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.