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ScienceQuest
Biology Simulator Undergraduate

Resting Membrane Potential Simulator

Set ion concentrations and permeabilities, and watch Nernst and Goldman settle the resting potential, with the driving force on each ion drawn.

Simulator

Resting potential, Vm
The Goldman-Hodgkin-Katz potential: the one voltage at which the net current summed over all three ions is zero.
-67.34 mV
E(K⁺)
Where potassium alone would sit. Vm is close to it because the resting membrane is far more permeable to potassium than to sodium.
-89.06 mV
E(Na⁺)
Where sodium alone would sit. Vm is nowhere near it at rest, and races towards it during an action potential.
60.63 mV
E(Cl⁻)
Negative even though chloride is more concentrated outside, because its charge is minus one. The same gradient in a cation would give a positive value.
-64.09 mV
Vm minus E(K⁺)
The gap that proves potassium does not set the potential entirely on its own. Sodium leaking inwards holds Vm above E(K⁺).
21.72 mV
Driving force on K⁺
Vm minus E(K⁺). Positive means outward current, so potassium leaves a resting cell even though the inside is already negative.
21.72 mV
Driving force on Na⁺
Large and inward. Nothing much happens at rest only because the sodium permeability is tiny, not because the force is small.
-128 mV
Nernst slope
What a tenfold concentration ratio is worth at this temperature. 61.5 mV at 37 °C, 59.2 mV at 25 °C.
61.54 mV/decade
Parameters
mmol/L

The one that moves in real life. Clinical range is roughly 2.5 to 7, and past 9 the heart stops.

mmol/L
mmol/L
mmol/L
mmol/L
mmol/L

Low in a neuron with an active extruder, high without one. This is why GABA inhibits an adult neuron and depolarises a neonatal one.

Relative to potassium at 1. Push it past 1 to see the action potential peak as a steady state.

Relative to potassium at 1. Chloride mostly stabilises the potential rather than setting it.

°C

The Nernst slope is 61.5 mV per decade at 37 °C and 59.2 at 25, which is why two textbooks appear to disagree.

Set the sodium permeability to 10 and watch Vm jump past zero towards E(Na⁺). That is the upstroke of an action potential, reached here as a steady state rather than as a trajectory. Set it back to 0.04 and the membrane returns to rest.

  • Membrane potential Vm (mV)
  • Potassium equilibrium E(K⁺) (mV)
Membrane potential and the potassium equilibrium potential against extracellular potassium, from 1 to 20 mmol/L. Both rise as potassium accumulates outside, which is why hyperkalaemia depolarises. The gap between them narrows as potassium rises, because potassium then carries a larger share of the membrane current and the cell behaves more like a pure potassium electrode.

Citing this tool

Last updated . Add the date you accessed it as well, which a citation of a page that can change asks for. If a specific result matters, cite the permalink from the tool’s share row instead of this page: it reproduces the exact parameters.

Teaching with this? You can put it on a class page or LMS for free, with no ads inside the frame. Get the embed code.

The equation

Vm=RTFln⁡PK[K+]o+PNa[Na+]o+PCl[Cl−]iPK[K+]i+PNa[Na+]i+PCl[Cl−]oV_m = \frac{RT}{F}\ln\frac{P_K[\text{K}^+]_o + P_{Na}[\text{Na}^+]_o + P_{Cl}[\text{Cl}^-]_i}{P_K[\text{K}^+]_i + P_{Na}[\text{Na}^+]_i + P_{Cl}[\text{Cl}^-]_o}

Nernst (1889), with Goldman (1943) and Hodgkin and Katz (1949)

Two equations, and the difference between them

The Nernst equation answers a question about one ion: at what voltage would this ion be at equilibrium, with its concentration gradient and the electrical gradient exactly cancelling so that no net current of it flows?

  • Eion = (RT / zF) × ln([ion]out / [ion]in)

Goldman, Hodgkin and Katz answer a question about the membrane: given that several ions are permeant at once and each is pulling towards its own equilibrium, what is the one voltage at which the currents sum to zero?

  • Vm = (RT / F) × ln[ (PK[K]o + PNa[Na]o + PCl[Cl]i) / (PK[K]i + PNa[Na]i + PCl[Cl]o) ]

So Nernst gives three separate answers and Goldman gives one, and the resting potential is the Goldman answer. Notice what the second equation does not contain: absolute permeabilities. Only the ratios appear, because a factor common to all three cancels top and bottom. That is why potassium’s permeability is fixed at 1 above and has no slider.

Chloride is written backwards, and that is not a typo

In the Goldman equation the inside chloride sits in the numerator with the outside cations, and the outside chloride sits in the denominator with the inside cations. This is the single most common error when people reproduce the equation from memory.

The reason is charge. A chloride ion moving inwards carries current in the opposite direction to a sodium ion moving inwards, because its charge is negative. Keeping the whole expression inside one logarithm, rather than splitting it into separate terms for cations and anions, requires inverting the chloride ratio. Write chloride the same way round as the cations and you get a resting potential that is wrong by tens of millivolts and looks entirely plausible.

The same charge appears as the z in the Nernst denominator, and it is why chloride’s equilibrium potential is negative at around minus 64 mV even though chloride is eleven times more concentrated outside than inside. An identical gradient in a cation would give plus 64 mV. If a calculation ever returns a positive equilibrium potential for chloride on a normal gradient, the sign of z has been dropped.

Why the resting potential is near minus 70

At textbook mammalian concentrations the three equilibrium potentials are roughly:

  • Potassium, 5 out and 140 in, gives about minus 89 mV.
  • Sodium, 145 out and 15 in, gives about plus 61 mV.
  • Chloride, 110 out and 10 in, gives about minus 64 mV.

Those span 150 mV, and the membrane settles at about minus 67 mV, which is close to the potassium end. Goldman is a weighted compromise and the weights are the permeabilities: at rest the membrane is roughly twenty five times more permeable to potassium than to sodium, so potassium dominates the outcome. The canvas above draws this as a position on a voltage scale rather than as a curve, because that is what the fact actually is.

The remaining 22 mV between Vm and E(K) is worth understanding rather than dismissing. It exists because sodium permeability is small but not zero. Any inward sodium current must be balanced by an equal outward potassium current, and potassium only carries an outward current at a potential positive to its own equilibrium. So the membrane is pulled up off E(K) by exactly enough to make the two leaks cancel, which is what the driving force readouts show: potassium plus 22 mV outward, sodium minus 128 mV inward.

Hyperkalaemia, and why a small number is an emergency

Extracellular potassium is the one concentration on this page that moves in real life, and the plot sweeps it deliberately. Raising it from 5 to 10 mmol/L reduces the potassium gradient, which raises E(K), and the membrane follows: minus 67 mV becomes about minus 60 mV.

Seven millivolts sounds trivial. It is not, because voltage gated sodium channels have three states rather than two. From rest they open on depolarisation, then inactivate, and they cannot reopen until the membrane repolarises to recover them. A membrane held 7 mV depolarised sits with a growing fraction of its sodium channels inactivated rather than closed, so fewer are available for the next stimulus. Conduction slows, the QRS widens, the T wave peaks, and past roughly 9 mmol/L conduction fails altogether.

Two things in the plot are worth noticing. Vm moves less than E(K) does, because sodium and chloride anchor it. And the gap between the two curves narrows as potassium rises, because more potassium outside means potassium carries a larger share of the total current, so the cell behaves more like a pure potassium electrode. That convergence is the reason hyperkalaemia is more dangerous than the raw voltage change suggests.

Hypokalaemia does the opposite and hyperpolarises: 2.5 mmol/L gives about minus 72 mV. The membrane is further from threshold, which is why the effect on excitable tissue is different in character rather than simply opposite in sign.

Reaching the action potential from here

Set the sodium permeability slider to 10 and the potential jumps past zero towards E(Na). That is the upstroke of an action potential, and it is the same equation with one ratio changed.

What this page cannot show is the timing, because Goldman has no time in it: everything here is the steady state the membrane would reach and hold. During a real action potential the sodium permeability rises several hundredfold and then falls again within a couple of milliseconds, and the membrane never quite arrives at E(Na) before the sodium channels inactivate and the delayed potassium channels open. The action potential simulator integrates Hodgkin and Huxley’s conductances through time and shows that trajectory. This page is the question that comes before it.

What this model leaves out

  • The sodium potassium pump. Goldman describes passive flux through channels and has no term for a pump. The pump appears here only as whatever maintains the gradients you are entering. In reality it also contributes directly: three sodium out for two potassium in is a net outward current, worth a few millivolts of hyperpolarisation on its own.
  • Divalent ions. This form of the equation assumes monovalent ions, because the constant field derivation does not collapse into a single logarithm once charges differ. Calcium matters enormously in cardiac and smooth muscle and cannot be added as another term here.
  • Anything that changes with voltage. Real permeabilities are voltage dependent, which is the whole basis of excitability. Here they are numbers you set, so the model tells you where the membrane would sit, not whether it would stay there.
  • The chord conductance equation. It is a legitimate alternative and is deliberately not computed alongside, because a permeability and a conductance are different quantities: conductance also depends on how many ions are present to carry the current. Feeding these permeability ratios into it gives about minus 78 mV where Goldman gives minus 67, and showing both from one set of numbers would imply an equivalence that does not hold.

Model and assumptions

Method
Exact expression, no time stepping
Repeatability
Deterministic. The same link gives the same numbers on any machine.

What it assumes

  • Nernst per ion and Goldman-Hodgkin-Katz for the membrane, both exact expressions evaluated on demand.
  • A resting potential is the steady state a membrane settles at, not a trajectory, so nothing is integrated.
  • Concentrations are fixed and uniform on each side, and only the ions listed contribute.

Where it stops holding. Anything active: GHK describes a passive steady state and does not include the sodium-potassium pump’s direct electrogenic contribution.

Numerical accuracy

No method error to report: the result is a closed-form expression evaluated directly, with no time stepping to accumulate error. What remains is double-precision rounding, of order one part in 10^16 per operation.

Common questions

Why is the resting potential close to E(K) and not halfway to E(Na)?

Because the resting membrane is roughly twenty five times more permeable to potassium than to sodium, and Goldman’s equation is a permeability weighted compromise rather than an average. At textbook concentrations potassium’s equilibrium potential is about minus 89 mV and sodium’s about plus 61 mV, and the membrane settles near minus 67 mV, which is 22 mV from potassium and 128 mV from sodium. Raise the sodium permeability on the slider and the potential moves towards sodium in proportion, which is exactly what happens during an action potential.

Why is chloride written the opposite way round in the Goldman equation?

Because it is an anion, so a chloride ion moving inwards carries current in the opposite direction to a cation moving inwards. Keeping the whole expression inside one logarithm requires putting the inside chloride in the numerator alongside the outside cations, and the outside chloride in the denominator. Writing chloride the same way round as sodium and potassium is the commonest error in reproducing this equation, and it gives a resting potential wrong by tens of millivolts that still looks plausible.

Why does hyperkalaemia depolarise the cell?

Because raising extracellular potassium reduces the potassium concentration ratio across the membrane, which raises the potassium equilibrium potential, and the membrane follows it. Going from 5 to 10 mmol/L moves the resting potential from about minus 67 to minus 60 mV. That sounds small, but a sustained depolarisation of that size leaves a growing fraction of voltage gated sodium channels inactivated rather than closed, so they cannot open on the next stimulus. The result is slowed conduction and, past roughly 9 mmol/L, cardiac arrest.

Is the Nernst slope 61.5 mV or 59.2 mV?

Both, at different temperatures. The slope is RT over F multiplied by the natural log of ten, so it scales with absolute temperature: 61.5 mV per tenfold concentration ratio at 37 degrees and 59.2 mV at 25 degrees. Textbooks quoting 59 or 60 are working at room temperature, which is where electrochemistry is usually taught, while physiology texts quote 61.5 because bodies are at 37. The temperature slider here shows the whole relation rather than either constant.

Where is the sodium potassium pump in this model?

Only in the gradients you are entering. Goldman’s equation describes passive flux through channels and has no term for a pump, so the potential computed here is what those gradients produce passively. The pump matters twice over in reality: it is what maintains the gradients in the first place, and because it moves three sodium out for every two potassium in it carries a small outward current of its own, contributing a few millivolts of hyperpolarisation directly. A model that reproduced the resting potential without a pump would be describing a cell that cannot sustain it.

Why does the potential not reach E(K) even when potassium permeability dominates?

Because sodium leaking inwards holds it above. Potassium permeability being large does not make the others zero, and any inward sodium current has to be balanced by an equal outward potassium current, which can only happen at a potential positive to E(K) where potassium has an outward driving force. That gap is the Vm minus E(K) readout, and it narrows as extracellular potassium rises, which the plot shows: with more potassium outside, potassium carries a larger share of the current and the cell behaves more like a pure potassium electrode.

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