How fast do inactivated sodium channels come back, and what shape does their recovery take? · Lobeworks/17

Exponentially.

How fast do inactivated sodium channels come back, and what shape does their recovery take?

Exponentially. Once the membrane is back near rest, the fraction of channels whose inactivation gate has reopened follows h(t)=h∞−(h∞−h0) e−t/τhh(t) = h_\infty - (h_\infty - h_0),e^{-t/\tau_h}h(t)=h∞​−(h∞​−h0​)e−t/τh​, so it climbs fast at first and then ever more slowly towards the resting availability h∞h_\inftyh∞​. The time constant τh\tau_hτh​ is a few milliseconds and depends on voltage and temperature: a more depolarised membrane or a colder one recovers more slowly. The threshold for a second spike falls along the same curve, which is why the relative refractory period has no sharp end.