The Goldman-Hodgkin-Katz Equation

The Nernst equation answers a hypothetical: what voltage would exactly balance the concentration gradient of one ion, given the membrane is permeable to it alone? Real membranes violate that premise constantly — they host potassium leak channels, voltage-gated sodium channels, chloride channels, and more, all conducting simultaneously. The Goldman-Hodgkin-Katz (GHK) voltage equation generalizes Nernst logic to this realistic, multi-ion situation. It gives the membrane potential (V_m) that results when several ions cross the membrane independently, each pulled toward its own equilibrium potential, with each ion's influence weighted by its relative permeability (P_ion). The critical conceptual move is recognizing that membrane potential is not "chosen" by any single ion. It is a steady-state compromise: at V_m, the net current across the membrane is zero, even though individual ion currents are almost never zero. Sodium ions still leak in down their electrochemical gradient; potassium ions still leak out. These fluxes exactly cancel in charge terms at steady state, maintained ultimately by the Na⁺/K⁺-ATPase pump restoring concentration gradients over the long run. The GHK equation describes this instantaneous voltage balance; it does not by itself describe the slow, energy-dependent restoration of ion gradients, which is the pump's job. The equation formalizes an intuitive rule: the ion(s) with the highest permeability dominate the resting voltage. If a membrane were permeable only to K⁺, V_m would equal E_K exactly (the Nernst potential for potassium). Add a little Na⁺ permeability and V_m shifts away from E_K, toward E_Na, by an amount proportional to how much Na⁺ permeability there is relative to K⁺ permeability. This weighting principle is the single mechanistic fact that explains both the resting potential and the dramatic voltage swing of the action potential, since the action potential is fundamentally a story of P_Na and P_K trading dominance over milliseconds.

Step 1 — Start from independent ion flow. The GHK derivation assumes each ion crosses the membrane independently (the "independence principle," itself experimentally tested by Hodgkin and Huxley), driven by both its concentration gradient and the local electric field, treated as constant across the membrane thickness (the "constant field assumption"). Solving the Nernst-Planck electrodiffusion equation under this constant-field assumption yields, for a membrane permeable to K⁺, Na⁺, and Cl⁻:

Note the asymmetry: cations appear as outside-over-inside, but chloride (an anion) appears inverted, inside-over-outside — because its equilibrium requires the opposite polarity relationship.

Step 2 — Interpret each permeability term as a weight. P_K, P_Na, and P_Cl are permeability coefficients (units of velocity, cm/s), each proportional to the number of open channels for that ion and their single-channel conductance. A term with near-zero permeability contributes almost nothing to V_m regardless of how steep that ion's concentration gradient is. This is why chloride, despite being far from equilibrium in many neurons, often has only a modest effect on resting V_m — its permeability is comparatively small.

Step 3 — Recognize the resting state. At typical mammalian neuron rest:

Because P_K ≫ P_Na, the GHK equation predicts V_m close to E_K but pulled a few millivolts positive by the small Na⁺ leak — landing near −65 mV rather than −90 mV. This tiny but persistent Na⁺ leak is also why the resting potential is metabolically expensive to maintain: the Na⁺/K⁺-ATPase must continuously pump back the Na⁺ that leaks in and the K⁺ that leaks out.

Step 4 — Follow the dynamic switch during an action potential. Voltage-gated Na⁺ channels open upon depolarization, increasing P_Na by roughly 500–600-fold within a millisecond. Plugging this into GHK, the P_Na term now dominates the numerator and denominator, swinging V_m dramatically toward E_Na (+60 mV). Milliseconds later, Na⁺ channels inactivate and delayed-rectifier K⁺ channels open, P_K again dominates, and GHK pulls V_m back toward E_K — repolarization. The equation thus explains, in one continuous framework, both the quiescent resting state and the explosive, transient state of excitation.

Step 5 — Note the limits of the model. GHK is a steady-state, single-instant description; it assumes constant field and ohmic-like independent ion movement, both simplifications. It does not include electrogenic pump currents directly (though the Na⁺/K⁺-ATPase's 3:2 stoichiometry contributes a small hyperpolarizing offset in real cells), and it treats permeabilities as given constants rather than deriving them from channel gating kinetics — that dynamic gating is the domain of the Hodgkin-Huxley model built on top of it.

The GHK equation was not merely a theoretical convenience — it was rigorously tested against real, measurable membrane potentials in an unusually tractable experimental preparation: the squid giant axon, whose diameter (up to ~1 mm) is large enough to allow direct insertion of intracellular electrodes, an impossibility in ordinary mammalian axons at the time.

David Goldman (1943) first derived the constant-field equation mathematically, generalizing Nernst-Planck electrodiffusion to multiple independently permeant ions under the simplifying assumption of a constant electric field across the membrane. But a derivation is only as good as its predictions.

Alan Hodgkin and Bernard Katz (1949), working at Plymouth Marine Laboratory and University College London, performed the decisive test. Their key manipulation: systematically vary extracellular sodium concentration ([Na⁺]_o) in seawater bathing an isolated squid giant axon, while measuring both the resting potential and the peak overshoot of the action potential with an intracellular microelectrode. Their logic was sharp. The Nernst equation for potassium alone predicts that resting potential should be completely insensitive to changes in external sodium — sodium doesn't even appear in that equation. But Hodgkin and Katz found that:

• Reducing [Na⁺]_o (replacing NaCl with an impermeant substitute like choline chloride) caused the resting potential to become slightly more negative (hyperpolarized), a small but real and reproducible shift — inconsistent with a pure-K⁺ Nernst model, but exactly what GHK predicts since lowering the Na⁺ term in the numerator/denominator pulls V_m toward E_K.
• Much more dramatically, the peak (overshoot) of the action potential, which had already been shown by Hodgkin and Huxley to approach E_Na, fell almost linearly with the logarithm of [Na⁺]_o, and did so with a slope close to the 58 mV per 10-fold change predicted by the Nernst/GHK relation at ~18–20 °C (the RT/F·ln(10) factor). Cutting external Na⁺ to about a third of normal seawater concentration reduced the overshoot by a predictable, quantitatively matched amount.
• Critically, at the peak of the action potential, when sodium permeability transiently dominates (P_Na ≫ P_K), the GHK equation reduces almost exactly to the Nernst equation for sodium — and the measured overshoot tracked E_Na, not a fixed value.

This was powerful because it demonstrated the same membrane, under the same equation, correctly predicting two very different voltages — a K⁺-dominated resting state and a transiently Na⁺-dominated peak — simply by allowing the weighting (permeability) to change between the two states, exactly as the constant-field equation formalizes. No single-ion Nernst model could explain both observations with one framework; GHK could, and its curve fit the data across a wide range of external Na⁺ concentrations, not just at one point. Later work by Hodgkin and Huxley (1952), using the voltage clamp technique they had developed, showed directly that P_Na and P_K (expressed as conductances g_Na and g_K) really do change dynamically and asymmetrically during the action potential — g_Na rising and falling within about a millisecond, g_K rising more slowly and staying elevated longer — providing the mechanistic underpinning for exactly the permeability shifts that the 1949 ionic-substitution experiments had already inferred indirectly.

Example 1 — Computing V_m for a neuron with three permeant ions. Take a mammalian central neuron at 37 °C (RT/F = 26.7 mV; using log base 10, RT/F·ln(10) ≈ 61.5 mV) with these values:

Numerator (cations outside + anion inside): P_K[K]_o + P_Na[Na]_o + P_Cl[Cl]_i = (1.00)(5) + (0.04)(145) + (0.45)(10) = 5 + 5.8 + 4.5 = 15.3 Denominator (cations inside + anion outside): P_K[K]_i + P_Na[Na]_i + P_Cl[Cl]_o = (1.00)(140) + (0.04)(15) + (0.45)(110) = 140 + 0.6 + 49.5 = 190.1 Ratio = 15.3 / 190.1 = 0.0805 V_m = 61.5 mV × log₁₀(0.0805) = 61.5 × (−1.094) ≈ −67.3 mV Compare to a pure-K⁺ Nernst prediction: E_K = 61.5 × log₁₀(5/140) = 61.5 × (−1.447) ≈ −89.0 mV The small Na⁺ and Cl⁻ terms pull V_m about 22 mV positive of E_K — this is precisely the gap a student measuring a real neuron's resting potential (typically −60 to −70 mV) will notice relative to the K⁺ equilibrium potential alone, and GHK, not Nernst, resolves it.

Example 2 — Predicting the effect of a channelopathy. Suppose a gain-of-function mutation in a persistent Na⁺ channel increases P_Na tenfold at rest (from 0.04 to 0.4), while P_K and P_Cl are unchanged (a simplified model of certain neurological channelopathies, such as some forms of paramyotonia or neuronal hyperexcitability syndromes). New numerator = (1.00)(5) + (0.4)(145) + (0.45)(10) = 5 + 58 + 4.5 = 67.5 New denominator = (1.00)(140) + (0.4)(15) + (0.45)(110) = 140 + 6 + 49.5 = 195.5 Ratio = 67.5 / 195.5 = 0.3453 V_m = 61.5 × log₁₀(0.3453) = 61.5 × (−0.462) ≈ −28.4 mV A tenfold increase in P_Na, still far short of dominating over P_K, nonetheless depolarizes the resting membrane by nearly 39 mV (from −67.3 to −28.4 mV) — enough to inactivate many voltage-gated Na⁺ channels and either trigger runaway firing or paradoxically silence the cell by depolarization block, depending on channel kinetics. This illustrates why even modest permeability changes, especially to an ion whose gradient is steep and opposite in sign to the resting potential, can have outsized clinical consequences — the logic behind many channelopathy-driven excitability disorders.

David E. Goldman (1910–1998) was an American biophysicist whose 1943 paper, "Potential, Impedance, and Rectification in Membranes," published in the Journal of General Physiology, provided the mathematical backbone later bearing his name. Goldman's derivation solved the electrodiffusion (Nernst-Planck) equation under the constant-field assumption for multiple ion species — an elegant piece of applied mathematics developed years before the ionic basis of the action potential itself was understood. His work initially attracted less attention than it deserved, since the tools to test it experimentally (intracellular microelectrodes in sufficiently large cells) did not yet exist. That changed with Alan Lloyd Hodgkin (1914–1998) and Bernard Katz (1911–2003), working at the Plymouth Marine Laboratory in the late 1940s, exploiting the uniquely large squid giant axon as an experimental preparation. Hodgkin, already deeply engaged in unraveling axonal excitability, and Katz, a German-Jewish émigré physiologist who had fled Nazi Germany in 1935 and would go on to major contributions on synaptic transmission and quantal neurotransmitter release, combined ionic-substitution experiments with the mathematical Goldman framework to produce the definitive 1949 paper, "The Effect of Sodium Ions on the Electrical Activity of the Giant Axon of the Squid," in the Journal of Physiology.

Patch-clamp electrophysiology relies on GHK logic constantly: when a researcher measures a channel's reversal potential in a mixed-ion recording solution, they use the GHK equation (or its simplified bi-ionic form) to infer the channel's relative permeability to different ion species — a routine step in characterizing any newly cloned or mutant ion channel.

Cardiac arrhythmia research and antiarrhythmic drug design depend on the same weighting logic: drugs that block specific K⁺ channels (e.g., HERG/Kv11.1 blockers implicated in drug-induced long QT syndrome) shift P_K downward, moving cardiac membrane potential and repolarization timing in a GHK-predictable direction, with direct consequences for arrhythmia risk assessment in drug safety testing.

Local anesthetics and antiepileptic drugs that block voltage-gated Na⁺ channels work by suppressing the P_Na surge that GHK shows is necessary for the depolarizing phase of action potentials — without that permeability shift, GHK predicts V_m simply cannot approach E_Na, and the action potential fails to propagate, the direct basis of nerve blocks used in regional anesthesia.

Channelopathy diagnosis, from periodic paralysis to certain epilepsies and cardiac Brugada/long-QT syndromes, is interpreted through exactly the kind of permeability-shift calculation shown in Worked Example 2 — clinicians and researchers reason from mutation-induced changes in channel gating or expression to predicted shifts in resting and threshold potentials.

Renal and epithelial physiology also use GHK-type multi-ion equations to model transepithelial potentials across kidney tubule cells and other ion-transporting epithelia, where several ion species cross in parallel through distinct channel and transporter populations, directly extending the same weighted-permeability logic beyond neurons.

Ask the tutor: How do voltage-gated channel kinetics (Hodgkin-Huxley gating variables) generate the time-varying permeabilities that the GHK equation converts into the action potential waveform?