Comparison of Asymptotics of Heart and Nerve Excitability Rebecca Suckley and Vadim N. Biktashev Department of Mathematical Sciences, Liverpool University, Mathematics & Oceanography Building, Peach Street, Liverpool, L69 7ZL, UK (Dated: Feb 11, 2003 (submitted); May 12, 2003 (revised)) We analyse 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: Hodgkin-Huxley’s 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 so- lutions, 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 fail- ure 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. PACS numbers: 87.10.+e I. INTRODUCTION The idea of the present study came from a 1972 paper by Zeeman [1], which was one in his series of works dedicated to possible applications of the then new catastrophe theory [2]. In that paper, Zeeman has analysed an apparent differ- ence between two sorts of biological excitable systems, nerve and heart, and conjectured that this difference may come as a consequence of them being described by singularly perturbed systems of equations, with the slow manifolds demonstrating catastrophes of different types. Amazingly, in the following 30 years, there were no published papers directly testing this conjecture. To fill in this gap, was one reason to undertake this study. The other reason was more practical. Mathematical mod- els describing biological excitable systems, particularly nerve and heart tissues, are historically the first, and so far the best, in terms of quantitative description of truly biological phe- nomena, based on solid experimental information. A special place in this set belongs to Hodgkin and Huxley’s [3] model of the squid giant axon, and Noble’s [4] model of the cells of Purkinje fibres of mammalian heart. These were historically the first and still the simplest in that family. Since then, the progress in development of realistic models of different kinds of cells has been enormous, and the current models achieve remarkable complexity and accuracy, particularly for cardiac cells [5, 6]. One disappointing, from a theoretical physicist’s point of view, feature of all these models is a seemingly ab- solute necessity of numerical treatment, since they are high- order (at least, of order four, as for both HH and N62) non- linear systems of differential equations, and do not admit ex- act analytical solution. Purely numerical study, however good the computers may be, always has well known disadvantages, e.g. lack of insight into dependence of the solutions on the pa- rameters. Thus, from the very beginning there were attempts to understand the behaviour of the solutions in these models by some asymptotic methods, and to devise simpler models that admit analytical treatment. The most prominent exam- ple of such study was the paper by FitzHugh [7], who has shown that a modification of the van der Pol’s nonlinear os- cillator can demonstrate qualitative properties very similar to those of the HH system, and that a collection of appropriate two-dimensional projections of the four-dimensional trajecto- ries of the HH system look similar to the phase portrait of the modified van der Pol system. When considered as a singu- larly perturbed system, FitzHugh’s system allowed a qualita- tive analysis explaining its main features without using a com- puter. Ever since, FitzHugh’s system and its numerous varia- tions are very popular as simple systems qualitatively similar to real excitable systems, and allowing both a better qualita- tive understanding, and a more efficient numerical treatment of large numbers of excitable cells, than detailed, realistic models. Yet, these simplified models are only in qualitative and not in quantitative agreement with the real systems. More- over, these simplified models are not in any way derived from the realistic systems, and therefore there is no way to be sure that they reproduce even the qualitative effects correctly. This makes a case for deriving simplified models from re- alistic models, by exploiting their real properties, via a clearly defined set of assumptions and transformations. One such at- tempt made as early as 1973 by Krinsky and Kokoz [8] who have considered the HH system as a singularly perturbed sys- tem to reduce its order to three, and an ad hoc empirical ob- servation to further reduce it to two, which ended up with a system whose phase portrait looked similar to that of the FitzHugh’s system, but already without any small parameters left. Although very interesting in a historical perspective, that paper failed to have a more lasting impact in its time, in par- ticular, because the ad hoc methods used there could not be transferred to more sophisticated models. With the advent of computational biology of extended bi- ological systems including large numbers of excitable ele- ments, such as large neural networks or whole heart, the ques-