# 6.021/Notes/2006-10-27

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## 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
• $J_{Na}(V_m,t) = G_{Na}(V_m,t) \cdot (V_m(t) - V_{Na})$
• $G_{Na}(V_m,t) = \frac{J_{Na}(V_m,t)}{V_m(t) - V_{Na}}$
• $G_{K}(V_m,t) = \frac{J_{K}(V_m,t)}{V_m(t) - V_{K}}$
• Vm(t) − VNa 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 Vm and has asymptote 1
• $h_\infty$ is the reverse. =1 for low Vm and 0 for high Vm
• τm (time constant for activating Na) is much smaller than other time constants