Nerve cells (Hodgkin-Huxley Equations)

$$C_m {d\phi\over dt} = - \bar g_{Na} m^3h(\phi - V_{Na}) - \bar g_K n^4(\phi - V_K) - g_L(\phi - V_L) + I_{\rm app},$$ (1)

and

$${dm \over dt} = \alpha_m(1-m) - \beta_m m,$$ (2)

$${dn \over dt} = \alpha_n(1-n) - \beta_n n,$$ (3)

$${dh \over dt} = \alpha_h(1-h) - \beta_h h.$$ (4)

The specific functions $\alpha$ and $\beta$ proposed by Hodgkin and Huxley were (in units of ms$^{-1}$)

$$\alpha_m = 0.1\, {25 - \phi \over \exp\left({25 -\phi \over 10}\right)-1},$$ (5)

$$\beta_m = 4\, \exp\left({-\phi \over 18}\right),$$ (6)

$$\alpha_h = 0.07\, \exp\left({-\phi\over 20}\right),$$ (7)

$$\beta_h = {1\over\exp({30-\phi\over 10})+1},$$ (8)

$$\alpha_n = 0.01\, {10-\phi\over \exp({10-\phi\over 10})-1},$$ (9)

$$\beta_n = 0.125\, \exp\left({-\phi\over 80}\right).$$ (10)

Equations for $m, n, h$ can be written as

$$\tau_n(\phi){dw \over dt} = w_\infty(\phi) - w, w = m, n, {\rm or} h$$ (11)