The four-dimensional model of Hodgkin-Huxley can be reduced to two dimensions
under the assumption that the *m*-dynamics is fast as compared to *u*, *h*,
and *n*, and that the latter two evolve on the same time scale.
Two-dimensional models can readily be visualized and studied in the phase
plane. In type II models oscillation onset for increasing input
starts with nonzero frequency as it is typical for
Hopf bifurcations. Type I models
exhibit oscillation onset with zero frequency.
This behavior can be obtained in two-dimensional models with
three fixed points, a stable one, an unstable one, and a
saddle point. Oscillations arise through a saddle-node bifurcation when the
stable fixed point merges with the saddle. Type I models have a sharp voltage
threshold whereas type II models have not. Nevertheless, type II models
exhibit a threshold-like behavior if the *u* dynamics is much faster than that
of the recovery variable *w*.

An in-depth introduction to dynamical systems, stability of fixed points, and (un)stable manifolds can be found, for example, in the books of Hale and Koçak (1991) and Verhulst (1996). The book of Strogatz (1994) presents the theory of dynamical systems in the context of various problems of physics, chemistry, biology, and engineering. A wealth of applications of dynamical systems to various (mostly non-neuronal) biological systems can be found in the comprehensive book of Murray (1993) which also contains a thorough discussion of the FitzHugh-Nagumo model. Phase plane methods applied to neuronal dynamics are discussed in the clearly written review paper of Rinzel and Ermentrout (1989). A systematic approach to reduction of dimensionality is presented in Kepler et al. (1992). For a further reduction of the two-dimensional model to an integrate-and-fire model, see the article of Abbott and Kepler (1990). The classification of neuron models as type I and type II can be found in Rinzel and Ermentrout (1989) and in Ermentrout (1996). For a systematic discussion of canonical neuron models based on their bifurcation behavior see the monograph of Hoppensteadt and Izhikevich (1997).

Cambridge University Press, 2002

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