Dynamical Systems in Neuroscience:

One-Dimensional Systems 8910.80.60.40.2h (V)0-140 -70 0Membrane Voltage (mV)Current200-20-40-60I (V)LI (V)Kir-80E KE L-100-140 -70 0Membrane Voltage (mV)210-1-2F(V)-100 -50 0Membrane Voltage (mV)Figure 3.38: The I Kir -model having injected current (I), leak current (I L ), and instantaneousK + inward rectifier current (I Kir ) and described by (3.11). Inactivation curveh ∞ (V ) is modified from Wessel et. al (1999). Parameters: C = 1, I = 6, g L = 0.2,E L = −50, g Kir = 2, E K = −80, V 1/2 = −76, k = −12 (see Fig. 2.20).10.80.60.40.2m (V)0-100 -50 0 50 100Membrane Voltage (mV)Current100500-50I (V)LE L-100E Na-150-100 -50 0 50 100Membrane Voltage (mV)INa,p(V)500F(V)-50-100 -50 0 50 100Membrane Voltage (mV)Figure 3.39: The I Na,p -model with leak current (I L ) and persistent Na + current (I Na,p ),described by (3.5) with the right-hand side function F (V ). Parameters: C = 1, I = 0,g L = 1, E L = −80, g Na = 2.25, E Na = 60, V 1/2 = −20, k = 15 (see Fig. 2.20).9. Draw the bifurcation diagram of the I Kir -modelC ˙V = I − g L (V − E L ) −instantaneous I Kir{ }} {g Kir h ∞ (V )(V − E K ) , (3.11)using parameters from Fig. 3.38 and treating I as a bifurcation parameter.10. Derive an explicit formula that relates the position of the equilibrium in theHodgkin-Huxley model to the magnitude of the injected dc-current I. Are thereany saddle-node bifurcations?11. Draw the bifurcation diagram of the I Na,p -model (3.5) using parameters fromFig. 3.39 and treating(a) g L as a bifurcation parameter,(b) E L as a bifurcation parameter.