Problem 1:
Show that, if k >1, then (1 - e ^{-x})^{k} has an inflection point, however (e ^{-x})^{k} does not.
Problem 2:
Plot the null clines of the Hodgkin-Huxley fast subsystem. Show that v_{r} and ve in Hodgkin-Huxley fast subsystem are steady states, while v_{s} is a saddle point. find out the stable manifold of saddle point and find out sample trajectories in the fast phase-plane, demonstrating threshold effect.
Problem 3:
Plot the null clines of fast-slow Hodgkin-Huxley phase-plane and find out a complete action potential.
Problem 4:
Assume that in the Hodgkin-Huxley fast-slow phase-plane, v is slowly diminish to v*< v_{0} (where v_{0} is the steady state), held there for a considerable time, and then released. Illustrate what happens in qualitative terms, i.e., without actually calculating the solution. This is called anode break excitation. What happens if v is instantaneously decreased to v*and then released immediately? Why do these two solutions differ?
Problem 5:
Solve the full Hodgkin-Huxley equations numerically with a variety of constant current inputs. For what range of inputs are there self-sustained oscillations? Construct bifurcation diagram.