the second hundred · metaphor 168
Some things stay perfectly quiet until they don't — a calm crowd, a patient temper, a level mood — and then they go off completely, all at once, and afterward need time before they can go off again.
Below a certain line, nothing shows. A small provocation lands and is absorbed without a trace; the crowd stays a crowd, the temper holds, the mood doesn't move. You could mistake the quiet for safety. But it isn't safety — it's just below the line. Push a little harder, past some threshold you can't see, and the response isn't a little bigger. It's total: a full, stereotyped outburst that looks the same whether you nudged an inch past the line or a mile.
And then — this is the part we forget — it can't happen again right away. The nerve, the crowd, the person who just blew up has spent something. For a while, even a real provocation lands on deaf ears. It has to reset before it can fire. Quiet, all-or-nothing, recover, repeat: mathematics calls a system like this excitable media, and the same three-beat shape runs through nerves, hearts, wildfires, and tempers.
The shape of the thing
An excitable system has one resting state it prefers and defends. Push it a little and it slides right back — a graded, forgettable wobble. But it hides a threshold: a line in its state space. Cross that line and the return route home no longer runs straight back. It runs the long way — a big, fixed loop — before it comes back to rest.
That loop is the spike: a fast surge up, a committed excursion, a fall. Its size doesn't depend on how far past the line you pushed — that's what all-or-nothing means. The system contributes the energy; the stimulus only decides whether to release it. And the loop leaves a residue: a slow recovery variable climbs during the spike and must drain away afterward. Until it does, the threshold is jacked sky-high and the cell is refractory — deaf to any push, however hard.
So the tell of an excitable system is not its average behaviour but this three-beat: long quiet, sudden total response, mandatory recovery. Watch for it and you stop being surprised by the quiet — and stop reading the size of the outburst as the size of the cause.
What to try
The instrument is a real FitzHugh–Nagumo cell — two coupled equations, integrated live, nothing scripted. Start with the stimulus low and press Stimulate: the state pops up and slides straight home. Nudge the slider up and try again. Nothing, nothing, nothing — then, past about 0.6, the state swings all the way around the big loop and the readout says fired. Push harder still and the spike is no bigger. It has one size.
Now the second lesson. Fire the cell, and while the readiness meter is still climbing back toward 100%, press Stimulate again. Same shove, no spike — it fizzles. That's the refractory period: the recovery variable hasn't drained, so the threshold is still out of reach. The Temperament dial sets how fast it drains; slow it down and the deaf spell lasts longer. Turn up the jitter and watch a jumpy cell occasionally ignite itself with no stimulus at all. The presets stage each of these.
The mapping
A calm crowd absorbs a hundred small slights and shows nothing — then one more, no larger than the rest, tips it past a threshold and it goes off as a whole, a single stereotyped surge of anger that no one in it chose. A patient person swallows provocation after provocation with no visible change, until the one that lands past the line — and the blow-up, when it comes, is full-sized regardless of how trivial the trigger looks from outside. That mismatch is the source of endless bad arguments about "overreacting": the size of the spike was never a readout of the size of the cause.
And the refractory truce explains the strange calm afterward. A crowd that has just discharged is briefly unrousable; a person who has just wept or raged can't be moved again for a while, even by something that would normally land. Push then and you'll swear they've "gone cold." They haven't — they're resetting. The same push that works tomorrow does nothing today, and the difference is not in the push but in the reserve that has to rebuild before anything can fire.
Read as life lessons
Cross the line by a hair or a mile — the spike is the same size. So the scale of a reaction tells you almost nothing about the scale of its cause. Ask where the line was, not how loud the last straw was.
Below threshold, a loaded system and an empty one look identical: both do nothing. The calm is real but uninformative. What matters is the distance to the line — and that is invisible from the outside.
After firing there is a refractory stretch when nothing lands. Read it as coldness and you'll misjudge people and crowds. It isn't a verdict; it's a reset — and it ends.
In the wild
The action potential is the original: sub-threshold inputs decay, a supra-threshold one fires a spike of fixed height, and a refractory window sets the maximum firing rate and keeps signals moving one way.
Cardiac tissue is excitable media in two dimensions. A wave of contraction sweeps through, trailed by a refractory tail — and when a wave re-enters tissue that hasn't reset, you get the spiral of an arrhythmia.
A forest builds fuel quietly, ignites all at once, then can't burn again until the fuel regrows. Rumours, panics, and outrages spread as the same excitable pulse across a coupled crowd.
The mapping, exactly
| Mathematics | Life |
|---|---|
| resting state | The calm baseline a person, mood, or crowd keeps returning to and defends against small pushes. |
| the threshold | The provocation line below which nothing shows — and which you cannot see from the quiet alone. |
| all-or-nothing spike | The full, stereotyped outburst — the same size whether the trigger was small or large. |
| recovery variable w | The reserve a blow-up spends: it rises during the outburst and must drain before another can happen. |
| refractory period | The recovery lull afterward, when even a real provocation lands on deaf ears — mistaken for coldness. |
| excitability ε | Temperament: how fast one resets, how easily a jittery, over-loaded system fires unprompted. |
The honest model
The cell obeys the FitzHugh–Nagumo equations, integrated live: a fast voltage-like variable and a slow recovery variable, v̇ = v − v³/3 − w + I and ẇ = ε(v + a − b·w), with a = 0.7, b = 0.8. Their two nullclines — the cubic curve where v̇ = 0 and the straight line where ẇ = 0 — cross once, at the stable resting state you see the point settle into. The middle branch of the cubic is the threshold: land left of it and the state slides back; land right of it and it must ride the whole cubic around before returning.
Every number on the panel is measured from that live trajectory, not typed in. Fired / fizzled is decided by whether the actual peak of v crosses the spike level; readiness is read straight off how far the recovery variable w still sits above rest; the jitter adds a genuine random term each step, so a noisy cell really can cross threshold on its own. Turn ε up or down and you change the timescale separation — a bigger ε drains w faster and shortens the refractory deaf spell, exactly as the meter shows.
Where the metaphor tears
The model's spike is identical every time; real outbursts vary in size, shape, and aftermath. All-or-nothing is a clean idealisation, and plenty of human responses are genuinely graded — proportional to the push, not blind to it. Use the metaphor for the systems that snap, not as a licence to flatten every reaction into a switch.
Here the line sits still. In a person it drifts with sleep, hunger, history, and context — a bad week lowers it, a good one raises it, so the "same" provocation crosses on Tuesday and not on Friday. Treating the threshold as a stable property of someone's character, rather than a moving state, is how you end up blaming a person for a bad week.
A resting cell has no urge to spike; left alone below threshold, it truly rests forever. "They were always going to blow" ignores that most of the time the system is quietly stable and the firing needed a real crossing. And a system stuck permanently refractory — depleted, unable to respond at all — is its own pathology, not a virtue of calm.