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