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.