Nullclines, Oscillations, and Excitability for the Two-Dimensional Model

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 $\nu = 0.1$. Using rhs_alpha.m and rhs_gamma.m, verify that the following hold:

• $\frac{d \alpha}{dt} > 0$ to the left of the $\alpha$-nullcline
• $\frac{d \alpha}{dt} < 0$ to the right of the $\alpha$-nullcline
• $\frac{d \gamma}{dt} > 0$ above the $\gamma$-nullcline
• $\frac{d \gamma}{dt} < 0$ below the $\gamma$-nullcline.

Note that the $\alpha$-nullcline intersects the $\gamma$ nullcline to the right of the local maximum. This figure also shows the superimposed stable periodic orbit. Compare this figure to Figure 6 of Tutorial 2. One of $\alpha$ and $\gamma$ can be thought of as a slow variable, and the other as a fast variable (although not as fast as $\beta$). Which is which?

Figure 6: The nullclines for equations (7,8) for $\nu = 0.1 s^{-1}$. The blue line is the $\alpha$-nullcline, and the red line is the $\beta$-nullcline. The stable periodic orbit is shown in green (the transient behavior is not plotted).

On the other hand, Figure 7 shows the nullclines for $\nu = 0.04$. Here the full transient behavior is included. Note that the $\alpha$-nullcline intersects the $\gamma$-nullcline to the left of the local maximum.

Figure 7: The nullclines for equations (7,8) for $\nu = 0.04 s^{-1}$. The blue line is the $\alpha$-nullcline, and the red line is the $\beta$-nullcline. The trajectory is shown in green.

Why is it that for $\nu = 0.1 s^{-1}$ stable, periodic oscillations are possible, while for $\nu = 0.04 s^{-1}$ they are not? The key is how the nullclines intersect. At this intersection, both $\alpha$ and $\gamma$ 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 $\nu = 0.04 s^{-1}$. 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.