# 6.021/Notes/2006-11-06

## Threshold in Hodgkin-Huxley model

• threshold is sharp
• change in $10^{-8}-10^{-14}$ can change AP to non-AP in model
• determine threshold in model
• asssume n & h are so slow that $n(V_m,t)=n_\infty(V_m^o)$ and $h(V_m,t)=h_\infty(V_m^o)$
• Also m is so fast that $m(V_m,t)=m_\infty(V_m)$
• 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 $V_m$
• To be at equilibrium, must be on isoclines
• $\frac{dm}{dt} = 0 \rightarrow m=m_\infty$
• $\frac{V_m}{dt} = 0$
• These two lines again cross 3 times, with one point being unstable
• The separatrix curve in $m-V_m$ space determines whether will go to rest or $V_{Na}$
• So threshold depends on both m and $V_m$
• If instead of fixing h to $h_\infty$, 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