Dynamical Systems in Neuroscience:

148 Conductance-Based Models1I=651I=700.80.8V-nullclineV-nullclineK + activation, n0.60.40.2n-nullcline0 50K + activation, n0.60.40.2n-nullcline000 50-80 -60 -40 -20 0 20membrane voltage, V-80 -60 -40 -20 0 20membrane voltage, V1I=681I=73V-nullcline0.80.8V-nullclineK + activation, n0.60.40.2n-nullcline0 100K + activation, n0.60.40.2n-nullcline000 100-80 -60 -40 -20 0 20membrane voltage, Va-80 -60 -40 -20 0 20membrane voltage, VbFigure 5.12: Possible intersections of nullclines in the I K +I Kir -model. Parameters:E K = −80 mV, g Kir = 20, g K = 2. Instantaneous I Kir with V 1/2 = −80 mV andk = −12. Slower I K with k = 5, τ(V ) = 5 ms, and V 1/2 = −40 mV (in a) orV 1/2 = −55 mV (in b).potential. If we increase I, the node and the saddle approach, coalesce, and annihilateeach other via a saddle-node bifurcation, and the model start to fire action potentialsperiodically.We see that I K +I Kir -model has essentially the same dynamic repertoire as the moreconventional I Na,p +I K -model or I Na,t -model, despite the fact that it is based on a ratherbizarre ionic mechanism for excitability and spiking.5.1.7 I A -modelThe last minimal voltage-gated model has only one transient K + current, often referredto as being A-current I A , yet it can also generate sustained oscillations. In some sense,the model is similar to the I Na,t -model. Indeed, each consists of only one transient