The FitzHugh-Nagumo Equations

A qualitative model which captures the behavior of the Hodgkin/Huxley equations (and equations (2) and (3)) is given by the FitzHugh-Nagumo equations:

$$\epsilon \frac{dv}{dt} = F(v) - w + I,$$ (4)

$$\frac{dw}{dt} = v - \gamma w,$$ (5)

where $F(v) = v (1-v) (v+a)$, $\epsilon \ll 1$, and $a$, $I$, and $\gamma$ are constants. These equations have the advantage over equations (2) and (3) that the righthand sides are simpler functions. The FitzHugh-Nagumo equations have been used to qualitatively model many biological phenomena (see, for example, Mathematical Physiology by J. Keener and J. Sneyd, on reserve in the library).

Using the programs given in this tutorial as models, use Matlab to draw the nullclines and solve the FitzHugh-Nagumo equations for $a = 0.1$, $\gamma = 0.5$, and different values of $I$.