Dynamical Systems in Neuroscience:

Conductance-Based Models 135Hodgkin-HuxleyTransient Na Current (m,h)Persistent K Current (n)Leak CurrentRemove nRemove hand LeakTransient Na Current (m,h)Leak CurrentPersistent Na Current (m)Persistent K Current (n)Minimal ModelshGating forInactivationof Na CurrentmGating forActivationof Na CurrentnGating forActivationof K CurrentLeak CurrentGating variablesFigure 5.1: The Hodgkin-Huxley model (top box) is a combination of minimal models(shaded boxes on second level). Each minimal model can oscillate at least for somevalues of its parameters.be reduced to planar systems, which are amenable to analysis using geometrical phaseplane methods. In Sect. 5.2 we discuss other methods of reduction of multi-dimensionalmodels, e.g., the Hodgkin-Huxley model, to planar systems.There are only few minimal models, and understanding their dynamics can shedlight on dynamics of more complicated electrophysiological models. However, thereader should be aware of limitations of such an approach: Understanding minimalmodels cannot provide exhaustive information about all electrophysiological models(the same way as understanding the zeros of the equations y = x, and y = x 2 does notprovide complete information about the zeros of the equation y = x + x 2 ).5.1.1 Amplifying and resonant gating variablesThe definition of the minimal models involves a top-down approach: Take a complicatedmodel and strip it down to minimal ones. It is unlikely that this could be done for all2 30 or so electrophysiological models. Instead, we employ here a bottom-up approach,which is based on the following rule of thumb: A mixture of one amplifying and oneresonant (recovery) gating variable (plus an Ohmic leak current) results in a minimalmodel. Indeed, neither of the variables alone can produce oscillation, but together they