Resting Membrane Potential (Goldman–Hodgkin–Katz) Calculator
Calculate the resting membrane potential of a cell from the ion concentrations on either side of its membrane and how permeable the membrane is to each ion. You can also find any single ion's Nernst equilibrium potential, work out the driving force and direction of ion flow, sweep the sodium-to-potassium permeability ratio to see an action potential happen, and test how raising extracellular potassium depolarizes a cell.
Background
Every living cell keeps a voltage across its membrane, with the inside typically 60–90 mV more negative than the outside. It arises because ion pumps build up concentration gradients and the membrane lets some ions leak through more easily than others. The Goldman–Hodgkin–Katz equation, Vm = (RT/F) × ln[(PK[K]o + PNa[Na]o + PCl[Cl]i) / (PK[K]i + PNa[Na]i + PCl[Cl]o)], predicts this voltage by weighting each ion by its permeability. It is the foundation of how neurons, muscles and heart cells generate electrical signals.
How to use this calculator
- Choose GHK Resting Potential to find the voltage of a cell that leaks K⁺, Na⁺ and Cl⁻ at the same time, given each ion's concentrations and relative permeability.
- Choose Nernst Potential to find the voltage at which a single ion is in perfect balance between diffusion and electrical force.
- Choose Driving Force to see which way an ion flows at a given membrane voltage, how hard it is pushed, the current through open channels, and (with a capacitance) how many ions that is and how fast Vm would move.
- Choose Permeability Sweep to watch the membrane voltage slide from EK toward ENa as sodium channels open, the heart of the action potential.
- Choose Potassium Shift to see how raising extracellular K⁺ depolarizes a cell, the mechanism behind hyperkalemia.
- Set the temperature, then click Calculate to see the diagram, the full step-by-step math, and a callout explaining what the result actually means.
How resting membrane potential works
A cell membrane is a thin insulator with ion-selective channels, so charge can build up on its two faces. The voltage between inside and outside, measured as inside relative to outside, is the membrane potential Vm.
Each ion that can cross the membrane has its own equilibrium potential, given by the Nernst equation: E = (RT/zF) × ln([out]/[in]). At that voltage the electrical force on the ion exactly cancels its diffusion down the concentration gradient.
A real membrane is permeable to several ions at once, so no single Nernst potential describes it. The Goldman–Hodgkin–Katz (GHK) voltage equation blends them, weighting each ion by its relative permeability.
Cations use [outside] in the numerator and [inside] in the denominator. Anions such as Cl⁻ are flipped, because a negative charge moving in has the same electrical effect as a positive charge moving out.
In most resting cells K⁺ is by far the most permeant ion, so Vm sits close to EK (around −90 mV) but is nudged upward by a small Na⁺ leak. That is why resting neurons sit near −70 mV rather than −90 mV.
The driving force on an ion is Vm − E. It tells you which way the ion will flow when its channels open and how hard it is pushed. Na⁺ at rest has a driving force of about −130 mV, the largest in the cell, which is why Na⁺ channels opening makes the cell fire.
Changing permeability, not concentration, is how nerves signal. Opening Na⁺ channels raises PNa/PK from about 0.04 to 20 or more, and Vm swings from about −70 mV toward +50 mV in a millisecond, even though almost no ions actually cross.
Because Vm depends on the logarithm of concentration ratios, changing outside K⁺ matters more than changing inside K⁺, which is huge by comparison. Even a few extra millimolar of K⁺ outside depolarizes cells measurably, and that is the basis of hyperkalemia's effect on the heart.
Formulas & Equations Used
Thermal voltage: RT/F = (8.3145 × T[K]) / 96,485 × 1000 mV, with T[K] = °C + 273.15
Nernst potential: E = (RT/zF) × ln([out] / [in])
Base-10 form: E = (2.303 RT/zF) × log10([out] / [in]) ≈ (61.5 mV / z) × log10 ratio at 37 °C
GHK voltage equation: Vm = (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) ]
Driving force: DF = Vm − E (ion flows out if z × DF > 0)
Ionic current: I = g × (Vm − E) (nS × mV = pA, outward positive)
Ions crossing & voltage change: ions per ms = |I| / (|z| e) / 1000, dVm/dt = −I / C, ions per mV = C × 1 mV / (|z| e) (e = 1.602 × 10−19 C; pA ÷ pF = mV/ms)
Two-ion GHK with ratio b = P_Na/P_K: Vm = (RT/F) × ln[ (Ko + b·Nao) / (Ki + b·Nai) ]
Ratio for Vm = 0: b = (Ki − Ko) / (Nao − Nai)
Example Problems & Step-by-Step Solutions
Example 1 — Nernst potential of K⁺
A neuron has 5 mM K⁺ outside and 140 mM inside at 37 °C.
Step: RT/F = 8.3145 × 310.15 ÷ 96,485 = 26.71 mV. EK = 26.71 × ln(5/140) = 26.71 × (−3.332).
Result: EK ≈ −89.0 mV. The inside has to be that negative to hold K⁺ back from leaking out.
Example 2 — Nernst potential of Ca²⁺
Ca²⁺ is 2 mM outside and about 100 nM (0.0001 mM) inside at 37 °C.
Step: Ratio = 2/0.0001 = 20,000, ln = 9.903. With z = 2: E = (26.71/2) × 9.903.
Result: ECa ≈ +132 mV. A 20,000-fold gradient is why Ca²⁺ rushes in whenever its channels open.
Example 3 — Full GHK resting potential
K⁺ 5/140 mM, Na⁺ 145/15 mM, Cl⁻ 110/10 mM (out/in), with PK:PNa:PCl = 1 : 0.04 : 0.45 at 37 °C.
Step: Numerator = 5 + 0.04×145 + 0.45×10 = 15.3. Denominator = 140 + 0.04×15 + 0.45×110 = 190.1. Vm = 26.71 × ln(15.3/190.1).
Result: Vm ≈ −67.3 mV, which sits between EK (−89 mV) and ENa (+60.6 mV) but much closer to EK.
Example 4 — Opening the Na⁺ floodgates
Same cell, but PNa rises to 20 × PK (PCl unchanged).
Step: Numerator = 5 + 20×145 + 0.45×10 = 2,909.5. Denominator = 140 + 20×15 + 0.45×110 = 489.5. Vm = 26.71 × ln(5.944).
Result: Vm ≈ +47.6 mV. The voltage flipped sign purely because permeability changed. That is the rising phase of an action potential.
Example 5 — Driving force on Na⁺ at rest
Vm = −70 mV, Na⁺ is 145 mM outside and 15 mM inside.
Step: ENa = 26.71 × ln(145/15) = +60.6 mV. Driving force = −70 − 60.6.
Result: Driving force ≈ −130.6 mV. Negative means Na⁺ is pushed inward, with enormous force.
Example 6 — Current through open channels
The same Na⁺ channels have a total conductance of 2 nS.
Step: I = g × (Vm − E) = 2 nS × (−130.6 mV), and nS × mV = pA.
Result: I ≈ −261 pA. The minus sign means inward current, which depolarizes the cell.
Example 7 — Where does Vm cross zero?
With K⁺ 5/140 mM and Na⁺ 145/15 mM, what PNa/PK gives Vm = 0 mV?
Step: Set Ko + b·Nao = Ki + b·Nai, so b = (Ki − Ko)/(Nao − Nai) = 135/130.
Result: b ≈ 1.04. Roughly equal permeability for Na⁺ and K⁺ puts the membrane at zero.
Example 8 — Hyperkalemia
K⁺ inside 140 mM, Na⁺ 145/15 mM, PNa/PK = 0.02. Compare outside K⁺ of 4 mM and 8 mM.
Step: Vm(4) = 26.71 × ln[(4 + 2.9)/(140 + 0.3)] = −80.5 mV. Vm(8) = 26.71 × ln[(8 + 2.9)/140.3] = −68.3 mV.
Result: Doubling outside K⁺ depolarizes the cell by about 12 mV, moving it much closer to threshold.
Frequently Asked Questions
What is the resting membrane potential?
It is the steady voltage across a cell's membrane when the cell is not being stimulated, measured inside relative to outside. Most neurons rest between about −60 and −75 mV, skeletal muscle near −90 mV.
What is the difference between the Nernst and GHK equations?
The Nernst equation gives the equilibrium voltage for one ion. The GHK equation combines several ions, each weighted by how permeable the membrane is to it, so it predicts the actual resting voltage of a real cell.
Why does chloride appear flipped in the GHK equation?
Chloride carries a negative charge, so moving it in has the same electrical effect as moving a positive ion out. Swapping inside and outside in the equation accounts for that sign.
Why is the resting potential closer to E_K than to E_Na?
Because at rest the membrane has many more open K⁺ channels than Na⁺ channels. The more permeable an ion is, the more it pulls Vm toward its own equilibrium potential.
Is resting potential an equilibrium?
No. Vm is a steady state, not an equilibrium: Na⁺ and K⁺ keep leaking down their gradients and the Na⁺/K⁺ pump uses ATP to put them back. If the pump stopped, the gradients would slowly run down.
What does a negative driving force mean?
Driving force is Vm − E. For a cation, a negative value means the ion is pushed into the cell and a positive value means it is pushed out. For an anion the movement is reversed, so the calculator uses z × (Vm − E) to report the ion's direction.
Why does temperature change the answer?
The thermal voltage RT/F scales with absolute temperature, from about 25.3 mV at 20 °C to 26.7 mV at 37 °C. Every potential in the calculator scales with it, so warm-blooded and cold-blooded cells differ slightly even with identical concentrations.
Why does high blood potassium cause heart problems?
Raising outside K⁺ shrinks the gradient that holds heart cells negative, so they depolarize toward threshold. Prolonged depolarization inactivates Na⁺ channels and slows electrical conduction. This calculator is an educational model, not medical advice.
How can so few ions change the membrane voltage?
A membrane acts like a capacitor, and a typical neuron body (about 20 pF) needs only around 125,000 net charges to shift 1 mV. That is a tiny fraction of the millions of ions inside, which is why an action potential changes voltage by about 100 mV without noticeably changing any concentration. The Driving Force mode shows this when you enter a conductance and capacitance.
Does the GHK equation work for Ca²⁺ or Mg²⁺?
The simple form used here is for monovalent ions. Divalent ions need a different GHK form, so this calculator handles Ca²⁺ only through the Nernst and driving-force modes, where a charge of +2 is built in.
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