6.021/Notes/2006-10-27

Hodgkin-Huxley

 * assumed conductances on depend on membrane potential and not concentrations
 * used this to determine contribution of Na and K currents by fixing membrane potential and changing concentrations which affect Nernst potentials only
 * persistent current primarily due to K
 * transient current due to Na
 * $$J_{Na}(V_m,t) = G_{Na}(V_m,t) \cdot (V_m(t) - V_{Na})$$
 * $$G_{Na}(V_m,t) = \frac{J_{Na}(V_m,t)}{V_m(t) - V_{Na}}$$
 * $$G_{K}(V_m,t) = \frac{J_{K}(V_m,t)}{V_m(t) - V_{K}}$$
 * $$V_m(t) - V_{Na}$$ is constant for $$t > 0$$ (step in potential). Same for K
 * Thus conductances are simply scaled versions of the current
 * Fit the current responses using following parameters
 * $$G_{K}(V_m,t) = \overline{G_K} n^4(V_m,t)$$ where
 * $$n(V_m,t) + \tau_n(V_m)\frac{dn(V_m,t)}{dt} = n_\infty(V_m)$$
 * $$G_{Na}(V_m,t) = \overline{G_{Na}} m^3(V_m,t)h(V_m,t)$$ where
 * $$m(V_m,t) + \tau_m(V_m)\frac{dm(V_m,t)}{dt} = m_\infty(V_m)$$
 * $$h(V_m,t) + \tau_h(V_m)\frac{dh(V_m,t)}{dt} = h_\infty(V_m)$$
 * $$n_\infty$$ and $$m_\infty$$ are activating functions
 * are about 0 at negative $$V_m$$ and has asymptote 1
 * $$h_\infty$$ is the reverse. =1 for low $$V_m$$ and 0 for high $$V_m$$
 * $$\tau_m$$ (time constant for activating Na) is much smaller than other time constants