= Solution
The equivalent circuit has <membrane capacitance> in parallel with <electrical conductance>–battery branches: voltage-dependent <sodium> and <potassium> conductances in series with their <reversal potentials>, a <chloride> <electrical conductance> in series with $E_{Cl}$, and other leak conductances. For externally supplied <electric current>,
$$
C_m\dot V=I_{\rm ext}-g_{Na}(V,t)(V-E_{Na})-g_K(V,t)(V-E_K)
-g_{Cl}(V-E_{Cl})-g_L(V-E_L).
$$
The model figure in <past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2018/2/c/solution> includes this circuit and an illustrative passive response. Although <chloride> <electric current> is small when $V\simeq E_{Cl}$, a perturbation produces the restoring <electric current> $\delta I_{Cl}=g_{Cl}\delta V$. A large <chloride> <electrical conductance> therefore shunts perturbations, lowers input resistance and shortens the passive <time constant>, $R_{\rm in}\simeq1/\sum g_j$ and $\tau_m\simeq C_m/\sum g_j$.
Reducing that <electrical conductance> increases the voltage response to a given <electric current> or extracellular-<potassium> disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if $E_{Cl}$ equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.<The skeletal muscle chloride channel in dominant and recessive human myotonia by Manuela C. Koch and others>[Channel defects associated with inherited muscle hyperexcitability]{ref}
Later cessation of contraction can be explained by sustained <depolarization> causing <voltage-gated sodium channel> <ion-channel inactivation> and failure to recover: fewer available channels support progressively smaller or less frequent spikes until <depolarization> block occurs. Accumulated extracellular <potassium> in restricted spaces and incomplete recovery between spikes can promote that <depolarization>. This requires the nonlinear voltage-gating dynamics; a passive <chloride> resistor and capacitor alone cannot generate or terminate an action-potential train. The stated <membrane permeability> change does not uniquely determine the time to block or prove one particular ionic accumulation history.
Back to article page
