1Physiology 472/572 - 2010 - Quantitative modeling of biological systems Lecture 9: Nerve impulse propagation - Part III Reference: Keener and Sneyd (2009), Mathematical Physiology, Volume 1, pp. 210-216 Behavior of conductance variables m, n and h during action potential • Em and m vary much faster than n and h • n + h stays almost constant, about 0.8 'Fast-slow' model of action potential • consider space-clamped behavior • let v(t) = Em - Eeq, vNa = ENa - Eeq, etc. where Eeq is resting potential • system consists of equations for m, n and h, together with ) v- (vg ) v- (vng ) v- h(vmg tdvdC-LLK4KmaxNa3Namax ++= • assume that m = m(v) because it varies fast • assume h = 0.8 - n, reduces to a system of two equations n)v()n1)(v(dtdn) v- (vg ) v- (vng ) v- n)(v - (0.8m(v)g tdvdC-nnLLK4KmaxNa3Namax βα+−=++= • this is a second-order dynamical system, v is a 'fast' variable and n is a slow variable2Phase plane representation for a second-order system • current state of system is represented by a point in the (v,n) plane • the vector (dv/dt,dn/dt) gives the direction of the trajectories in the (v,n) plane • v-nullcline is the curve where dv/dt = 0, similarly n-nullcline • equilibrium point of system is where nullclines cross • behavior can be deduced by following trajectories in phase plane Phase plane for 'fast-slow' model of action potential • rapid changes of v can occur for constant n • n can only change slowly, trajectories with changing n lie close to the dv/dt nullcline • starting near the equilibrium point, the system makes a rapid excursion and returns to the equilibrium point during an action potential • with a slight change in conditions, this system can become oscillatory • an example of a limit-cycle
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