The firing of action potentials has been successfully described by the Hodgkin-Huxley model, originally for the spikes in the giant axon of the squid but also, with appropriate modifications of the model, for other neuron types. The Hodgkin-Huxley model is defined by four nonlinear differential equations. The behavior of high-dimensional systems of nonlinear differential equations is difficult to visualize – and even more difficult to analyze. For an understanding of the firing behavior of the Hodgkin-Huxley model, we therefore need to turn to numerical simulations of the model. In Section 4.1 we show, as an example, some simulation results in search of the firing threshold of the Hodgkin-Huxley model. However, it remains to show whether we can get some deeper insights into the observed behavior of the model.
Four equations are in fact just two more than two: In Section 4.2 we exploit the temporal properties of the gating variables of the Hodgkin-Huxley model so as to approximate the four-dimensional differential equation by a two-dimensional one. Two-dimensional differential equations can be studied in a transparent manner by means of a technique known as ‘phase plane analysis’. Section 4.3 is devoted to the phase plane analysis of generic neuron models consisting of two coupled differential equations, one for the membrane potential and the other one for an auxiliary variable.
The mathematical tools of dimension reduction and phase plane analysis that are presented in Sections 4.2 and Section 4.3 will be repeatedly used throughout this book, in particular in Chapters 5, 6, 16 and 18. As a first application of phase plane analysis, we study in Section 4.4 the classification of neurons into type I and type II according to their frequency-current relation. As a second application of phase plane analysis, we return in Section 4.5 to some issues around the notion of a ’firing threshold’, which will be sketched now.
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