# 6.021/Notes/2006-11-06

## Threshold in Hodgkin-Huxley model

• threshold is sharp
• change in $\displaystyle{ 10^{-8}-10^{-14} }$ can change AP to non-AP in model
• determine threshold in model
• asssume n & h are so slow that $\displaystyle{ n(V_m,t)=n_\infty(V_m^o) }$ and $\displaystyle{ h(V_m,t)=h_\infty(V_m^o) }$
• Also m is so fast that $\displaystyle{ m(V_m,t)=m_\infty(V_m) }$
• The potassium current is constant as the the potassium conductance doesn't change
• Find that there are 2 stable equilibrium points and 1 unstable point
• The unstable point is the threshold voltage
• We can relax assumption that m is instant and instead obeys the standard HH model for m
• Make phase plane showing m vs $\displaystyle{ V_m }$
• To be at equilibrium, must be on isoclines
• $\displaystyle{ \frac{dm}{dt} = 0 \rightarrow m=m_\infty }$
• $\displaystyle{ \frac{V_m}{dt} = 0 }$
• These two lines again cross 3 times, with one point being unstable
• The separatrix curve in $\displaystyle{ m-V_m }$ space determines whether will go to rest or $\displaystyle{ V_{Na} }$
• So threshold depends on both m and $\displaystyle{ V_m }$
• If instead of fixing h to $\displaystyle{ h_\infty }$, we set it to another value, as h decreases, the isoclines change such that thresholds increase until a point when the curves only intersect once at rest
• This explains the relative and absolute refractory period
• The relative refractory period is characterized by higher threshold
• During the absolute refractory period it is impossible to reach threshold no matter the amount of stimulus