Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 3 411Unstableequilibria10-1StableequilibriuaxBifurcation DiagramaaSaddle-node(fold)bifurcation-2 -1 0 0.5 1aRepresentative Phase Portraits1.51xa+x 21.51x1.5a+x2xa+x210.50x0.50x0.50x-0.5-0.5-0.5-1-2 -1 0 1 2a=-1-1-2 -1 0 1 2a=0-1-2 -1 0 1 2a=0.5Figure 10.5: Saddle-node (fold) bifurcation diagram and representative phase portraits of the systemẋ = a + x 2 (see Chap. 3, Ex. 6).where all gating variables assume their asymptotic values. The solutionI = g K n 4 ∞(V )(V − E K ) + g Na m 3 ∞(V )h ∞ (V )(V − E Na ) + g L (V − E L )is depicted in Fig. 10.9. Since the curve in Fig. 10.9 does not have folds, there are no saddle-nodebifurcations in the Hodgkin-Huxley model (with the original values of parameters).11. The curvesand(b)are depicted in Fig. 10.7.12. The curves13.and(a) g L (V ) = −g Na m ∞ (V )(V − E Na )/(V − E L )E L (V ) = V + g Na m ∞ (V )(V − E Na )(V − E L )/g L(a) g L (V ) = {I − g Kir h ∞ (V )(V − E K )}/(V − E L )(b) g Kir (V ) = {I − g L (V − E L )}/{h ∞ (V )(V − E K )}are depicted in Fig. 10.10. Notice that the curve in Fig. 10.10a does not have the S shape.F ′ (V ) = −g L − g K m 4 ∞(V ) − g K 4m 3 ∞(V )m ′ ∞(V )(V − E K ) < 014.because g L > 0, m ∞ (V ) > 0, m ′ ∞(V ) > 0, and V − E K > 0 for all V > E K .F ′ (V ) = −g L − g h h ∞ (V ) − g h h ′ ∞(V )(V − E h ) < 0because g L > 0, h ∞ (V ) > 0, but h ′ ∞(V ) < 0 and V − E h < 0 for all V < E h .