Traveling Waves
or, your heart, your brain, and a chemical reaction in a Petri dish
In a glass dish in a Russian laboratory in 1951, a chemist named Boris Belousov was watching colored liquid do something it was not allowed to do.
The mixture in his dish — sodium bromate, malonic acid, ferroin indicator, in a citric acid solution — was supposed to settle into a single uniform color and stay there. The second law of thermodynamics, as everyone understood it, said so. Reactions go forward. Concentrations equalize. Entropy increases. Nothing chemical oscillates. Belousov’s mixture refused.
It cycled. Red, blue, red, blue, on a clock he could time with a wristwatch. He stirred it; it kept cycling. He let it sit; it kept cycling. He took photographs. He wrote up his findings. He submitted to the Soviet chemistry journals. And the journals rejected him, repeatedly, on the grounds that what he was reporting could not happen.
He kept the work in a drawer for ten years. He died in 1970 without seeing his discovery accepted in print. Three years later — on the strength of follow-up work by a graduate student named Anatol Zhabotinsky — the chemistry community quietly admitted that the Belousov-Zhabotinsky reaction was real, that oscillating chemical reactions were real, and that a great deal of physical and biological science needed updating to accommodate them.
The most striking thing about the BZ reaction is not that it oscillates in time. That part is a curiosity. The most striking thing is what it does in space. If you let the dish sit unstirred and watch through the bottom, you see waves. Concentric ringed waves, like a stone dropped into a pond, except instead of one stone there are dozens, distributed across the dish, each generating its own slowly expanding ring of color. When two rings collide they cancel. When a ring is interrupted — by a piece of dust, an air bubble, a speck — the open end can curl back on itself and form a spiral. The dish, viewed from above, becomes a slow-motion galaxy of chemical spirals.
This is a traveling wave in an excitable medium. It is not the same trick as synchronization, exactly — in synchronization, every element of the population pulses together; here, the pulse moves through the population in a particular direction at a particular speed. They are cousins. The universe seems to want both.
An excitable medium is any system whose individual elements have three modes:
- Resting — quiet, ready to be excited.
- Excited — firing, doing whatever firing means in this system.
- Refractory — just fired, recovering, currently incapable of firing again.
Each element, while excited, can trigger its neighbors to become excited. But each element, immediately after firing, spends a refractory period — milliseconds in cardiac tissue, minutes in chemical solutions, days in some forest-fire dynamics — during which it cannot fire again no matter how hard a neighbor tries.
This three-state design is what produces the wave. When element A excites neighbor B, B fires and excites C in turn, and so on. But B does not also re-excite A backward, because A is now refractory. The wave can only move forward into resting territory. It cannot go back. It travels.
The wave has a particular shape. At the front, an advancing band of excited elements. Right behind the front, a thicker band of refractory elements that just fired and are recovering. Far behind that, resting elements that are once again ready. The wave is a moving stripe of excitement towed by a stripe of inexcitability, with quiet on both sides. It is not unlike a thrown stone in a pond, except in a pond the energy that drives the wave came from the stone, and in an excitable medium the energy is being supplied locally by every element as the wave passes through it.
You contain at least two excitable media. The first is your heart.
The cells of your heart muscle are excitable: each one fires (briefly contracts) when nudged, and then spends a few hundred milliseconds being refractory before it can fire again. But a heart full of identical excitable cells would just sit there, since none of them would have any reason to fire first. The trick of the heart is that a small cluster of cells in the upper right atrium — the sinoatrial node, your natural pacemaker — is not merely excitable but also spontaneous. SA-node cells fire on their own, on a regular cycle of around once a second, with no input required. They are the metronome. Each spontaneous firing of the SA node launches a wave that propagates across the atria, pauses momentarily at a relay called the atrioventricular node, then sweeps through the ventricles via specialized conduction pathways and triggers the squeeze that pumps blood. Your heartbeat is not a synchronized clench. It is a directed wave with a particular shape and a particular path, fired by a small clock at the top.
(When the SA node fails — through age, disease, or injury — an implanted pacemaker takes over its job, firing electrical pulses on a regular schedule into the same conduction pathways. The implant doesn’t replace the heart. It replaces the metronome.)
This matters because the directionality is what makes the heart work. If all the cells fired at once, the chambers would clench simultaneously, no blood would move, and you would be dead. The wave has to travel — atria first, then ventricles — for the pump to function. The heart’s whole architecture, from the placement of the SA node to the geometry of the conduction pathways, is engineered to produce one specific traveling wave shape, again and again, around once a second, for several billion repetitions over a human lifetime.
And when the wave breaks, you die.
If a wave front sweeping across the heart hits a small patch of tissue that is briefly inexcitable — a scarred region from an old heart attack, a transient electrical glitch — the wave gets interrupted on one side. The unbroken half can curl back into the gap left by the broken half, and now the wave is going in a circle rather than across. A circulating wave in heart tissue is called a reentrant arrhythmia. Multiple circulating waves at once is called fibrillation. In the atria it is unpleasant but survivable. In the ventricles it stops the pump immediately. This is what defibrillators are for: they fire a single overwhelming pulse that pushes every cell in the heart into refractory simultaneously, breaking the spirals and giving the SA node a clean substrate to start a new wave on. They don’t restart the heart. They silence the heart, briefly, so the heart can restart itself.
The second excitable medium you contain is your brain. Slow oscillations of cortical activity propagate across the surface of the brain during sleep, sweeping front to back. Migraines, the leading theory says, may be a slow wave of cortical depression — an excited band followed by a long refractory zone — moving across the visual cortex at roughly three millimeters per minute, which is about the speed at which the visual aura crosses a sufferer’s field of view. Seizures, in some models, are runaway traveling waves: regions of cortex firing into other regions in a self-sustaining pattern that recruits a large fraction of the brain. The picture of the awake cortex that has been emerging over the past two decades is not one of synchronized chunks but of a forest of overlapping local waves, each carrying information across populations of neurons in a particular direction at a particular speed.
Outside biology, the same pattern shows up in the most mundane places.
A particularly clean example is the phantom traffic jam — a slowdown on a busy highway that has no apparent cause. There is no accident, no road work, no obstacle. The jam exists; the reason for it does not. The Japanese physicist Yuki Sugiyama demonstrated in a 2008 experiment that you can produce a phantom jam at will by putting a couple dozen cars on a closed circular track and asking the drivers to maintain a steady speed. They cannot. Small fluctuations in driver reaction time amplify into a self-sustaining backward-propagating jam wave that travels around the track even though every individual driver is doing their best to keep moving forward. The jam outlives any one driver’s contribution to it; it is a structure, with a velocity and a shape, made of slowdowns. Stopped cars are excited; cars that have just escaped the jam are refractory (they accelerate back up to speed and won’t immediately stop again); approaching cars are resting. This is the same mathematics as the heart’s beat, on different scales and substrates. (A single car braking once and the few cars behind it tapping their brakes in turn is something simpler — a one-time backward ripple, not a self-sustaining wave. The phantom jam is the wave-shaped phenomenon.)
A forest fire is a wave: burning trees are excited, recently-burned ground is refractory (it cannot burn again until vegetation regrows), healthy trees are resting. The fire front sweeps forward into the resting forest and does not loop back into the ash. An epidemic moving through a population of susceptible-infectious-recovered individuals is, in the simplest model, the same kind of wave. A round of applause that starts at one corner of an auditorium and crosses the room before everyone is clapping is a wave. The pulses that propagate through some slime mold colonies are waves. Wound healing happens, in part, as a wave of activity radiating inward from the wound’s edge.
Synchronization gets you all elements of a system pulsing together. Traveling waves get the pulse moving through the system in a particular direction at a particular speed. They are different design patterns for different jobs. Synchronization is good when you need every element to act simultaneously: a school of fish flicking left as one, an audience clapping in unison, electrical generators feeding the same grid. Traveling waves are good when you need information or activity to move through a population — to recruit cells in sequence, to deliver a contraction wave that pumps blood, to sweep a forest fire across a landscape, to carry a piece of news through a crowd.
The universe seems to find both irresistibly cheap. Both require nothing more than local interaction rules; neither requires a central coordinator. Both fall out of the math when you have a population of elements with the right kind of internal dynamics. The difference between them is whether the elements all fire at the same instant (sync) or in a cascade where each excites the next (wave).
And there are systems that do both at once. The cortex synchronizes large populations into rhythmic bands while also passing traveling waves of activity over those bands; the heart fires its pacemaker cells almost in sync, then propagates the wave from there. The patterns are not exclusive. They are two of the universe’s favorite tools for getting many elements to do something coordinated, and many real systems use both, depending on what they need.
The Experiment
Below is a small excitable medium — a grid of cells, each with a state that follows the resting / excited / refractory cycle described above. The rule is the simplest possible cellular automaton: a resting cell becomes excited if any of its eight neighbors is currently excited; an excited cell immediately becomes refractory; refractory cells count down through several recovery states until they return to resting. That is the whole rulebook. Three lines.
You start a wave by clicking. The wave propagates outward. Click again somewhere else and you have two waves; watch what happens when they collide. Drag through the grid to paint a streak of excited cells — a wave front you have shaped. Hit the spiral button to seed a half-wave; with luck, it will curl into a rotating spiral that pumps out wave after wave from the curl, the way some BZ reactions do, the way fibrillating hearts do.
Things to try:
Click somewhere in the middle of the grid. A circular wave radiates outward, expanding until it hits the walls. It does not bounce back — the cells just inside the wave are refractory, so the wave can only move into resting territory.
Click in two places at the same time (or quickly enough that both waves are still propagating). Watch where they collide. The collision zone goes dark — both waves are running into each other’s refractory tails, and the wave fronts annihilate. The universe lets these waves cross each other only by destroying both.
Click and drag a curving line across the grid. You have painted a wave front. It propagates perpendicular to the line. Now you understand how a traveling wave is shaped.
Hit Make spiral. The simulation seeds a half-wave (a partial wave with one open end). The open end curls into a rotating spiral that emits waves outward from its hub, repeatedly, indefinitely. This is exactly what a fibrillating heart looks like to a cardiologist watching electrical activity, and what a particularly active BZ dish looks like in time-lapse. It is also stable: spirals are an attractor of this system, and once you have one it is hard to clear without using the Clear button.
Crank up Spontaneous fire. The grid lights up with little firing events all over, each one trying to start its own wave but mostly being interrupted by other waves. The medium becomes a low chaotic boil rather than producing clean fronts. This is the regime in which wave dynamics fight noise; many real systems sit in some version of this regime, with a steady stream of small disturbances against which the wave structure has to organize itself.
Slide Refractory length to its minimum. Waves become very thin (the tail is short), travel fast, and collide messily — cells recover and fire again before the previous wave has cleared, so you get re-excitation everywhere. Now slide it to maximum. Waves become thick, travel cleanly, and spirals are extremely stable. The geometry of an excitable medium is set in large part by how long it takes a cell to recover.
Watch a spiral run for a few minutes and let it sink in. The medium has no goal. The cells have no plan. Every cell is following the same three-line rule. And out of that rule comes a structure that pumps wave after wave from a fixed center, indefinitely, at a steady rate, in a particular direction of rotation. Whatever the universe wants when it is making a heartbeat, or a chemical clock, or a forest fire, or a wave of news through a crowd, it is making it out of cells doing this same trivial thing in parallel. The structure is not in any cell. The structure is what shows up between them.