Dynamical Systems in Neuroscience:

156 Conductance-Based Models1h=0.89-1.1n0.8h0.60.40.200 0.2 0.4 0.6 0.8 1nFigure 5.19: The relationship between n(t) andh(t) in the Hodgkin-Huxley model can better bedescribed by h = 0.89 − 1.1n.the gating variables n(t) and h(t), namely,n(t) + h(t) ≈ 0.84 ,as shown in Fig. 5.18. In fact, plotting the variables on the (n, h) plane, as we do inFig. 5.19, reveals that the orbit is near the straight lineh = 0.89 − 1.1n .We can use this relationship in the voltage equation to reduce the Hodgkin-Huxleymodel to a three-dimensional system. If, in addition, we assume that the activationkinetics of the Na + current is instantaneous, i.e., m = m ∞ (V ), then the Hodgkin-Huxley model can be reduced to the two-dimensional systemI Kinstantaneous IC ˙V{ }} {NaI{ }} { { }} L{= I − g K n 4 (V −E K ) − g Na m 3 ∞(V )(0.89−1.1n)(V −E Na ) − g L (V −E L ) ,ṅ = (n ∞ (V ) − n)/τ n (V ) ,whose solutions retain qualitative and some quantitative agreement with the originalfour-dimensional Hodgkin-Huxley system; see Fig. 5.20.The first step in the analysis of any two-dimensional system is to find its nullclines.The V -nullcline can be found by solving numerically the equationI − g K n 4 (V −E K ) − g Na m 3 ∞(V )(0.89−1.1n)(V −E Na ) − g L (V −E L ) = 0for n. The nullcline has the familiar N-shape depicted in Fig. 5.21. Notice that ithas only one intersection with the n-nullcline n = n ∞ (V ), hence there is only oneequilibrium, which is stable when I = 0. When the parameter I increases, the equilibriumloses stability via subcritical Andronov-Hopf bifurcation, as discussed in the nextchapter. When I is sufficiently large (e.g. I = 12 in Fig. 5.21), there is a limit cycleattractor corresponding to periodic spiking. In Ex. 2 we discuss what happens when Ibecomes very large.