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 vr and ve in Hodgkin-Huxley fast subsystem are steady states, while vs is a saddle point. Calculate the stable manifold of saddle point and calculate sample trajectories in the fast phase-plane, demonstrating threshold effect.
Problem 3:
Plot the null clines of fast-slow Hodgkin-Huxley phase-plane and calculate a complete action potential.
Problem 4:
Assume that in the Hodgkin-Huxley fast-slow phase-plane, v is slowly diminish to v*< v0 (where v0 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.