6 - Mathematical analysis of physiological models

Summary

Introduction

We have already described the numerical analysis of the Hodgkin–Huxley equations (in Chapter 3) and other models (in Chapter 4). In view of the high success of much of this numerical analysis, it is natural to ask why any further mathematical analysis is required. That is, when a model of an electrically excitable cell has been obtained, why not answer all mathematical questions that arise by simply carrying out a numerical analysis? The answer to this question lies partly in the nature of the models that we have considered and partly in the kind of results that can be obtained from numerical analysis. As we have seen, the models considered are empirical descriptions and the constants that appear in them are, at best, reasonably good approximations. In fact, for some purposes, it is well to regard these constants as parameters that have various values. Moreover, the very functions that appear in the models are, in some cases, quite tentatively proposed [see Chapter 2 and Hodgkin and Huxley (1952d, p. 510).] In carrying out a numerical analysis, an entirely specific model must be considered. That is, the forms of the functions and the values of the parameters must be completely specified. Moreover, the numerical analysis gives no information about whether solutions of similar models have the same or similar behavior. As a simple example, suppose that the numerical analysis of one model suggests that there is a unique asymptotically stable periodic solution.