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FitzHugh's Reduction of the Hodgkin Huxley Equations

Then we have a two variable system
$\displaystyle C{d\phi\over dt}$ $\textstyle =$ $\displaystyle \bar g_{Na} m_\infty^3(\phi)(N-n)(\phi-v_{Na})+ \bar g_Kn^4(\phi-v_K) + \bar g_l(\phi-v_L),$ (1)
$\displaystyle {dn \over dt}$ $\textstyle =$ $\displaystyle \alpha_n(1-n) - \beta_n n.$ (2)

where
$\displaystyle \alpha_m$ $\textstyle =$ $\displaystyle 0.1\, {25 - \phi \over \exp\left({25 -\phi \over 10}\right)-1},$ (3)
$\displaystyle \beta_m$ $\textstyle =$ $\displaystyle 4\, \exp\left({-\phi \over 18}\right),$ (4)
$\displaystyle \alpha_n$ $\textstyle =$ $\displaystyle 0.01\, {10-\phi\over \exp({10-\phi\over 10})-1},$ (5)
$\displaystyle \beta_n$ $\textstyle =$ $\displaystyle 0.125\, \exp\left({-\phi\over 80}\right).$ (6)

and
\begin{displaymath} m_\infty(\phi) = {\alpha_m\over \alpha_m+\beta_m} \end{displaymath} (7)


Phase portrait: