Pour a uniform random noise of integer states 0…K−1 into a grid and apply a single rule: a cell advances to its next state if any neighbour already carries that successor state. Within a few hundred steps, order spontaneously emerges — rotating spiral waves paint the grid in concentric rings of colour, self-perpetuating indefinitely. This is the cyclic cellular automaton (CCA), the simplest model of excitable media, and a masterclass in emergence.

1. The Greenberg-Hastings Model

The Greenberg-Hastings model (1978) was the first rigorous study of an excitable cellular automaton. Each cell carries a state s ∈ {0, 1, …, K−1}. State 0 is the resting state, states 1…K−1 are successive excited/refractory stages, and cells periodically cycle.

Update rule (synchronous): s'(c) = (s(c) + 1) mod K if any neighbour has state (s(c)+1) mod K s'(c) = s(c) otherwise Neighbourhood: Moore (8 cells) or von Neumann (4 cells) Threshold θ: require at least θ neighbours in the successor state

With von Neumann neighbourhood and K ≥ 3, Fisch, Gravner, and Griffeath (1991) proved that CCA starting from random initial conditions form persistent spiral waves with probability 1 — the spirals are the unique attractors of the dynamics.

2. Phase Transitions and the (K, θ, N) Parameter Space

Three parameters govern the qualitative behaviour of CCA:

Parameter Symbol Effect
Number of states K More states → longer refractory period → larger spirals, slower rotation
Threshold θ Higher θ → harder to excite → fewer but larger spirals; θ = 1 is most excitable
Neighbourhood N (vN or Moore) Moore neighbourhood → faster propagation; more complex tips
Typical phase diagram in K-θ space (Moore neighbourhood): θ = 1: Active regime — dense spirals, rapid colonisation θ = 2: Marginal regime — spirals form but may self-annihilate θ = 3: Sparse regime — only occasional waves survive θ ≥ 4: Frozen regime — no excitation propagates (for K ≤ 3)

3. Spiral Wave Mechanics

A CCA spiral is a rotating wave around a central organising defect — the tip. The tip traces a closed curve (often a circle) or meanders ergodically. Spiral formation follows a precise sequence:

Step 1 — Nucleation

In a random initial condition, local patches occasionally form an asymmetric sequence: a small region where states increase monotonically in one direction. This "seed wave" expands as a target ring.

Step 2 — Breakup and Tip Formation

When a planar wave front encounters a region in a different phase, the front breaks and the free ends curl inward. The curling tips are topological defects — locations where all K states meet at a point — and they become the spiral cores.

Step 3 — Annihilation

Spirals of opposite chirality (clockwise vs counter-clockwise) annihilate upon collision. Long-lived spirals are those that open up enough territory to avoid incoming waves from competitors.

Spiral rotation period: T ≈ K (in synchronous CA steps) Wavelength (ring spacing): λ ≈ K cells (for Moore neighbourhood, θ=1) Wave speed: c = λ/T ≈ 1 cell/step

4. Connection to Excitable Media in Biology

The same spiral wave dynamics appears across radically different physical substrates:

System Rest state Excitation Refractory
Belousov-Zhabotinsky reaction Oxidised ferroin Reduced ferroin (blue→red) Return to oxidised (seconds)
Heart muscle Resting (−80 mV) Action potential Absolute refractory (200 ms)
Retinal spreading depression Normal neural activity Depolarisation wave Suppressed activity (minutes)
Slime mould (Dictyostelium) Individual cells cAMP pulse propagation Desensitisation to cAMP
Wildfire on terrain Unburnt fuel Burning front Burnt (no fuel)

5. Gliders and Composite Structures

With carefully chosen parameters, CCA can produce gliders — spatially compact patterns that translate across the grid — analogous to Conway's Game of Life gliders. In the "Cyclic 255" variant (K=255, θ=2, Moore), Gravner and Griffeath (1996) documented an extraordinary zoo of persistent structures:

Structure Description
Turbulent regime Dense, irregular spiral tips, chaotic breakup and merging
Droplet Quasi-circular rotating wave, self-contained, collides elastically
Macaroni Long thin rotating wave that wraps around itself
Curlicue Spiralling arm that self-wraps into a tightly wound glider
Yin-yang Two interlocked counter-rotating spirals forming a stable pair

6. Topological Analysis: Winding Number

The state field s(x, y) ∈ {0, …, K−1} on the grid can be interpreted as a discretised scalar phase field on a circle. Around each spiral tip, the phase winds by exactly ±2π — the topological charge (winding number) ±1:

Winding number of a spiral core: w = (1/2π) ∮ ds (integral around a small loop enclosing the tip) Conservation: total topological charge is conserved modulo K Annihilation: w=+1 and w=−1 spirals must meet to annihilate Spontaneous creation: only as +1/−1 pairs from a uniform background

This topological protection explains why spirals are so long-lived: you cannot destroy a single spiral without an opposite-chirality partner. It is the same mathematics as vortices in superfluids and dislocations in crystals.

7. Complexity Classes: When CCA Dies Out

Not all parameter choices lead to spirals. Gravner and Griffeath (1993) established three phases (for large K):

Phase 1 — Dying (θ > K/4 approximately): Almost all initial conditions lead to a static, uniform state. Phase 2 — Chaotic (θ ≈ K/4): Transient complex activity; eventual fixation possible. Phase 3 — Persistent (θ ≤ K/4 approximately): Spiral waves form inevitably and persist forever.

8. Interactive: Cyclic CA Spiral Explorer

Choose K (states), threshold θ, and neighbourhood type, then press Randomise. Watch spiral waves emerge from noise. Increase K for slower, larger spirals. Try θ=2 for larger core regions.

Gen: 0

Each colour represents a state in the cycle 0…K−1. A cell advances when a neighbour has its successor state. Spirals emerge spontaneously from random noise within ~200 generations.

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