Past exam of the biological physics course of the University of Oxford / 2017 / 4 / e / Solution

A two-state closed–open scheme at fixed voltage relaxes monotonically and cannot by itself produce a sodium-current transient during a maintained step with constant driving force. A minimal sodium scheme therefore includes an unavailable state: $C\rightleftarrows O\rightleftarrows I$, with voltage-dependent activation, inactivation and recovery; closed-state recovery pathways may be added when needed. Opening is initially rapid, occupancy then accumulates in $I$, and recovery on repolarization restores an activatable population.

For the illustrated potassium current, a closed–open model gives sustained opening, but several activation steps can account for its initial delay. A conventional illustrative coarse-grained description iswith each gate following $\dot{q}={\alpha }_q(V)(1-q)-{\beta }_q(V)q$. Fast $m$ activation, subsequent decrease of $h$, and slower increase of $n$ produce the two qualitative records. The powers are phenomenological gate assumptions, not values uniquely inferred from these traces or an exact structural count for every cardiac channel.

A 1-Hz cardiac rhythm has about a second between successive activations, whereas a 100-Hz neuronal rhythm has only $10\mathrm{ms}$; the available resting interval is shorter still after subtracting the action potential. Recovery or deactivation with a timescale comparable to that interval can then leave residual inactivation or potassium activation, giving use-dependent changes and refractoriness. Millisecond or faster relaxation usually completes much more readily during a long diastolic interval. The particular two-state A and B in d would both nearly equilibrate over a full $10\mathrm{ms}$ at their stipulated constant rates; their values alone do not demonstrate high-frequency failure. Real comparison requires voltage-dependent recovery rates and the different cardiac and neuronal waveforms. A frozen-rate model cannot quantitatively predict either firing pattern from frequency alone.