# 6.021/Notes/2006-10-30

## Hodgkin-Huxley

• state variables: $\displaystyle{ m, n, h, V_m }$
• using $\displaystyle{ m(t_0), n(t_0), h(t_0), V_m(t_0) }$ and the input $\displaystyle{ J_m(t) }$ for $\displaystyle{ t\gt t_0 }$, we can propagate into the future to calculate all of the variables
• For example, $\displaystyle{ \frac{dm}{dt}=\frac{m_\infty(V_m)-m(V_m,t)}{\tau_m(V_m)} }$
• To calculate next value of the membrane potential, solve the circuit model
• If you run the HH model by appyling a current, you get an action potential!
• Response to current pulse:
1. $\displaystyle{ J_m \rightarrow \Delta V_m }$
2. $\displaystyle{ V_m\uparrow \rightarrow m\uparrow \rightarrow G_{Na}\uparrow \rightarrow V_m\uparrow }$ (positive feedback)
• Both $\displaystyle{ m }$ and $\displaystyle{ V_m }$ increase about exponentially until $\displaystyle{ V_m }$ about the max ($\displaystyle{ V_{Na} }$)
3. Negative feedback until membrae potential drops to below rest
• $\displaystyle{ V_m \gt V_m^o \rightarrow n\uparrow \rightarrow G_K\uparrow \rightarrow V_m\downarrow }$
• $\displaystyle{ V_m \gt V_m^o \rightarrow h\downarrow \rightarrow G_{Na}\downarrow \rightarrow V_m\downarrow }$
4. n & h need to be reset to original values. Explains why action potential is refractory
• Put HH model of membrane behavior into core conductor model
• assume constant speed of propagation
• As speed of propagation not part of HH model, guess/fit
• The Hodgkin-Huxley model can account for decrement-free conduction