# 6.021/Notes/2006-10-27

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## 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 ${\displaystyle t>0}$ (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