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).

The work of Hodgkin and Huxley on nerve conduction has long been recognized as an outstanding scientific achievement. Their papers were published in 1952 [Hodgkin and Huxley (1952a, b, c, d)] and they received a Nobel prize in physiology for their research in 1961. Hodgkin and Huxley's work was at once a triumphant culmination of many years of theoretical and experimental work by research physiologists and a pioneering effort that set the direction and defined the goals for much of the ensuing research in biophysics.

The purpose of this book is, first, to provide an introductory description of the work of Hodgkin and Huxley and the later work that is based on the techniques that they introduced. Our main emphasis is on the theoretical aspect of the Hodgkin–Huxley work, that is, the derivation and analysis of their mathematical models (nonlinear ordinary and partial differential equations); the second purpose of this book is to describe some of the mathematics that is used to study these differential equations.

The hope is that this book will indicate to some biologists the importance of the mathematical approach and will serve as an introduction for mathematicians to the mathematical problems in the field. However, this discussion is bound to be unsatisfactory to many readers. The biologists will find the description of the physiology simplistic, crude, if not actually misleading, and they may also be dubious about the value of conclusions that can be drawn from the mathematical analysis.

The physiological problem

We start by describing in some detail the physiological problem to be studied. An impulse that, say, carries a command from the brain to a muscle travels along a sequence of neurons that can be portrayed roughly as in the sketch in Fig. 2.1. When the impulse arrives at the dendrites on the left-hand side of the neuron, the stimuli given the dendrites are integrated at the cell body to form a nerve impulse. The nerve impulse travels along the axon to the branches of axons on the right-hand side of the neuron. The impulse then jumps to another set of dendrites and the process just described is repeated. Neurons vary considerably in size. The sciatic nerve of the giraffe contains an axon that may be several meters in length; many other axons are much shorter. The diameter of the squid axon, which has been the subject of many experimental studies, is ~ 0.5 mm but it can be as much as 1 mm. Its length is several centimeters.

The process by which an impulse travels along a sequence of neurons is quite complicated. For example, when the impulse jumps from one set of dendrites to another, both chemical and electrical processes, which are not well understood even today, play an important role. The subject of Hodgkin and Huxley's work is the process by which the impulse travels along the axon in the giant axon of the squid.

Introduction

The techniques developed by Hodgkin and Huxley for the study of the squid axon have been applied in the ensuing years by many other researchers. In a certain sense, the techniques remain the same, that is, voltage-clamp experiments followed by a quantitative analysis in which activation and inactivation variables are introduced and then described by differential equations. But there are many obstacles to these applications of Hodgkin and Huxley's techniques: The analysis of the ionic current becomes much more complicated because the descriptions of some of the currents are more intricate and because the number of distinct components of the current is in some cases much larger than for the squid axon. (Later we shall describe a mathematical model for the cardiac Purkinje fiber in which the ionic current has nine components.) Also the use of voltage-clamp methods is much more difficult in some cases. For example, voltage-clamp techniques were used successfully in the study of cardiac fibers for the first time in 1964, and the voltage-clamp technique used in the study of striated muscle fibers was not developed until the late 1960s.

The purpose of this chapter is to describe mathematical models (systems of nonlinear ordinary differential equations) of a number of electrically excitable cells that can be investigated by using Hodgkin–Huxley techniques. Since our primary concern is the derivation and study of these mathematical models that stem from the experimental studies, it is easy to forget or lose sight of the extensive and taxing work that goes into successful experiments.

The quantitative study of electrically active cells received its principal impetus from the remarkable work of Hodgkin and Huxley, in 1951, on nerve conduction in the squid giant axon. Hodgkin and Huxley used voltage-clamp methods to obtain extensive quantitative experimental results and proposed a system of ordinary differential equations that summarized and organized these data. Since then, their experimental methods have been extended and adapted to the study of other electrically active cells. Also, numerous mathematical studies of the Hodgkin–Huxley equations have been made. The results, experimental and mathematical, are scattered through the literature in research papers, and the first purpose of this book is to provide an organized account of some of these results. This account is intended to be accessible to mathematicians with little or no background in physiology.

In Chapter 2, a fairly detailed account is given of the experimental results of Hodgkin and Huxley, and similar detail is provided for the derivation of the Hodgkin–Huxley equations. It is not necessary to study the experimental results or the derivation of the equations in order to understand the equations themselves, which are a four-dimensional system of autonomous differential equations containing messy nonlinear functions. (The functions are, however, quite well-behaved: They are, indeed, real analytic functions, and the usual existence theorems can be applied to the differential equations.)

It is tempting to the mathematician to disregard the derivation of the equations and plunge ahead, instead, to the mathematical analysis of the equations, a familiar activity made additionally attractive, in this case, by the fact that the equations model an important system in the “real” world.

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.

Functionals of types D′ and Z′

In order to give meaning to the Fourier transform of any locally integrable function it is necessary to generalize beyond the functionals in S′, and we describe now the functionals in classes D′ and Z′ which provide the necessary generalization. We give examples in section 14.2, but start in this section with the basic concepts.

In summary, the functionals of class D′ are based on the use of test functions in class D (the good functions of bounded support), whilst the functionals in class Z′ are based on the use of test functions in class Z (the Fourier transforms of functions in D). Much, but not all, of chapter 12 can simply be adapted by replacing S by D (or Z) and S′ by D′ (or Z′). For instance, functionals of type D′ and Z′ are defined as follows.

Definition An association of exactly one real or complex number with each ϕ∈D is said to be a functional in class D′ if there exists at least one sequence of ordinary functions such that fnϕ∈L for each ϕ∈D and each n, and such that for each ϕ∈D the number associated with ϕ is equal to limn→∞ int; fnϕ.