The Belousov–Zhabotinsky (BZ) reaction is a family of oscillating chemical reactions discovered by Boris Belousov in 1951 and extended by Anatol Zhabotinsky in the 1960s, representing one of the most dramatic examples of non-equilibrium self-organisation in chemistry. In a typical BZ mixture — malonic acid, bromate, sulfuric acid, and a metal catalyst such as ferroin — the solution spontaneously oscillates between oxidised (blue) and reduced (red) states, producing concentric target waves or rotating spiral waves visible to the naked eye. These patterns emerge because the reaction acts as an excitable medium: a resting cell can be triggered by neighbouring excited cells, undergoes a refractory period, and then returns to rest — the same behaviour exhibited by cardiac tissue and nerve fibres.
This simulator uses a three-state cellular automaton (the Greenberg–Hastings model) to approximate the BZ reaction's excitable dynamics on a 2D grid. Click the canvas to plant asymmetric "sparks" that seed spiral centres; adjust the excitation threshold, refractory steps, and spontaneous firing rate to explore different wave modes — from stable spirals to chaotic target waves. Four colour schemes (Fire, Ocean, Spectrum, Mono) visualise the different cell states, with excited cells shown brightest transitioning through refractory states to dark resting background.
What makes the BZ reaction oscillate?
The BZ reaction oscillates because it contains a negative feedback loop with delay: bromate oxidises the metal catalyst, producing bromous acid (HBrO₂) which autocatalytically amplifies itself — a positive feedback. When catalyst concentration peaks, bromomalonic acid accumulates and inhibits the autocatalysis, resetting the system to its reduced state. This bromide-inhibited autocatalysis creates a limit cycle in chemical concentration space, analogous to the predator–prey Lotka–Volterra cycle. The characteristic period in a well-stirred BZ mixture is typically 30–120 seconds at room temperature.
What is an excitable medium and how does it differ from an oscillator?
An excitable medium has a stable resting state that requires a threshold perturbation to fire — unlike a true oscillator which fires continuously without external input. Once triggered, an excitable cell produces a large-amplitude response and then enters a refractory period during which it cannot fire again, before returning to rest. The cardiac muscle is the classic biological excitable medium; nerve axons (described by the Hodgkin–Huxley equations) are another. The BZ reaction is excitable in sub-threshold conditions and oscillatory above a parameter boundary — a Hopf bifurcation.
Why do spirals form rather than simple expanding rings?
Spirals arise from asymmetric initial conditions — in this simulator, a half-disc ignition seed creates the required asymmetry. An expanding circular wave annihilates when its wave fronts collide; but if part of the wave front is blocked or arrives later (creating a "broken end"), that free tip curls inward, initiating a self-sustaining spiral. Spiral waves have a characteristic rotation period determined by the medium's excitability; they are attractors — perturbing them tends to restore the spiral, making them very robust. Spiral breakup and turbulence occurs when the medium's refractory period approaches the spiral period.
The Oregonator is a simplified three-variable ODE model of the BZ reaction, proposed by Field and Noyes in 1974, distilling the complex FKN (Field–Körös–Noyes) mechanism into three key species: HBrO₂ (X), the oxidised catalyst (Y), and bromomalonic acid (Z). The equations dX/dt = s(qY + X(1−X) − XY)/ε, dY/dt = (−qY − XY + fZ)/ε', dZ/dt = X − Z describe the positive (X) and negative (Y, Z) feedback loops. Despite its simplicity, the Oregonator reproduces the oscillation period and wave speed of real BZ experiments quantitatively.
In the Greenberg–Hastings CA used here, the threshold determines how many excited Moore neighbours (out of 8) a resting cell needs to fire. A low threshold (1 neighbour) means even a single excited neighbour triggers firing — waves spread rapidly, tend to fill the domain, and spirals spin fast with short wavelength. A high threshold (5–6) means only densely excited regions trigger neighbours, producing widely spaced, slow-moving concentric rings that are easily extinguished. Real BZ media have an effective excitability threshold tunable by changing bromate or acid concentration.
Yes — excitable spiral waves appear widely in biology. Cardiac fibrillation (both atrial and ventricular) involves rotating spiral waves of electrical activation that disrupt the normal periodic heartbeat; defibrillation terminates these spirals by simultaneously depolarising all tissue. The social amoeba Dictyostelium discoideum uses spiral cAMP waves to coordinate aggregation during starvation. Cortical spreading depression — the neurological basis of migraine aura — propagates as a slow depolarisation wave at ~3 mm/min. The BZ reaction has been used as a physical model to study all these phenomena.
The refractory period determines how long a cell must wait after firing before it can be re-excited. A longer refractory period (more steps) increases the spatial wavelength of the spirals — cells behind the wave front take longer to recover, so the next wave crest must travel further before it encounters a recovered region. Very long refractory periods can suppress spiral formation entirely, allowing only isolated target wave emission from pacemaker regions. This parameter maps directly to the relative refractory period in cardiac electrophysiology, which governs arrhythmia vulnerability.
Yes — BZ reaction–diffusion systems have been proposed and demonstrated as unconventional computing substrates. Adamatzky and colleagues have used BZ waves in gel media to solve maze problems (the wave finds the shortest path), implement logical gates (wave collision → AND/OR), and even perform image processing. The system's inherent parallelism — every spatial location evolves simultaneously — offers potential advantages for certain massively parallel computations. More practically, reaction–diffusion computing informs understanding of morphogenetic computation in developmental biology.
The spontaneous firing rate controls how often resting cells ignite without sufficient excited neighbours — representing thermal noise or heterogeneous nucleation sites in a real BZ medium. At zero spontaneous rate, the grid will eventually reach a silent fixed point unless manually seeded. Low spontaneous rates maintain the pattern by occasionally nucleating new spirals when existing ones decay. High spontaneous rates create dense, chaotic firing that destroys organised spirals and produces disordered target waves — analogous to a BZ experiment run at high temperature where thermal fluctuations dominate.
Alan Turing's 1952 paper "The Chemical Basis of Morphogenesis" predicted that two diffusing chemical species — an activator and an inhibitor — with the inhibitor diffusing faster than the activator, could spontaneously break spatial symmetry and generate periodic patterns (spots, stripes, labyrinths). The BZ reaction demonstrates this class of Turing instability in three dimensions: spiral and target waves are spatio-temporal Turing patterns. The same mathematical framework explains animal coat patterns (leopard spots, zebra stripes), digit spacing in developing limbs, and photoreceptor spacing in the eye.
Wave annihilation upon collision is a hallmark of excitable media and arises from the refractory period. When two wave fronts meet, the cells at the collision point have just been excited by both approaching fronts; instead of generating a new excitation, these cells enter the refractory state. Behind each front is a refractory zone that cannot propagate excitation in the reverse direction, so neither wave continues beyond the collision — they annihilate. This is fundamentally different from oscillatory media where two waves pass through each other, and from linear waves where constructive/destructive interference occurs.