The Single Neuron

Abstract

In our book, the study of neural nets composed of many neurons is in the foreground of our interest. To this end, we had to make a number of simplifying assumptions on the behavior of individual neurons. In this chapter we want to get acquainted with more precise models of individual neurons, whereby we will focus our attention on the generation of axonal pulses. This will also allow us to judge the approximations we had to make in our network analysis. The fundamental equations describing the generation of action potentials and their spiking property had been established by Hodgkin and Huxley as early as 1952. In Sect. 13.1 we will discuss these equations and some simplifying approaches to them. In order to elucidate the origin of the spiking behavior of the solutions, we will discuss the FitzHugh–Nagumo equations in Sect. 13.2 and then return in Sect. 13.3 to the Hodgkin–Huxley equations and especially to generalizations of them. The axon membrane is a lipid bilayer that may be considered as a thin insulating sheet that can store electric charges like a conventional capacitor.