Dynamical Systems in Neuroscience:

Conductance-Based Models 165500I slow (V) = g(V-E K )500I slow (V) = g(V-E K )400400membrane current, Imembrane current, I3002001000-100 E K -60 -20 0 20membrane voltage, V (mV)5004003002001000 15I-I fast (V)0 15I-I fast (V)I slow (V) = g(V-E K )membrane current, I membrane current, I3002001000E K-100 -60 -20 0 20membrane voltage, V (mV)5004003002001000 15I-I fast (V)0 15I-I fast (V)I slow (V) = g(V-E K )00-100 E K -60 -20 0 20membrane voltage, V (mV)-100 E K -60 -20 0 20membrane voltage, V (mV)Figure 5.25: Ex. 5: The (V, I)-phase plane of the I Na,p +I K -model (compare withFig. 5.4).6. Show that I Cl +I K -model can have oscillations. (Hint: inject negative dc-currentso that the voltage-gated Cl − current becomes inward/amplifying).7. (NMDA+I K -model) Show that a neuronal model consisting of an NMDA currentand a resonant current, say I K , can exhibit excitability and periodic spiking.8. The Nernst potential of an ion is a function of its concentration inside/outside thecell membrane, which may change. Consider the I Na,p +E Na ([Na + ] in/out )-modeland show that it can exhibit excitability and oscillations on a slow time scale.9. Determine when the I A -model has a limit cycle attractor without assumingτ h (V ) ≪ τ m (V ).10. [Ph.D.] There are Na + -gated and Cl − -gated currents besides the Ca 2+ -gatedcurrents considered in this book. In addition, the Nernst potentials may changeas concentrations of ions inside/outside the cell membrane change. This maylead to new minimal models. Classify and study all these models.