E.2 The Goldman–Hodgkin–Katz equation and the resting potential

A real membrane passes more than one ion. Each ion has its own Nernst potential, and the membrane cannot sit at all of them at once. It settles instead at the voltage where the net current — summed over every permeant ion — vanishes. That compromise voltage is the resting potential, and the Goldman–Hodgkin–Katz equation gives it.

A weighted compromise

When several ions cross a membrane with different permeabilities, the steady-state voltage at which their currents cancel is the Goldman–Hodgkin–Katz (GHK) equation:

V_\text{rest} \;=\; \frac{RT}{F}\ln\frac{P_K c_K^\text{out} + P_\text{Na} c_\text{Na}^\text{out} + P_\text{Cl} c_\text{Cl}^\text{in}}{P_K c_K^\text{in} + P_\text{Na} c_\text{Na}^\text{in} + P_\text{Cl} c_\text{Cl}^\text{out}}.

where

$V_\text{rest}$: resting membrane potential V
$P_X$: membrane permeability to ion $X$ m/s
$c_X^\text{in/out}$: intra/extracellular concentration of $X$ mol/m³

The chloride terms appear inverted (in over out) because its charge is negative. Each ion contributes in proportion to its permeability: the resting potential is a permeability-weighted average of the individual Nernst potentials, pulled toward whichever ion the membrane passes most freely.

P_K = 1.00 P_Na = 0.04 P_Cl = 0.45

The resting potential sits *between* the individual Nernst potentials, pulled toward the most-permeable ion. At a typical neuron's rest (P_K ≫ P_Na, P_Cl), V_rest is near E_K ≈ −90 mV. During an action potential, P_Na shoots up by ~100× and V_rest swings toward E_Na ≈ +60 mV — the depolarisation phase. The membrane is a *voltage-dependent permeability filter*, and the GHK equation is its statement.

Rest and excitation

The GHK equation makes the membrane a voltage-dependent permeability filter. At rest the potassium permeability dominates, $P_K \gg P_\text{Na}, P_\text{Cl}$ , so the resting potential sits close to the potassium Nernst potential, $E_K \approx -90\,\text{mV}$ . When voltage-gated sodium channels open during an action potential, $P_\text{Na}$ rises roughly a hundredfold, and the resting potential swings toward $E_\text{Na} \approx +60\,\text{mV}$ — the depolarising upstroke. Repolarisation restores the potassium dominance and the voltage falls back.

This permeability-weighted voltage is the baseline on which every hair cell and neuron operates. The cochlea then adds a twist the GHK equation alone does not capture: a compartment held at a large positive voltage by active pumping, which supplies the energy the hair cell uses to detect sound. That is the endocochlear potential of the next lesson.