Dynamical Systems in Neuroscience:

90 One-Dimensional Systems10.80.60.40.2m (V)0-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)Current100500E KE LI (V)LI (V)K-50-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)500F(V)-50-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)Figure 3.40: The I K -model with leak current (I L ) and persistent K + current (I K ),described by (3.12). Parameters: C = 1, g L = 1, E L = −80, g K = 1, E K = −90,V 1/2 = −53, k = 15 (see Fig. 2.20).10.80.60.40.2h (V)0-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)Current500-50I (V)LI (V)hE LE h-100-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)100500-50F(V)-100-120 -100 -80 -60 -40 -20 0Membrane Voltage (mV)Figure 3.41: The I h -model with leak current (I L ) and “hyperpolarization-activated”inward current I h , described by (3.13). Parameters: C = 1, g L = 1, E L = −80, g h = 1,E h = −43, V 1/2 = −75, k = −5.5 (Huguenard and McCormick 1992).12. Draw the bifurcation diagram of the I Kir -model (3.11) using parameters fromFig. 3.38 and treating(a) g L as a bifurcation parameter,(b) g Kir as a bifurcation parameter.13. Show that the I K -model in Fig. 3.40C ˙V = −g L (V − E L ) −instantaneous I K{ }} {g K m 4 ∞(V )(V − E K ) . (3.12)cannot exhibit saddle-node bifurcation for V > E K . (Hint: show that F ′ (V ) ≠ 0for all V > E K .)14. Show that the I h -model in Fig. 3.41C ˙V = −g L (V − E L ) −cannot exhibit saddle-node bifurcation for any V < E h .instantaneous I h{ }} {g h h ∞ (V )(V − E h ) (3.13)