Dynamical Systems in Neuroscience:

410 Solutions to Exercises, Chap. 3TopologicallyEquivalentVTopologicallyEquivalentF1(V)VTopologicallyEquivalentVF (V) 2F (V)1VF (V) 2VF (V) 2F (V)1VabcTopologically NOTEquivalentF1(V)F (V) 2?VVTopologically NOTEquivalentF1(V)LeftF (V) 2Topologically NOTF1(V)Equivalent? ??RightUnstableStableRightLeftVVdeFigure 10.4: Answer to Chap. 3, Ex. 5.f9. Recall that the current I Kir is turned off by depolarization and turned on by hyperpolarization.The dynamics of the I Kir -model is similar to that of the I Na,p -model in many respects. In particular,this system can also have co-existence of two stable equilibria separated by an unstableequilibrium, which follows from the N-shaped I-V relation. Indeed, when V is hyperpolarized,the current I Kir is turned on (deinactivated) and it pulls V toward E K . In contrast, when V isdepolarized, the current is turned off (inactivated) and does not obstruct further depolarizationof V .We use (3.11) to find the curveI = g L (V − E L ) + g Kir h ∞ (V )(V − E K ) ,in Fig. 10.8. (The curve might not be S-shaped if a different bifurcation parameter is used, asin Ex. 12a).The bifurcation diagram of the I Kir -model (3.11) in Fig. 10.8 has three branches correspondingto the three equilibria. When the parameter I is relatively small, the outward I Kir currentdominates and the system has only one equilibrium in the low voltage range — the “downstate”.When the parameter I is relatively large, the injected inward current I dominates, andthe system has one equilibrium in the intermediate voltage range — the “up-state”. Whenthe parameter I is in neighborhood of I = 6, the system exhibits bistability of the “up-” and“down-states”. The states appear and disappear via saddle-node bifurcations. We see thatthe behavior of the I Kir -model is conceptually (and qualitatively) similar to the behavior ofthe I Na,p -model (3.5) even though the models have completely different ionic mechanisms forbistability.10. The equilibrium satisfies the one-dimensional equation0 = I − g K n 4 ∞(V )(V − E K ) − g Na m 3 ∞(V )h ∞ (V )(V − E Na ) − g L (V − E L ) ,