A neuron does not send a bigger signal for a bigger stimulus. It sends the same signal, or it sends nothing, and it decides which at a threshold so sharp that a four per cent change in the input flips the output by a hundred millivolts. Below is the model Alan Hodgkin and Andrew Huxley fitted to a squid axon in 1952 — four coupled differential equations, integrated here in your browser — which won them a Nobel Prize and is still what every computational neuroscientist starts from.
Potassium is thirty times more concentrated inside the cell than outside; sodium is about ten times more concentrated outside than in. Each imbalance is a battery, and the Nernst equation says what voltage it holds: the point at which the pull of concentration exactly cancels the pull of charge. Nothing about a nerve impulse makes sense until you see that the cell is not generating energy when it fires — it is discharging batteries that a pump has already charged.
| ion | outside | inside | ratio | charge | equilibrium potential |
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Three things happen in about two milliseconds. Sodium gates fly open — that is m, and it is fast — so sodium rushes in and drags the voltage up toward its own battery at +50 mV. Then two slower things catch up: sodium gates inactivate (that is h, falling) and potassium gates finally open (that is n, rising), and together they haul the voltage back down past where it started. The undershoot at the end is not a mistake; it is potassium overshooting because its gates are slow to close, and it is what makes the neuron briefly deaf.
Sweep the stimulus from nothing to plenty and plot how high the response gets. There is no ramp. Below threshold the membrane sags a few millivolts and gives up; above it, the same full-size spike every time, and pushing harder makes it very slightly taller and no more. The step happens between 13.0 and 13.5 microamps per square centimetre — a four per cent change in the input for a ninety-four millivolt change in the output. This is why nerves carry information in when they fire and not in how hard.
Fire the neuron, then try to fire it again after a chosen gap. For the first few milliseconds nothing you do works at all — sodium channels are inactivated and no stimulus of any size will open them. That is the absolute refractory period, and it is what stops a nerve impulse running backwards down the axon it just came from. After that comes a stretch where a spike is possible but needs a bigger push. Drag the gap and watch the second spike appear.
If every spike is identical, intensity has to be carried some other way, and it is carried in rate. Hold a steady current on and the neuron fires repeatedly; the stronger the current, the shorter the gap between spikes. Below about 6.2 microamps it will not fire at all, then it starts abruptly at around sixty per second — note that it does not creep up from zero, so there is a range of firing rates this cell simply cannot produce. Push far too hard and it stops again, jammed permanently depolarised with its sodium channels inactivated.
These are the actual Hodgkin–Huxley equations with their published constants — g̅Na = 120, g̅K = 36 and g̅L = 0.3 mS/cm², ENa = +50, EK = −77 and EL = −54.387 mV, membrane capacitance 1 µF/cm² — integrated by plain forward Euler, on the voltage and on all three gating variables alike, at a 0.005 ms timestep for the traces and 0.01 ms for the sweeps. They are not curve-fitted to look right: the resting potential comes out at −64.996 mV without being told to, the peak of a spike at +39.45 mV, and the threshold for a 0.5 ms pulse sits between 13.25 µA/cm² (peak −55.80 mV, no spike) and 13.50 (peak +35.69 mV, a full spike) — the figures section 3 prints, which you can reproduce.
Be careful with one of those digits. Resting potential and firing rate are not sensitive to the timestep at all: rest is −64.9964 mV and the rate 79.2 Hz at every dt from 0.0005 to 0.05 ms, because rest is found by relaxation rather than by the integrator. The peak is. Forward Euler overshoots as dt grows — 39.33 mV at dt = 0.0005, 39.45 at the 0.005 this page uses, and 40.51 at 0.05, a drift of 1.18 mV across that range. So the peak quoted here carries about 0.12 mV of integration error, and a reader who reimplements this with a coarser step will get a slightly taller spike. That is a property of the method, not of the model; an exponential-Euler scheme for the gates would hold the peak far steadier at large dt, and plain Euler was kept only because it is what the equations look like when you write them out. The Nernst potentials in section 1 are computed from RT/zF ln(out/in), giving −89.06 mV for potassium at 5 and 140 mM and +60.63 mV for sodium at 145 and 15 mM, at 37 °C.
Two honest mismatches. Those mammalian Nernst values are not the reversal potentials the 1952 model uses, because Hodgkin and Huxley worked on a squid giant axon at 6.3 °C with quite different concentrations — their potassium equilibrium is about −72 mV, not −89. You can switch the model to the mammalian values with the button at the top; it still fires, but rest falls to −67.7 mV and the spike peaks near +52. Neither set is wrong; they are different animals. And this is one patch of membrane, not a neuron. There is no axon here, so nothing propagates and there is no conduction velocity and no myelin; there are no synapses, no dendrites summing hundreds of inputs, and no calcium, which does most of the interesting work in a real cell. Real mammalian neurons at 37 °C also fire far faster than this model does — the rate constants scale steeply with temperature — and many cortical neurons adapt, slowing down during a sustained input, which these four equations have no mechanism for at all.