6.021/Notes/2006-11-06

Threshold in Hodgkin-Huxley model

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