The equations that turned the nerve impulse into biophysics. Four coupled ODEs — one for voltage, three for the gates that open and close the ion channels — and out of them falls the action potential: an all-or-none spike that a sub-threshold nudge cannot make and a supra-threshold nudge cannot make bigger. Down the center, current goes in, the membrane integrates, the spike comes out. The blue team builds and defends it; the red team tries to break it.
source Hodgkin & Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve (1952), J. Physiol. 117, 500–544 — doi:10.1113/jphysiol.1952.sp004764. Rendered, not quoted.
The membrane is a capacitor in parallel with three conductances. Charge balance gives one line:
C dV/dt = −g̅Na·m³h·(V−ENa) − g̅K·n⁴·(V−EK) − gL(V−EL) + I
Each gate obeys dx/dt = αx(V)(1−x) − βx(V)x — voltage-dependent opening and closing, staying in [0,1]. m is fast Na activation, h slow Na inactivation, n slow K activation.
| channel | g̅ (mS/cm²) | E (mV) | gate |
|---|---|---|---|
| Na⁺ | 120 | +50 | m³h |
| K⁺ | 36 | −77 | n⁴ |
| leak | 0.3 | −54.4 | — |
AMBER Illustrative squid-giant-axon parameters, modern −65 mV resting convention. C = 1 µF/cm². Not medical advice.
1952: Hodgkin and Huxley voltage-clamped the squid axon, measured Na and K currents separately, and fit the rate constants that predicted the spike shape and conduction velocity. Four ODEs, no fudge factor.
It is the biophysical root of the-action-potential, and the source it reduces to — collapse m,h,n into one recovery variable and you get the excitable phase plane of the-fitzhugh-nagumo. Each sphere is the next one's premise.
The blue team's live check: re-integrate the model at a sub-threshold and a supra-threshold current and confirm the all-or-none signature. If red removes the sodium (window 6), this badge is where it shows.
A brief depolarizing current pulse I (µA/cm², 0.5 ms) is injected into a resting membrane at −65 mV. That is the whole input — one number. Below a threshold the membrane just leaks the charge back; above it, the sodium gates catch and the spike ignites itself.
Slide the current below to feed the panel. The membrane does the rest — nothing about the spike shape is drawn by hand; it is integrated from the four equations above.
Move the slider — the trace is integrated (forward Euler, dt = 0.01 ms) on the spot from C dV/dt and the three gate equations, never looked up.
What the machine produces, proven: below threshold, only a small passive bump; above it, a full ~100 mV action potential whose peak barely changes with how hard you push (all-or-none). Sodium conductance peaks first (m rises, then h falls); potassium follows and repolarizes past rest. Every gate stays in [0,1].
The blue team's witness (left) re-checks this signature live; the red team (right) tries to make it wrong.
The parameters are illustrative, not a patient. Euler integration adds its own small error; a finer solver shifts the peak by a fraction of a millivolt. The model is a triumph of form — it does not diagnose, prescribe, or replace measurement.
"The spike is the neuron amplifying its input." Cut. The energy is the membrane's own ionic gradients; the current only triggers a release. Peak height is nearly independent of stimulus — that is the whole point of all-or-none.
"Threshold is a fixed voltage." Cut, corrected. It is a moving boundary set by the race between m opening and h closing; it drifts with history (accommodation) and prior spikes (refractoriness).
"Potassium causes the spike." Cut. Na+ influx makes the upstroke; K+ efflux ends it. The engine proves the order: gNa peaks before gK.
The red team's move: remove the sodium current (set g̅Na = 0). Without regenerative Na⁺ influx there is no upstroke — no current, however large, produces an all-or-none spike. The blue team's witness (window 7) is watching.
Delete the sodium channel and the membrane only charges passively — the witness re-integrates, finds no spike where one is required, and turns red. Nothing is faked; the attack is real and it is caught.