Dynamical Systems in Neuroscience:

Conductance-Based Models 15510050V(t)010.5m(t)n(t)h(t)02n(t)+h(t)0.8400 10 20 30 40 50 60 70 80 90 100time, t (ms)Figure 5.18: The sum n(t) + h(t) ≈ 0.84 in the Hodgkin-Huxley model. Parameters asin Chap. 2 and I = 8.5.2 Reduction of multi-dimensional models5.2.1 Hodgkin-Huxley modelLet us consider again the Hodgkin-Huxley modelC ˙V = I −I K{ }} {g K n 4 (V − E K ) −ṅ = (n ∞ (V ) − n)/τ n (V ) ,ṁ = (m ∞ (V ) − m)/τ m (V ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) ,I Na{ }} {g Na m 3 h(V − E Na ) −I L{ }} {g L (V − E L ) ,with the original values of parameters presented in Chap. 2. How can we understandthe qualitative dynamics of this model? One way, discussed above, is to throw awayvariable h or n and to reduce this model to the I Na,p +I K -model or I Na,t -model, respectively.Although the reduced minimal models can tell a lot about the behaviorof the original model, they are not equivalent to the Hodgkin-Huxley model from theelectrophysiological or the dynamical systems point of view. Below we discuss anothermethod of reduction of multidimensional electrophysiological models to planar systems.The Hodgkin-Huxley model has four independent variables. Early computer simulationsby Krinskii and Kokoz (1973) have shown that there is a relationship between