A neuron at rest sits at −65 mV, held there by a balance of ion gradients and leak channels. Disturb it gently and it recovers. Disturb it past a threshold — around −55 mV — and something irreversible happens: sodium channels open, sodium floods in, the voltage spikes to +40 mV, potassium channels open to bring it back, then overshoot slightly before recovering. The whole event takes about 3 milliseconds.
Hodgkin and Huxley fit four differential equations to voltage-clamp data from squid giant axons in 1952. The model has three gating variables: m (Na activation, fast), h (Na inactivation, slow), and n (K activation, medium). The spike is all-or-nothing: below threshold, nothing; above threshold, always the same shape, regardless of stimulus strength. That uniformity is what lets spikes encode information in their timing rather than their amplitude.
Adjust the step current to find the threshold. Apply a brief pulse to trigger a single spike. Watch the gating variables: m rises first (Na rushes in), h falls (Na channels close), n rises last (K drives repolarization). Try applying a second pulse immediately after a spike — during the refractory period, h is too low to allow another full action potential.
The threshold near −55 mV is not a hard line in the model — it emerges from the competition between the autocatalytic Na current (depolarization opens more Na channels, which depolarize further) and the restoring K and leak currents. Below threshold the restoring currents win. Above it the Na cascade wins, briefly, until inactivation (h) shuts it down and K finishes the job.
The refractory period comes from two sources: absolute refractory (h ≈ 0, Na channels cannot open regardless of stimulus) for about 1 ms, then relative refractory (h partially recovered, n still elevated, so a stronger stimulus is needed) for another 2–3 ms. This imposes a maximum firing rate of roughly 300–500 Hz — the biophysical ceiling on how fast information can travel in a single spike train.
What it can't show: individual channel stochasticity (each Na channel is either open or closed — the smooth m³h is a population average over thousands of channels). Spatial propagation along a myelinated axon. The variety of channel types neurons actually use — HH is fit to squid axon; mammalian neurons have a dozen or more channel subtypes. The model also can't show synaptic integration, dendritic computation, or why some neurons fire in bursts while others adapt and go silent. It shows the spike. The spike is real. Everything else is a different question.