The driving forces
Ions respond to chemical and electrical gradients together
What you should already have
About 45 minutes
Plus the time it takes to redraw this lesson’s figures from memory, which is the fastest way to find out what you have not understood.
Where this shows up
Two equations in this lesson explain most of what a disordered potassium does to a patient. The Nernst and Goldman-Hodgkin-Katz equations answer different questions, and knowing which one applies is the difference between predicting the effect of hyperkalaemia and guessing at it. Everything in neurophysiology and cardiac electrophysiology builds directly on this.
Learning outcomes
By the end of this lesson you should be able to:
- Explain how ions respond to chemical and electrical gradients together, and what an electrochemical equilibrium is.
- Use the Nernst equation to convert a concentration ratio into a voltage, defining every symbol and its units.
- Explain how Goldman-Hodgkin-Katz differs from Nernst by weighting concentrations by permeability, and say which question each answers.
- Account for the resting potential as a permeability-weighted diffusion potential, and state the pump's contribution to it.
- Explain why hyperkalaemia is not uniformly excitatory.
- Describe the action potential as regenerative channel behaviour, and explain how local circuit current propagates it along an axon.
Together these settle one syllabus objective: Membrane potentials, the Nernst and GHK equations, and the action potential. Tick it on the Physiology objective list once you can do all of the above without notes.
| Gradient | Rule | Example |
|---|---|---|
| Chemical | Movement from higher to lower concentration | K⁺ tends to leave most cells |
| Electrical | Cations move toward negative potential; anions toward positive potential | Negative cell interior attracts cations |
| Electrochemical | Net effect of chemical and electrical forces | At rest, both forces drive Na⁺ inward |
A membrane potential requires only a minute separation of charge immediately at the membrane. The bulk intracellular and extracellular solutions remain nearly electrically neutral throughout. At electrochemical equilibrium for a given ion, the electrical force exactly opposes the chemical force, so there is no further net movement of that ion — this equilibrium condition is exactly what the Nernst equation calculates.
One ion at a time
The Nernst equation converts a concentration ratio into a voltage
| Symbol | Meaning | Units |
|---|---|---|
| Eion | Equilibrium potential for one ion | mV (millivolts) |
| R | Universal gas constant | J mol⁻¹ K⁻¹ |
| T | Absolute temperature | K |
| z | Valence (charge) | dimensionless |
| F | Faraday constant | C mol⁻¹ |
| [ion]o/[ion]i | Extracellular/intracellular concentration | any consistent units — the ratio is what matters |
Worked example — potassium
[K⁺]o = 4 mmol L⁻¹, [K⁺]i = 140 mmol L⁻¹, z = +1. EK = 61.5 × log₁₀(4/140) = 61.5 × log₁₀(0.0286) = 61.5 × (−1.544) ≈ −95 mV. Sanity check: potassium’s chemical gradient favours efflux (high inside, low outside), so at equilibrium the inside must be negative enough to hold that efflux back — a negative EK is the expected sign.
Worked example — calcium (note the divalent valence)
[Ca²⁺]o ≈ 1.2 mmol L⁻¹, [Ca²⁺]i ≈ 0.0001 mmol L⁻¹ (100 nmol L⁻¹), z = +2. ECa = (61.5/2) × log₁₀(1.2/0.0001) = 30.75 × log₁₀(12 000) = 30.75 × 4.08 ≈ +125 mV. Two points to verify: dividing by z = 2 halves the voltage needed for the same concentration ratio, and the steep 12 000-fold gradient is why a small calcium leak produces a large, fast signal — the physiological basis for calcium’s role as an intracellular messenger (developed in the next lesson).
Several permeant ions
Goldman-Hodgkin-Katz weights concentrations by permeability
| Symbol | Meaning | Units |
|---|---|---|
| Vm | Membrane potential | mV (millivolts) |
| PK, PNa, PCl | Relative membrane permeability to that ion | dimensionless (ratio between the three) |
| [ion]o / [ion]i | Extracellular / intracellular concentration of that ion | any consistent units — mmol L⁻¹ below |
Chloride’s terms are inverted (intracellular on top, extracellular on the bottom) because it is an anion — its electrical response to a given voltage is opposite to a cation’s. Use Nernst for an individual ion’s equilibrium potential; use Goldman-Hodgkin-Katz for the actual membrane potential of a membrane permeable to several ions simultaneously. Both equations describe a snapshot at specified concentrations, permeabilities and temperature; neither one, by itself, explains how those concentration gradients are built or maintained in the first place — that is the pump’s job, covered next.
Worked example — representative resting values
Using illustrative relative permeabilities PK = 1, PNa = 0.04, PCl = 0.45 with [K]o=4, [K]i=140, [Na]o=145, [Na]i=12, [Cl]i=4, [Cl]o=120 (mmol L⁻¹): numerator = (1×4)+(0.04×145)+(0.45×4) = 4+5.8+1.8 = 11.6. Denominator = (1×140)+(0.04×12)+(0.45×120) = 140+0.48+54 = 194.48. Vm = 61.5 × log₁₀(11.6/194.48) = 61.5 × log₁₀(0.0596) = 61.5 × (−1.225) ≈ −75 mV. Interpretation: this sits close to EK (≈ −95 mV) but is pulled less negative by the small Na⁺ and Cl⁻ contributions — exactly the qualitative behaviour the resting-potential section below describes.
Change permeability or extracellular potassium
This simplified Goldman-Hodgkin-Katz model uses representative concentrations. It illustrates direction and relative effect, not a universal value for every cell.
Representative resting conditions: potassium permeability dominates, so Vm lies near EK but not exactly at it.
Two separate jobs
The Na⁺/K⁺-ATPase creates the gradients; permeability decides the instant voltage
Without the Na⁺/K⁺-ATPase (an ATPase is an enzyme that hydrolyses adenosine triphosphate, ATP, to release usable energy), the K⁺ and Na⁺ gradients that Nernst and Goldman-Hodgkin-Katz depend on would run down within minutes as passive leak dissipated them — there would be no concentration ratio left to convert into a voltage. The pump’s continuous, ATP-consuming work (three Na⁺ out, two K⁺ in, per ATP) is what keeps [K⁺]i high and [Na⁺]i low over the long term, against constant passive leak in the opposite direction.
But at any single instant, Vm is set by the Goldman-Hodgkin-Katz relationship above — by which channels happen to be open and how permeable the membrane currently is to each ion — not by the pump acting directly on the voltage. The pump is slow and continuous; permeability changes (a channel opening or closing) are fast and can shift Vm within milliseconds without the underlying gradients changing at all. This is why a channel-blocking drug or a change in extracellular K⁺ concentration can alter membrane potential immediately, while pump inhibition (for example by a cardiac glycoside) changes the gradients only gradually.
At rest
Resting potential is a permeability-weighted diffusion potential
At rest, potassium leak conductance is high. Potassium tends to leave down its concentration gradient, making the cell interior negative and drawing voltage toward EK. Small but non-zero sodium permeability pulls voltage away from EK, in the depolarising direction. Chloride’s contribution varies with cell type and how it is itself distributed. Intracellular impermeant anions (proteins, phosphates) also contribute, via Gibbs-Donnan effects, to the asymmetric ion distribution that Nernst and Goldman-Hodgkin-Katz are calculated from.
An ion channel carries current only if it is open (conductance, gion, exists) and a driving force exists (Vm ≠ Eion) — both conditions are necessary. With Vm = −70 mV and EK = −95 mV, the potassium driving force is +25 mV (outward); opening a potassium channel therefore drives current outward and pulls Vm toward EK.
Excitable cells
An action potential is regenerative channel behaviour
Nerve action potential: voltage, phases and refractory periods
Representative neuron, schematic timescale. Landmark voltages are the source texts' own stated values — rest −70 mV, firing level (threshold) −55 mV, peak +30 mV. The spike runs toward the sodium equilibrium potential (+60 mV) without reaching it; the undershoot runs toward the potassium equilibrium potential (≈ −95 mV, the value derived from the Nernst equation above) without reaching that either. The gradual rise before the firing level is the local (electrotonic) response to the stimulus; the steep regenerative upstroke only begins once threshold is crossed. The absolute refractory period spans almost the whole spike, the relative period the undershoot.
- Rest: high K⁺ conductance; voltage-gated Na⁺ channels are closed but available (not inactivated).
- Upstroke: rapid Na⁺-channel activation causes inward current, which further depolarises the membrane and opens more Na⁺ channels — a positive-feedback (regenerative) loop.
- Repolarisation: Na⁺ channels inactivate (a distinct state from closed) while delayed-rectifier K⁺ conductance rises, producing outward current.
- After-hyperpolarisation: K⁺ conductance transiently remains above its resting level, briefly pulling Vm below the normal resting potential.
| Period | Mechanism | Functional result |
|---|---|---|
| Absolute | Na⁺ channels are inactivated, not merely closed | No second action potential can be generated, however strong the stimulus |
| Relative | Some Na⁺ channels have recovered, but K⁺ conductance remains elevated | A second action potential is possible, but only with a stronger-than-usual depolarising stimulus |
Predict what happens if: a local anaesthetic blocks a fraction of voltage-gated Na⁺ channels without inactivating them all — fewer channels are available to contribute to the regenerative upstroke, so the rate of rise and peak of the action potential fall, and if enough channels are blocked, threshold can no longer be reached at all (conduction block), even though the resting potential itself is unchanged.
Propagation
Local circuit current regenerates the signal along an axon
| Factor | Effect | Reason |
|---|---|---|
| Larger axon diameter | Faster | Lower axial (internal) resistance to local circuit current |
| Myelination | Faster | Less transmembrane current leak and less capacitive charging along internode; current jumps node to node (saltatory conduction) |
| Lower temperature | Slower | Channel gating kinetics slow |
| Demyelination | Slower, or conduction block | Leak current increases and may fail to reach the next node above threshold |
| Local-anaesthetic Na⁺-channel block | Slower, or conduction block | Regenerative inward Na⁺ current is reduced or abolished |
In continuous conduction (unmyelinated axons), every adjacent membrane segment regenerates the action potential in turn. In saltatory conduction (myelinated axons), current spreads passively and rapidly beneath the high-resistance myelin, and active regeneration occurs only at the exposed nodes of Ranvier. Propagation is normally unidirectional because the membrane immediately behind the advancing wavefront is still refractory.
Take-home points for this lesson. Nernst answers “what voltage would this one ion be at equilibrium”; Goldman-Hodgkin-Katz answers “what is the actual membrane potential given every permeant ion right now.” The Na⁺/K⁺-ATPase maintains the gradients those equations depend on; permeability decides the instant value of Vm. The action-potential upstroke is regenerative Na⁺ entry; the absolute refractory period exists because inactivated channels cannot reopen until repolarisation. Conduction velocity is set by axial resistance (diameter) and membrane properties (myelination) — both local-anaesthetic block and demyelination slow or stop conduction by increasing effective leak or reducing available regenerative current, not by changing the resting potential itself.