🧠 Tier: HSC / Early Undergraduate · Foundation of Neurophysiology
§ 01
Interactive Simulation
E_Na
+50.0
mV
E_K
-77.0
mV
E_Cl
-65.0
mV
V_rest (GHK)
-65.0
mV
P_Na/P_K ratio
0.040
—
Temperature
37.0
°C
RT/F
26.7
mV
Preset Cell Type
Ion Concentrations
[K⁺]_out (mM)20
[K⁺]_in (mM)400
[Na⁺]_out (mM)440
[Na⁺]_in (mM)50
[Cl⁻]_out (mM)560
[Cl⁻]_in (mM)52
Permeabilities
P_K (rel. = 1.0)1.000
P_Na / P_K0.040
P_Cl / P_K0.450
Temperature (°C)37
Overlays
Quick Actions
§ 02
The Idea, Step by Step
▸ A Neuron Is a Tiny Battery — and Salt Charges It
Touch a small battery and you feel nothing, yet the charge is real, sitting ready between its terminals. A neuron is exactly this: a sliver of membrane holding about $-70$ millivolts, "inside negative." Where does that voltage come from? Not from a power supply — from salt. The fluid inside the cell is packed with potassium ($K^+$); the fluid outside has very little. Open a few potassium-sized doors in the membrane and the $K^+$ does what any crowd does: it spills from the packed side toward the empty side. Each potassium ion that leaves carries one positive charge out, so the inside is left slightly negative. That tiny charge imbalance is the resting potential.
STEP 1 — The Leak Stops Itself (high-school view)
The potassium does not drain forever. As the inside turns negative, that negativity starts pulling the positive $K^+$ back. Diffusion pushes potassium out; the growing voltage pulls it in. They reach a standoff at one special voltage — the equilibrium, or Nernst, potential. At body temperature it is simply $E_X = \dfrac{61.5\ \text{mV}}{z}\,\log_{10}\dfrac{[X]_\text{out}}{[X]_\text{in}}$, where $z$ is the ion's charge. Put in real potassium numbers ($[K^+]_\text{out}=5$ mM, $[K^+]_\text{in}=140$ mM): $E_K = 61.5\,\log_{10}(5/140) \approx -89$ mV — the exact voltage at which potassium stops leaking.
STEP 2 — Why a Real Cell Rests at −70, Not −89 (AP / college view)
A real membrane is not a perfect potassium door — a thin trickle of sodium ($Na^+$) leaks the other way, dragging the voltage up from $E_K$. The Goldman–Hodgkin–Katz equation blends every ion, weighting each by how permeable the membrane is to it: $$V_\text{rest} = \frac{RT}{F}\ln\!\frac{P_K[K^+]_o + P_{Na}[Na^+]_o + P_{Cl}[Cl^-]_i}{P_K[K^+]_i + P_{Na}[Na^+]_i + P_{Cl}[Cl^-]_o}.$$ At rest $P_{Na}/P_K \approx 0.04$, so $V_\text{rest}$ sits close to $E_K$ but a touch above it — around $-70$ mV. In the sim, the P_Na / P_K slider is exactly that permeability ratio, and the six concentration sliders set the bracketed terms.
▸ Try This in the Sim Above
Drag P_Na / P_K down to its minimum and watch $V_\text{rest}$ slide all the way onto $E_K$ — a near-perfect potassium membrane, just like Step 1.
Push [K⁺]_out from 5 up toward 12 mM and watch the resting voltage climb toward zero. This is hyperkalemia — in heart muscle that creeping depolarisation can stop the beat.
Press ▶ AP state to make $P_{Na}$ huge and see the voltage flip and chase $E_{Na}$ — the opening move of every action potential.
Derived by Walther Nernst (1888) from thermodynamic equilibrium: the membrane potential at which the electrical gradient exactly balances the chemical (diffusion) gradient for a single ion species.
When the membrane is permeable to multiple ions simultaneously, no single Nernst potential applies. The GHK equation (Goldman 1943, Hodgkin & Katz 1949) gives the steady-state voltage where the net ionic current is zero:
Note Cl⁻ appears with swapped inside/outside because its valence is −1. At rest, \(P_K : P_{Na} : P_{Cl} \approx 1 : 0.04 : 0.45\) for squid axon. As P_Na → 0: GHK → E_K (Nernst for K⁺). During AP peak: P_Na ≫ P_K, GHK → E_Na.
The chord conductance model (alternative formulation) expresses resting potential as a conductance-weighted average of reversal potentials:
For an ion at equilibrium across a membrane, the electrochemical potential must be equal on both sides. The electrochemical potential of ion X is \(\mu_X = \mu_X^0 + RT\ln[X] + zFV\). Setting \(\mu_X^{\text{in}} = \mu_X^{\text{out}}\): \(RT\ln[X]_i + zFV_i = RT\ln[X]_o + zFV_o\). Rearranging: \(zF(V_i - V_o) = RT\ln([X]_o/[X]_i)\), giving the Nernst equation.
STEP 2 — Physical Meaning of the Nernst Potential
E_K ≈ −89 mV means: if the membrane were permeable only to K⁺, K⁺ would flow outward (down its concentration gradient) until the inside is 89 mV negative relative to outside — at which point the electrical force pulling K⁺ back in exactly balances the concentration force pushing it out. At rest (V ≈ −65 mV), V > E_K, so there is still a small net outward K⁺ current that is balanced by Na⁺ leak inward.
STEP 3 — Why Multiple Ions: The GHK Derivation
Assume each ion follows the constant-field approximation (linear voltage drop across membrane). The current density for ion X is \(J_X = P_X z_X^2 \frac{F^2 V}{RT} \frac{[X]_i - [X]_o e^{-z_X FV/RT}}{1 - e^{-z_X FV/RT}}\). Setting total current \(J_K + J_{Na} + J_{Cl} = 0\) and solving for V gives the GHK equation. The key assumption is independence of ion flows (ions do not interact).
STEP 4 — Role of the Na⁺/K⁺-ATPase Pump
The GHK equation predicts a V_rest that is less negative than E_K because Na⁺ leaks inward continuously. This would gradually depolarise the cell. The Na⁺/K⁺-ATPase pump restores concentrations by moving 3 Na⁺ out and 2 K⁺ in per ATP — a net outward current that slightly hyperpolarises the membrane (electrogenic contribution ≈ −2 to −4 mV). Without the pump, V_rest would depolarise by ~0.1 mV/min due to Na⁺ leak; with normal pump activity, concentrations are maintained indefinitely.
STEP 5 — Temperature Dependence
The Nernst factor RT/F scales linearly with absolute temperature: at 0°C (273 K): RT/F = 23.5 mV; at 25°C (298 K): 25.7 mV; at 37°C (310 K): 26.7 mV. This means E_K becomes ~3 mV more negative at body temperature vs room temperature — a small but measurable effect. More importantly, membrane permeabilities themselves are temperature-dependent (Q10 ≈ 2–3 for ion channels), so V_rest shifts nonlinearly with temperature.
STEP 6 — From GHK to the Hodgkin-Huxley Model
The GHK equation describes the resting state. When voltage-gated Na⁺ channels open (during an AP), P_Na jumps ~500-fold — the GHK equation then gives V near E_Na (+60 mV), i.e., the AP peak. The HH model replaces the static permeabilities with time- and voltage-dependent conductances g(t,V), recovering the full action potential dynamics. The Nernst and GHK equations are thus the starting point from which all of neural electrophysiology is derived.
▸ Worked Example — Computing V_rest with GHK (Mammalian Neuron)
Given: [K⁺]_o = 5 mM, [K⁺]_i = 140 mM, [Na⁺]_o = 145 mM, [Na⁺]_i = 12 mM, [Cl⁻]_o = 110 mM, [Cl⁻]_i = 10 mM. Permeability ratios: P_Na/P_K = 0.04, P_Cl/P_K = 0.45. T = 37°C (RT/F = 26.7 mV).
Compare with E_K (Nernst): \(61.5 \log_{10}(5/140) = 61.5 \times (-1.447) = -89\) mV. The GHK value (−67 mV) is depolarised relative to E_K because of the Na⁺ and Cl⁻ permeabilities pulling V toward their respective reversal potentials.
▸ Primary References — Section 3
[Ner1888]
Nernst — Zur Kinetik der in Lösung befindlichen Körper, Z. Phys. Chem. 2:613, 1888. Original Nernst equation.
[Gol43]
Goldman — Potential, impedance, and rectification in membranes, J. Gen. Physiol. 27:37–60, 1943. GHK derivation.
[HK49]
Hodgkin & Katz — The effect of sodium ions on the electrical activity of the giant axon of the squid, J. Physiol. 108:37–77, 1949.
[Hil01]
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001. Ch. 1–2.
🔬 SimulationWhat does the Goldman tab show, and how does moving the P_Na slider change V_rest?▼
The Goldman tab displays the GHK equation result as a function of the P_Na/P_K permeability ratio, with all ion concentrations fixed at the preset values. The x-axis spans from P_Na/P_K = 0 (membrane impermeable to Na⁺ — V_rest → E_K ≈ −89 mV) to P_Na/P_K = 1 (equal permeability — V_rest moves toward the average of E_Na and E_K, roughly −20 mV). The current operating point is highlighted. At rest, P_Na/P_K ≈ 0.04 (squid) or 0.01–0.05 (mammalian), giving V_rest ≈ −65 to −70 mV — well below E_Na because K⁺ permeability dominates. During an action potential peak, Na⁺ channels open and P_Na/P_K can exceed 20, pushing V toward E_Na. The Nernst Potentials tab shows all four ion equilibrium potentials as horizontal bars on a voltage axis — adjust concentration sliders and watch E_K and E_Na shift in real time.
Key takeaway: V_rest is a weighted compromise between E_K (−89 mV) and E_Na (+67 mV), with the weights determined by relative permeabilities — K⁺ dominates at rest, Na⁺ dominates at the AP peak.
🧠 ConceptualWhy is the resting potential negative, and why is it NOT exactly equal to E_K?▼
The resting potential is negative because K⁺ is ~30× more concentrated inside the cell than outside. K⁺ diffuses outward down its concentration gradient, carrying positive charge out and leaving behind negatively charged proteins (A⁻, impermeant anions). This charge separation creates an inside-negative potential. V_rest is NOT exactly E_K (−89 mV) for two reasons: (1) The membrane has a small but non-zero permeability to Na⁺ (which has E_Na ≈ +67 mV), pulling V positive; (2) The Na⁺/K⁺ pump makes a small electrogenic contribution (−2 to −4 mV) by pumping 3 Na⁺ out for 2 K⁺ in. The net result is V_rest ≈ −65 to −70 mV — between E_K and E_Na, much closer to E_K because K⁺ permeability is ~25× greater than Na⁺ permeability at rest.
Key takeaway: V_rest = E_K only if the membrane is perfectly K⁺-selective — real membranes have a Na⁺ leak that depolarises V_rest ~15–25 mV above E_K.
🌍 AppliedHow do changes in extracellular K⁺ concentration affect the resting potential — why is hyperkalemia dangerous?▼
Increasing [K⁺]_o (hyperkalemia) shifts E_K less negative (Nernst equation: higher outside concentration = smaller outward gradient = less negative equilibrium). Since V_rest follows E_K closely, the resting potential depolarises. Normal [K⁺]_o ≈ 3.5–5 mM (E_K ≈ −89 mV, V_rest ≈ −70 mV). At [K⁺]_o = 8 mM (mild hyperkalemia): E_K ≈ −78 mV, V_rest ≈ −63 mV — Na⁺ channels partially inactivated, reduced excitability. At [K⁺]_o = 12 mM (severe): E_K ≈ −68 mV, V_rest ≈ −56 mV — near threshold, spontaneous firing, fatal cardiac arrhythmias. The simulation "Concentration Profile" tab shows this: drag [K⁺]_o from 5 to 12 mM and watch V_rest depolarise. This is why potassium chloride is used in lethal injections — it causes cardiac arrest by eliminating the resting potential gradient in heart muscle.
Key takeaway: E_K ∝ log([K⁺]_o/[K⁺]_i) — every 10-fold increase in [K⁺]_o shifts E_K by +61.5 mV (Nernst), depolarising the cell and eventually causing uncontrolled firing or arrest.
💡 Non-ObviousWhy does the Nernst equation use the LOG of concentration ratios — what is the physical origin of the logarithm?▼
The logarithm comes directly from the thermodynamics of ideal solutions. The chemical potential of an ion in solution is \(\mu = \mu^0 + RT\ln[X]\) — this ln[X] term arises from the entropy of mixing (Boltzmann distribution: the probability of finding a particle at energy E is proportional to \(e^{-E/kT}\), so the energy is \(-kT\ln P\), and for concentration, P ∝ [X]). At equilibrium, \(\Delta\mu_{\text{chemical}} + \Delta\mu_{\text{electrical}} = 0\), i.e., \(RT\ln([X]_o/[X]_i) = zFE_X\), giving the Nernst equation. The logarithm means the relationship is highly nonlinear: doubling [K⁺]_o from 5 to 10 mM shifts E_K by +18.5 mV, but doubling it again to 20 mM adds only another +18.5 mV. This is why E_K changes dramatically with small [K⁺]_o at low concentrations but becomes less sensitive at high concentrations.
Key takeaway: The ln in the Nernst equation is a direct consequence of the Boltzmann entropy of ion distributions — it is the same logarithm that appears in the Gibbs free energy, pH, and every other thermodynamic equilibrium expression.
📐 ComputationalWhat is the difference between the Nernst equation and the GHK equation, and when should you use each?▼
The Nernst equation gives the equilibrium potential for a single ion assuming the membrane is permeable only to that ion and ionic concentrations don't change. Use it to calculate E_K, E_Na, E_Ca etc. as reference potentials. The GHK (Goldman-Hodgkin-Katz) equation gives the steady-state membrane potential when the membrane is permeable to multiple ions simultaneously, with concentrations held constant (pumps maintaining gradients). It assumes (1) the electric field is uniform across the membrane (constant field approximation), (2) ions move independently (no ion-ion interactions), and (3) concentrations are held fixed by pumps. The GHK is more realistic for resting potential because real membranes are simultaneously permeable to K⁺, Na⁺, and Cl⁻. Neither equation is valid during an action potential (concentrations change transiently, currents are transient, constant-field assumption breaks down) — the Hodgkin-Huxley model is needed instead.
Key takeaway: Nernst = single ion equilibrium; GHK = multi-ion steady state. Use Nernst for reversal potentials, GHK for resting potential — neither for dynamic AP analysis.
🎓 Deep / AdvancedHow does Donnan equilibrium relate to the resting potential, and what role do impermeant anions play?▼
The Donnan equilibrium (F.G. Donnan, 1911) describes the equilibrium distribution of ions across a membrane that is permeable to some ions but not others, in the presence of impermeant charged molecules. In neurons, large negatively charged proteins (A⁻: aspartate, glutamate, phosphorylated proteins) are trapped inside the cell. These create a negative fixed charge that (1) attracts K⁺ inward (contributing to high [K⁺]_i), (2) repels Cl⁻ inward (low [Cl⁻]_i), and (3) requires electrical negativity inside to balance the charge. In the Donnan equilibrium, the product of permeable ion concentrations is equal on both sides: [K⁺]_i[Cl⁻]_i = [K⁺]_o[Cl⁻]_o. Real cells are not at Donnan equilibrium (pumps keep them out of true equilibrium), but the Donnan principle explains why high [K⁺]_i and low [Cl⁻]_i arise naturally from the impermeant anions — no active pumping of K⁺ or Cl⁻ is required.
Key takeaway: The high intracellular K⁺ concentration is partly maintained by Donnan attraction to impermeant proteins — the Na⁺/K⁺-ATPase maintains the Na⁺ gradient, not primarily the K⁺ gradient.
🧠 ConceptualDoes the resting potential represent a large or small charge separation across the membrane?▼
Counterintuitively, the charge separation responsible for the resting potential is vanishingly small relative to the total ionic content. A voltage of −70 mV across a membrane with capacitance C_m ≈ 1 μF/cm² requires a surface charge of \(Q = C_m V = 10^{-6} \times 0.07 = 70\) nC/cm². Converting to moles: \(Q/F = 7 \times 10^{-8}/96485 \approx 7 \times 10^{-13}\) mol/cm². For a spherical neuron of radius 25 μm, this is ~1.4 × 10⁻¹⁷ mol of charge — about 8,500 K⁺ ions out of ~10¹² K⁺ ions in the cell (1 part in 10⁸). The ionic concentrations ([K⁺], [Na⁺]) are essentially unchanged by the membrane potential — only an infinitesimal number of ions need to redistribute to create the −70 mV. This justifies using constant concentrations in the GHK equation and explains why the Na⁺/K⁺-ATPase can restore gradients so quickly after activity.
Key takeaway: The resting potential requires only ~1 in 10⁸ K⁺ ions to redistribute — concentrations are effectively constant during normal signalling, validating the constant-concentration assumptions of the Nernst and GHK equations.
§ 04 Best Resources
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001. Ch. 1–2
Purves et al. — Neuroscience (6th ed.), Ch. 2 — best introductory treatment
Kandel et al. — Principles of Neural Science (6th ed.), Ch. 6–7
Johnston & Wu — Foundations of Cellular Neurophysiology, MIT Press, 1995. Ch. 1–3
Khan Academy — Resting membrane potential series (excellent HSC-level visual)
§ 05
Misconceptions & Common Errors
Sub-block A — Conceptual Misconceptions
❌"The resting potential of −70 mV is maintained by the Na⁺/K⁺-ATPase pump actively pumping ions every millisecond."
✅The resting potential is primarily maintained by passive K⁺ diffusion through leak channels, not by active pumping. The pump is essential for long-term maintenance of concentration gradients (it restores ions lost during activity), but it operates on a seconds-to-minutes timescale. At any given moment, the −70 mV is set by K⁺ ions diffusing outward through open K⁺ leak channels (creating charge separation) until balanced by the electrical force. Blocking the pump with ouabain causes only a slow (~0.1 mV/min) depolarisation — the neuron can fire thousands of action potentials before gradients are meaningfully depleted. The pump's electrogenic contribution is only −2 to −4 mV of the total −70 mV.
📖 Hille — Ion Channels (3rd ed.), Ch. 1; Purves et al. — Neuroscience, Ch. 2
❌"The intracellular K⁺ concentration is high because the Na⁺/K⁺ pump pumps K⁺ in — remove the pump and [K⁺]_i immediately drops."
✅High [K⁺]_i is primarily maintained by Donnan equilibrium with impermeant intracellular anions (negatively charged proteins attract K⁺). The pump only needs to compensate for the small Na⁺/K⁺ leak. In red blood cells (which have no Na⁺/K⁺ pump at all), [K⁺]_i remains high because impermeant proteins dominate. In neurons, blocking the pump causes a slow rise in [Na⁺]_i and fall in [K⁺]_i over minutes, but [K⁺]_i stays near-normal for thousands of action potentials because only a tiny fraction of intracellular K⁺ is exchanged per AP.
❌"The Nernst equation gives the resting potential of the neuron."
✅The Nernst equation gives the equilibrium potential for a single ion assuming the membrane is permeable to that ion only. It is NOT the resting potential unless the membrane is perfectly selective for one ion. The actual resting potential requires the GHK equation (multiple ions, multiple permeabilities). For a real neuron: E_K ≈ −89 mV (Nernst for K⁺ alone) ≠ V_rest ≈ −65 to −70 mV (GHK with K⁺, Na⁺, Cl⁻). Using E_K as V_rest overestimates the resting hyperpolarisation by ~20 mV and gives wrong predictions for threshold and AP amplitude.
❌Using z = +1 for Cl⁻ in the Nernst equation: E_Cl = (RT/F)*ln([Cl]_out/[Cl]_in), giving +61.5 mV instead of −61.5 mV.
✅Cl⁻ has valence z = −1. The correct formula is \(E_{Cl} = \frac{RT}{(-1)F}\ln\frac{[Cl^-]_o}{[Cl^-]_i} = -\frac{RT}{F}\ln\frac{[Cl^-]_o}{[Cl^-]_i}\). With [Cl⁻]_o = 110, [Cl⁻]_i = 10 (mammalian): \(E_{Cl} = -61.5\log_{10}(11) = -61.5 \times 1.041 = -64\) mV. Note the sign: high [Cl⁻]_o means Cl⁻ would flow IN (inward current for a negative ion is actually hyperpolarising), consistent with a negative E_Cl. Equivalently: \(E_{Cl} = +\frac{RT}{F}\ln\frac{[Cl^-]_i}{[Cl^-]_o}\) (swap numerator and denominator to remove the negative sign).
🔍 Why students do this: Forget to apply z = −1 for anions; the Nernst formula is often memorised with the concentration ratio in the "outside/inside" direction assuming z = +1.
❌Using the Nernst 61.5 mV factor (valid at 37°C) in experiments conducted at room temperature (22°C), underestimating E_K by ~5 mV.
✅The Nernst factor \(2.303 RT/F\) is temperature-dependent. At 22°C (295 K): \(2.303 \times 8.314 \times 295 / 96485 = 58.9\) mV. At 37°C (310 K): 61.5 mV. Always use the temperature of your experiment. For in vitro electrophysiology (often done at 22–25°C), use 58–59 mV. This 2.5 mV difference in the prefactor translates to ~3–4 mV error in E_K. In patch-clamp experiments where junction potentials are 5–15 mV, this error matters.
🔍 Why students do this: The "61.5 mV at 37°C" or "58 mV at room temperature" constants are memorised without checking which one applies to a given experiment.
❌In the GHK equation, swapping [Cl⁻]_i and [Cl⁻]_o compared to K⁺ and Na⁺ — forgetting that Cl⁻ appears inverted due to its negative valence.
✅In the GHK equation, Cl⁻ appears with inside and outside swapped vs K⁺ and Na⁺. The numerator contains \(P_{Cl}[Cl^-]_{\mathbf{i}}\) and the denominator contains \(P_{Cl}[Cl^-]_{\mathbf{o}}\) — the opposite of K⁺ and Na⁺ (which use outside in numerator). This inversion arises because Cl⁻ carries a negative charge; the mathematical derivation of the GHK equation automatically produces this inversion. A common student error writes all ions with [outside]/[inside], which gives the wrong sign for V_rest when Cl⁻ permeability is significant.
🔍 Why students do this: Memorise the GHK formula pattern for cations and apply it identically to Cl⁻, forgetting the sign inversion for anions.
§ 05 References
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001
Purves et al. — Neuroscience (6th ed.), Ch. 2
Johnston & Wu — Foundations of Cellular Neurophysiology, MIT Press, 1995
Goldman (1943) — J. Gen. Physiol. 27:37–60
Hodgkin & Katz (1949) — J. Physiol. 108:37–77
Kandel et al. — Principles of Neural Science (6th ed.), Ch. 6