Dynamical Systems in Neuroscience:

414 Solutions to Exercises, Chap. 4Voltage V40200-20-40-60thresholdup-statedown-statesaddle-node(fold)bifurcation-800 0.1 0.2 0.3 0.4E LVoltage V-30-40-50-60saddle-node(fold)bifurcationup-statethresholddown-statesaddle-node(fold)bifurcation-701.7 1.8 1.9 2 2.1 2.2leak conductance g LaKir conductance g KirbFigure 10.10: Bifurcation diagrams of the I Kir -model (3.11), I = 6, with bifurcation parameters (a)g L and (b) g Kir (see Chap. 3, Ex. 12).17. (Gradient systems) For ˙V = F (V ) takewhere c is any constant.∫ VE(V ) = − F (v) dv ,ca. E(V ) = 1 b. E(V ) = −V c. E(V ) = V 2 /2d. E(V ) = V − V 3 /3 e. E(V ) = −V 2 /2 + V 4 /4 f. E(V ) = − cos V18. (c) implies (b) because |x(t) − y| < exp(−at) implies that x(t) → y as t → ∞. (b) implies (a)according to the definition.(a) does not imply (b) because x(t) might not approach y. For example, y = 0 is an equilibriumin the system ẋ = 0 (any other point is also an equilibrium). It is stable, since |x(t)−0| < ε for all|x 0 −0| < ε and all t ≥ 0. However, it is not asymptotically stable because lim t→∞ x(t) = x 0 ≠ 0regardless of how close x 0 is to 0 (unless x 0 = 0).(b) does not imply (c). For example, the equilibrium y = 0 in the system ẋ = −x 3 is asymptoticallystable (check by differentiating that x(t) = (2t + x −20 )−1/2 → 0 is a solution withx(0) = x 0 ), however x(t) approaches 0 with a slower than exponential rate, exp(−at), for anyconstant a > 0.Solutions to Chapter 41. See figures 10.11 through 10.15.2. See Fig. 10.16.3. See figures 10.17 through 10.21.4. The diagram follows from the form of the eigenvaluesλ = τ ± √ τ 2 − 4∆2.