`Real' neurons are extremely complex biophysical and biochemical entities. Before designing a model it is therefore necessary to develop an intuition for what is important and what can be savely neglected. The Hodgkin-Huxley model describes the generation of action potentials on the level of ion channels and ion current flow. It is the starting point for detailed neuron models which in general include more than the three types of currents considered by Hodgkin and Huxley.
Electrophysiologists have described an overwhelming richness of different ion channels. The set of ion channels is different from one neuron to the next. The precise channel configuration in each individual neuron determines a good deal of its overall electrical properties. Synapses are usually modeled as specific ion channels that open for a certain time after presynaptic spike arrival.
The geometry of the neuron can play an important role in synaptic integration because the effect of synaptic input on the somatic membrane potential depends on the location of the synapses on the dendritic tree. Though some analytic results can be obtained for passive dendrites, it is usually necessary to resort to numerical methods and multi-compartment models in order to account for complex geometry and active ion channels.
A nice review of the Hodgkin-Huxley model including some historical remarks can be found in the book of Nelson and Rinzel (1995). Mathematical aspects of the Hodgkin-Huxley equations are discussed in the Monograph of Cronin (1987).
A comprehensive and readable introduction to the biophysics of single neurons is provided by the book of Christof Koch (Koch, 1999). Even more detailed information on ion channels and non-linear effects of the nervous membrane can be found in B. Hille's book on `Ionic channels of excitable membranes' (Hille, 1992). A practical guideline for do-it-yourself experiments with computer simulations of detailed neuron models is the book of Bower and Beeman (Bower and Beeman, 1995).
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