A single set of equations explains spiral waves in a beaker of chemicals, in cardiac tissue, and in half a dozen other places where rest, excitation, and recovery chase each other around a plane.
An excitable medium has parts that rest, fire when pushed past threshold, and stay refractory until recovery completes. Couple them in space and you get traveling pulses that annihilate on collision — and one persistent pathology: the rotating spiral.
The Belousov-Zhabotinsky reaction (cerium, malonic acid, bromate, sulfuric acid) oscillates spontaneously and, spread into a thin film, organizes into turning spirals. The same dynamics appear in the heart, where spirals cause ventricular tachycardia and, if they shatter, fibrillation.
The FitzHugh-Nagumo equations are the simplest model. An activator \(u\) drives firing; an inhibitor \(v\) shuts it down. In BZ chemistry these are a fast autocatalytic species and a slow bromide intermediate; in heart muscle they are membrane voltage and a lumped recovery variable. Same bookkeeping, different molecules.
Plot \(u\) against \(v\) and the trajectory traces a limit cycle — one lap per oscillation. The dashed nullclines mark where \(du/dt = 0\) and \(dv/dt = 0\); their intersection's stability decides whether the cell ticks on its own or rests until pushed.
Figure 1. Phase plane of a single excitable cell. Left: the trajectory through \((u, v)\) space. The dashed curves are nullclines; their intersection is the fixed point. When it sits on the middle branch of the \(u\)-nullcline the cell oscillates; near the outer branches it rests until perturbed. Right: \(u\) (solid) and \(v\) (dashed) as time series. The swatch below shows the current "color" of the cell — translate that color into chemistry or voltage as you prefer.
The BZ solution oscillates: yellow, clear, yellow, ticking back and forth like a chemical clock. The oscillation is built into the kinetics, and the FHN model is the qualitative reduction of the full Oregonator scheme.
Spread that chemistry across a thin film and each point becomes its own FHN oscillator coupled to its neighbors by activator diffusion. A firing point triggers its neighbors; a wave rolls outward.
Two BZ waves do not pass through each other. Each front leaves a refractory trail that cannot fire again until the inhibitor decays. Colliding fronts hit refractory tissue and die — wave annihilation, the defining difference from linear water waves.
Click anywhere to start a circular wave. Click two spots to watch them collide.
Figure 2. BZ dish. A 2D excitable medium colored with the characteristic BZ palette: bright orange where the activator has spiked, dark purple behind the wave (refractory), deep blue at rest. Click to deposit activator and launch a circular wave. Watch two wavefronts annihilate when they collide.
Heart muscle uses the same excitable machinery. A myocyte rests at a negative voltage; a threshold stimulus opens sodium channels; voltage spikes; calcium enters; the cell contracts; potassium channels open; voltage falls; the cell sits refractory while pumps reset the gradients.
Line up such cells and poke one end. Each excited cell triggers its neighbor and a pulse rolls down the row, never reversing because the trail behind is refractory. Poke both ends and the pulses meet in the middle and annihilate.
Figure 3. A 1D cable of cardiac cells. "Poke both ends" sends two pulses toward each other. They collide and die, for exactly the reason BZ waves die when they collide.
Spread the cells across a 2D sheet to model the atrial or ventricular wall. A pacemaker region in the corner fires at a steady rate; every cell fires once per heartbeat.
Pace faster than the refractory period and the tissue cannot keep up — some pulses fail to propagate. This is the simplest case of rate-dependent conduction block.
Figure 4. Normal cardiac conduction. The pacemaker in the corner fires periodically and waves sweep across the tissue as orderly fronts. The same FHN equations as the BZ dish, displayed in a cardiac color scheme.
Circular waves die at the boundaries. But break a wavefront and the severed end curls back on itself, pivoting around the break point until it forms a rotating spiral. Once formed, a spiral emits wave after wave from its tip indefinitely.
Multiple spirals coexist with definite chiralities, competing for territory. In the heart, a single spiral on the ventricular wall fires the tissue faster than the sinus node — ventricular tachycardia. If the spiral fragments, the tissue fibrillates.
The S1-S2 protocol creates reentry in a lab preparation: launch a planar wave (S1), then drop a premature stimulus (S2) onto the recovering tail. The premature beat can only spread sideways into recovered tissue, and a spiral is born.
Click to deposit a stimulus. Switch to "Quench" to erase activity locally.
Figure 5. Spiral formation and defibrillation. "Break wavefront" cuts a planar wave so the severed end curls into a spiral. "Seed spiral pair" plants two counter-rotating spirals that tile the medium. "S1-S2 protocol" launches a planar wave then drops a premature stimulus onto its recovering tail. "Induce fibrillation" shatters the medium into turbulence. "Defibrillate" resets every cell to refractory simultaneously, erasing the gradients that sustain re-entry.
Zhabotinsky and Winfree reported BZ spirals in the 1970s; Winfree's When Time Breaks Down later treated the heart as an excitable medium and re-entry as a topological inevitability. A defibrillator does not restart the heart — it depolarizes every cell at once, wiping out the phase gradients spirals need.
The same mathematics governs slime-mold aggregation, calcium waves on egg surfaces, cortical spreading depression in migraines, and forest-fire fronts. The heart is just the case where the spirals can kill you.