# 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
• $\displaystyle{ J_{Na}(V_m,t) = G_{Na}(V_m,t) \cdot (V_m(t) - V_{Na}) }$
• $\displaystyle{ G_{Na}(V_m,t) = \frac{J_{Na}(V_m,t)}{V_m(t) - V_{Na}} }$
• $\displaystyle{ G_{K}(V_m,t) = \frac{J_{K}(V_m,t)}{V_m(t) - V_{K}} }$
• $\displaystyle{ V_m(t) - V_{Na} }$ is constant for $\displaystyle{ t \gt 0 }$ (step in potential). Same for K
• Thus conductances are simply scaled versions of the current
• Fit the current responses using following parameters
• $\displaystyle{ G_{K}(V_m,t) = \overline{G_K} n^4(V_m,t) }$ where
• $\displaystyle{ n(V_m,t) + \tau_n(V_m)\frac{dn(V_m,t)}{dt} = n_\infty(V_m) }$
• $\displaystyle{ G_{Na}(V_m,t) = \overline{G_{Na}} m^3(V_m,t)h(V_m,t) }$ where
• $\displaystyle{ m(V_m,t) + \tau_m(V_m)\frac{dm(V_m,t)}{dt} = m_\infty(V_m) }$
• $\displaystyle{ h(V_m,t) + \tau_h(V_m)\frac{dh(V_m,t)}{dt} = h_\infty(V_m) }$
• $\displaystyle{ n_\infty }$ and $\displaystyle{ m_\infty }$ are activating functions
• are about 0 at negative $\displaystyle{ V_m }$ and has asymptote 1
• $\displaystyle{ h_\infty }$ is the reverse. =1 for low $\displaystyle{ V_m }$ and 0 for high $\displaystyle{ V_m }$
• $\displaystyle{ \tau_m }$ (time constant for activating Na) is much smaller than other time constants