Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / B / 5 / a

The nullclines are the cubic $w=f(v)$ and the straight line $w=\gamma v$. At an equilibrium either $v=0$, or $\gamma =(a-v)(v-1)$. The latter quadratic has maximum $(1-a{)}^2/4$, so $\gamma >(1-a{)}^2/4$ leaves only the resting state $(0,0)$. Its Jacobian matrix isBoth eigenvalues have negative real parts; the rest state is linearly stable and asymptotically stable.

On the fast time $t/ϵ$, the recovery variable $w$ is approximately fixed. At $w\simeq 0$, the scalar fast equation has stable roots $v=0,1$ and an unstable root $v=a$. A small kick below $a$ decays towards rest; a kick above it moves rapidly towards the right attracting branch. For finite $ϵ$, the threshold is displaced from the frozen root by slow recovery, but the leading threshold mechanism is this unstable middle branch.

After the fast excitation, $w_t=\gamma v-w>0$ raises $w$ while $v$ follows the right attracting branch of the cubic. At its upper fold the branch ends and $v$ makes a fast downward jump to the left branch. Recovery then lowers $w$ slowly, with a negative undershoot in $v$, before return towards $(0,0)$. Thus the action potential time trace has rapid depolarization, a slower high-voltage excursion, rapid repolarization, and slow recovery. On the fast segments $w$ scarcely changes; on slow segments $w\simeq f(v)$ and $f^{\prime }(v)<0$. The figure shows a numerically integrated superthreshold excursion and its time trace, rather than just arrows on the nullclines.