Dynamical Systems in Neuroscience:

Conductance-Based Models 163v peak , and the voltage reset value is c. The parameter d describes the total amount ofoutward minus inward currents activated during the spike and affecting the after-spikebehavior. All these parameters can be easily fit to any particular neuron type, as weshow in Chap. 8.Review of Important Concepts• Amplifying gating variables describe activation of an inward currentor inactivation of an outward current. They amplify voltage changes.• Resonant gating variables describe inactivation of an inward currentor activation of an outward current. They resist voltage changes.• To exhibit excitability, it is enough to have one amplifying and oneresonant gating variable in a neuronal model.• Many models can be reduced to two-dimensional systems with oneequation for voltage and instantaneous amplifying currents, and oneequation for resonant gating variable.• The behavior of a two-dimensional model depends on the positionof its nullclines. Many models have an N-shaped V -nullcline and asigmoid shaped nullcline for the gating variable.• There is a relationship between nullclines and I-V curves.• Quite different electrophysiological models can have similar nullclines,and hence essentially the same dynamics.• The spike-generation mechanism of detailed electrophysiologicalmodels depends on the dynamics near the left knee of the fast V -nullcline, and it can be captured by a simple model (5.5, 5.6).Bibliographical NotesRichard FitzHugh pioneered the usage of phase planes and nullclines to study theHodgkin-Huxley model (FitzHugh 1955). Later, he suggested a simple model withN-shaped cubic V -nullcline and a straight-line slow nullcline, known as the FitzHugh-Nagumo model, to illustrate the mechanism of excitability of the Hodgkin-Huxleysystem. However, it was Krinskii and Kokoz (1973) who first discovered the relationshipn(t) + h(t) ≈ const and were thus able to reduce the four-dimensional Hodgkin-Huxleymodel to a two-dimensional system. Since then, the phase plane analysis of neuronalmodels became standard, at least in Russian language literature.Current awareness of the geometrical methods of phase plane analysis of neuronal