Goldman Equation Membrane Potential

The Goldman-Hodgkin-Katz (GHK) equation calculates the resting membrane potential when multiple ions with different permeabilities are present simultaneously. It is a generalization of the Nernst equation.

For three ions (K+, Na+, Cl-), the membrane potential at 37 degrees C is:

V_m = (RT/F) * ln( (P_K[K]_o + P_Na[Na]_o + P_Cl[Cl]_i) / (P_K[K]_i + P_Na[Na]_i + P_Cl[Cl]_o) )

Note that chloride concentrations appear swapped (inside numerator, outside denominator) because Cl- is an anion with a negative charge.

At 37 degrees C, RT/F = 26.73 mV.

Typical mammalian neuron values. Extracellular: [K+]_o = 5 mM, [Na+]_o = 145 mM, [Cl-]_o = 110 mM. Intracellular: [K+]_i = 140 mM, [Na+]_i = 15 mM, [Cl-]_i = 10 mM. Relative permeabilities at rest: P_K : P_Na : P_Cl = 1 : 0.04 : 0.45. These values produce a resting potential of roughly -70 mV.

Why not just use the Nernst equation? The Nernst equation gives the equilibrium potential for one ion in isolation. Real membranes are permeable to several ions, none of which is at its individual equilibrium. The Goldman equation weights each ion by its permeability.

The GHK equation also underlies the Hodgkin-Huxley model: during an action potential, P_Na spikes dramatically, shifting V_m toward the sodium reversal potential (+60 mV), before K+ permeability rises to restore the resting state.