What you're seeing
A sheet of cells, each of which is normally resting — quiet, but primed. When enough of a cell's neighbors are excited, it fires too, flashing bright for a single instant. Then, crucially, it cannot fire again right away: it enters a refractory period, a forced recovery during which it is deaf to its neighbors, before it fades back to rest. Excitation therefore sweeps across the medium as a traveling wave, with a glowing front and a fading tail.
Because the tail is refractory, a wave can only move forward — it can never reverse into the ground it just covered. A complete, unbroken wave simply expands and dies at the edges. But a broken wave — one with a loose free end — has nothing to stop that end from curling around the recovering tissue behind it. It winds up into a rotating spiral that, once formed, pumps out waves indefinitely. A periodic source instead emits target waves: concentric rings expanding like ripples from a dripping tap.
The rule
This is the Greenberg–Hastings cellular automaton (1978), the canonical discrete model
of an excitable medium. Each cell holds one of N states: 0 = resting,
1 = excited, and 2 … N−1 = a queue of refractory states it must march through.
Every tick, all cells update at once:
- A resting cell (state 0) becomes excited (state 1) if at least threshold of its four neighbors are currently excited.
- Any non-resting cell in state s simply advances to s+1, and from the last state N−1 wraps back to 0 (rested, ready again).
So excitation lasts exactly one tick; then the cell is refractory for N−2 ticks — immune to re-excitation — before it rests. That refractory period is the essential ingredient: it makes waves one-directional and it is what lets a spiral persist instead of immediately re-igniting its own wake. Lengthen it (raise N) and the wave's tail thickens and the spiral's pitch changes.
It is worth contrasting this with two neighbors in this Atlas that also make spirals. The spirals of spatial rock-paper-scissors come from cyclic dominance — each species chases the next around a loop — not from excitability and recovery. And the patterns of reaction–diffusion are stationary Turing structures set by competing diffusion rates. Here the engine is different: quiet → fire → recover.
Why it matters
Excitable media appear wherever a system can sit quietly, fire when poked past a threshold, and then needs time to recover before it can fire again. Remarkably, the same spiral-wave mathematics governs systems with nothing else in common: oscillating chemical reactions, signaling cells, and — most consequentially — the heart.
Cardiac tissue is an excitable medium: a wave of electrical excitation normally sweeps once across the heart to drive each beat, with the refractory period ensuring it passes cleanly and dies out. But if a wave is broken — by scar tissue, an ill-timed extra beat, or a region of slowed conduction — its free end can curl into a rotating spiral. A spiral wave anchored in the heart is a re-entrant arrhythmia that drives the tissue far faster than the normal pacemaker (tachycardia); when such spirals break up into many drifting, colliding wavelets, the result is the disorganized, lethal electrical chaos of fibrillation. Understanding how these waves form, drift, anchor, and can be terminated is a genuine and active problem in cardiology — and the spiral-wave phenomenology studied in dishes of the Belousov–Zhabotinsky reaction is directly relevant to it.
In the wild
- The Belousov–Zhabotinsky reaction. A self-oscillating chemical reaction in a shallow dish spontaneously forms expanding target rings and rotating spirals of colored chemical activity — the textbook laboratory excitable medium (Zaikin & Zhabotinsky 1970; Winfree 1972).
- Slime-mould signaling. Starving Dictyostelium discoideum amoebae relay pulses of the messenger cAMP outward as target and spiral waves, and chase those waves inward to aggregate — an excitable medium built from living cells.
- Calcium waves inside and across cells, and spreading depression — slow waves of depolarization that travel across cortical and retinal tissue and are implicated in migraine aura — are further biological excitable media.
- Cardiac tissue, where rotating spiral waves are re-entrant tachycardia and their breakup is fibrillation (Davidenko et al. 1992).
An honest caveat: this cellular automaton is a caricature. Real excitable dynamics are continuous and are classically modeled by differential equations such as FitzHugh–Nagumo; the CA throws away the real chemistry and electrophysiology and keeps only the bare skeleton — a firing threshold and a refractory wait. What it gets genuinely right, and what is genuinely shared across all these systems, is the phenomenology: directional waves, target patterns from periodic sources, and self-sustaining spirals from broken fronts.
Try this
- Switch between Spiral, Target waves, and Random. Watch the single rotating spiral, the concentric rings from one pacemaker, and the turbulent sea of colliding waves and spontaneous spirals.
- Paint a wavefront with the mouse, then drag across it to break it — the free end you create curls into a new spiral. This is exactly how a spiral nucleates in a real excitable medium.
- Lengthen the refractory period (raise states N) and watch the wave tails thicken and the spiral's pitch open up; raise the threshold and waves propagate more reluctantly.
References
- Greenberg, J. M. & Hastings, S. P. (1978). "Spatial Patterns for Discrete Models of Diffusion in Excitable Media." SIAM Journal on Applied Mathematics 34(3), 515–523.
- Winfree, A. T. (1972). "Spiral Waves of Chemical Activity." Science 175(4022), 634–636.
- Zaikin, A. N. & Zhabotinsky, A. M. (1970). "Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System." Nature 225, 535–537.
- Davidenko, J. M., Pertsov, A. V., Salomonsz, R., Baxter, W., & Jalife, J. (1992). "Stationary and drifting spiral waves of excitation in isolated cardiac muscle." Nature 355, 349–351.