable, two fast variables, and one superfast variable. The singular perturbation theory of these models adequately reproduces some features of the accurate numeric solutions, such as excitability and the shape of the voltage upstroke, but fails to reproduce other features, such as the relatively slow return from the excited state, compared to the speed of the upstroke. We present arguments towards the viewpoint that contrary to the conjecture proposed by Zeeman (1972), for these two models this failure is an inevitable consequence of the Tikhonov-style appearance of the small parameters, and a more adequate asymptotic description may only be achieved with small parameters entering the equations in a significantly different way.