As for the FitzHugh's simplification of the Hodgkin/Huxley equations, let's plot the nullclines for the two-dimensional system (7,8). This is done in Matlab with the following programs. First nullclines.m:

global nu sigma k kt L q h g nu = 0.1; %nu = 0.04; sigma = 1.2; k = 0.4; kt = 0.4; L = 10^6; q = 100; h = 10; for i=1:250 g = 0.1*i; gamma(i) = g; null_alpha(i) = fzero('rhs_alpha',70); null_gamma(i) = fzero('rhs_gamma',80); end figure(2) hold on; plot(gamma,null_alpha,'b'); plot(gamma,null_gamma,'r'); axis([0 25 40 100]); xlabel('\gamma'); ylabel('\alpha');Text version of this program

Next, rhs_alpha.m:

function r = rhs_alpha(alpha) global nu sigma k L q h g r = nu - sigma*phi(alpha,g);Text version of this program

Finally, rhs_gamma.m:

function r = rhs_gamma(alpha) global nu sigma k L q h g lambda = q*sigma/h; r = lambda*phi(alpha,g) - k*g;Text version of this program

Figure 6 shows the nullclines for . Using rhs_alpha.m and rhs_gamma.m, verify that the following hold:

- to the left of the -nullcline
- to the right of the -nullcline
- above the -nullcline
- below the -nullcline.

On the other hand, Figure 7 shows the nullclines for . Here the full transient behavior is included. Note that the -nullcline intersects the -nullcline to the

Why is it that for stable, periodic oscillations are possible, while for they are not? The key is how the nullclines intersect. At this intersection, both and are constant. In the language of dynamical systems, this is a fixed point. But this fixed point could be stable or unstable. Convince yourself that when the nullclines intersect as in Figure 6 the fixed point is unstable, whereas when the nullclines intersect as in Figure 7 the fixed point is stable.

A system which has a stable fixed point but under sufficiently large
perturbation undergoes a large excursion before returning to the fixed
point is called **excitable**. This is the case for
.
Referring to Figure 7, what is the simple graphical
way to determine if a given perturbation will lead to a large excursion
before returning to the fixed point?

Comparing the material covered in this tutorial and that covered in Tutorial 2, we see that the dynamics of squid giant axons and slime molds apparently have some very nice qualitative similarities.