Dynamical Systems in Neuroscience:

408 Solutions to Exercises, Chap. 310.90.8h (V)987τ (V) h0.760.60.50.40.30.20.1n (V)m (V)0-40 -20 0 20 40 60 80V (mV)54321τ (V)m0-40 -20 0 20 40 60 80 100V (mV)τ (V) nFigure 10.1: Open dots: The steady-state (in)activation functions and voltage-sensitive time constantsin the Hodgkin-Huxley model. Filled dots: steady-state Na + activation function m ∞ (V ) inthe squid giant axon (experimental results by Hodgkin and Huxley, 1952, Fig. 8). Continuous curves:Approximations by Boltzmann and Gaussian functions. See Ex. 4.Solutions to Chapter 31. Consider the limit case: (1) activation of Na + current is instantaneous, and (2) conductancekinetics of the other currents are frozen. Then, the Na + current will result in the nonlinearterm g Na m ∞ (V ) (V − E Na ) with the parameter h ∞ (V rest ) incorporated into g Na , and all theother currents will result in the linear leak term.In Fig. 3.15, the activation of the Na + current is not instantaneous, hence the sag right afterthe pulses. In addition, its inactivation, as well as the kinetics of the other currents are notslow enough, hence the membrane potential quickly reaches the excited state and then slowlyrepolarizes back to the resting state.2. See Fig. 10.2. The eigenvalues are negative at each equilibrium marked as filled circle (stable),and positive at each equilibrium marked as open circle (unstable). The eigenvalue at thebifurcation point (left equilibrium in Fig. 10.2b) is zero.F(V) F(V) F(V)V V Va b cFigure 10.2: Phase portraits of the system ˙V = F (V ) with given F (V ).3. Phase portraits are shown in Fig. 10.3.(a) The equation 0 = −1 + x 2 has two solutions: x = −1 and x = +1, hence there aretwo equilibria in the system (a). The eigenvalues are the derivatives at each equilibrium,λ = (−1 + x 2 ) ′ = 2x, where x = ±1. Equilibrium x = −1 is stable because λ = −2 < 0.Equilibrium x = +1 is unstable because λ = +2 > 0. The same fact follows from thegeometrical analysis in Fig. 10.3.