5 - Mathematical theory

Summary

Introduction

So far we have described a number of mathematical models of electrically excitable cells and, at least for some of the models, we have indicated the kind of mathematical questions and analysis that arise in work on these models. The next major step is to describe in detail how the mathematical analysis can be carried out. In order to obtain this description, we will first give some theory of differential equations. By doing this, we will introduce the mathematical language that is appropriate for discussing the problems that are of concern to us. We also describe some mathematical techniques and results that will be useful in the study of the models.

Basic theory

In this section we describe some basic properties of solutions of differential equations. These are used very frequently in the analysis of all the physiological models.

Existence theorems and extension theorems

It is reasonable that our first concern should be for the existence of solutions. To a reader whose experience with differential equations is limited to an introductory course following calculus, such a concern may seem unnecessarily fussy. In a first course in differential equations various techniques for computing solutions are described and it might be expected that our chief concern would be to summarize such computational techniques; but all the physiological models in which we are interested are nonlinear systems, and, consequently, it is usually impossible to get explicit expressions for the solutions (i.e., closed solutions).