Excitable Waves

A pulse that travels because recovery follows. Quiet tissue ignores weak taps, but a suprathreshold kick launches a front that leaves a refractory wake; broken fronts can curl into spirals. Global wave geometry grows from a short local cycle—rest, excitation, recovery, readiness—not from a pre-drawn path.

Excitability is not ordinary oscillation

An excitable medium has a stable resting state but can make a large excursion when a perturbation crosses a threshold. After activation it must recover before it can respond again. This differs from an autonomous oscillator, which cycles repeatedly without being triggered, and from a passive diffusive field, in which a disturbance simply spreads and weakens.

The canonical local sequence is:

1. resting: the site is ready but inactive;
2. threshold crossing: a sufficiently strong input causes rapid activation;
3. excited: the activator rises strongly;
4. refractory recovery: an inhibitor or recovery variable suppresses immediate reactivation;
5. restored readiness: the site can be excited again.

Because each location is coupled to neighbours by diffusion, one site’s excitation can push the next site across threshold. The local cycle becomes a travelling wave.

The Barkley equations

AL.Floys implements a common two-variable Barkley form:

Compact form: (∂ u)/(∂ t) = (1)/(\epsilon)u(1-u) \left(u-(v+b)/(a)\right) +D_u∇^2u,

Compact form: (∂ v)/(∂ t)=u-v+D_v∇^2v.

The fast field (u) acts as activation. The slower field (v) acts as recovery or inhibition. Parameter (\epsilon) separates their timescales. Parameters (a) and (b) shape the threshold and excitability regime. Diffusion carries activation into neighbouring cells and, if enabled for (v), also spatially couples recovery.

This equation is phenomenological. It was designed to reproduce generic wave dynamics efficiently, not to encode a chain of chemical reactions.

Why waves collide and disappear

Two ordinary water waves can pass through one another because the medium supports linear superposition. Excitable waves behave differently. Each front leaves refractory material behind it. When two fronts meet, each encounters a region that the other has just excited. Neither can propagate through that refractory wake, so the fronts annihilate.

This collision rule is one of the most informative things to observe. It reveals that the visible line is not a passive contour moving across a background. It is the boundary between resting, excited and recovering states.

How a broken front becomes a spiral

A complete circular wave expands outward and leaves a recovering interior. A planar front advances across the field. If part of a front is removed, the remaining open end has rested medium on one side and refractory medium on the other. Propagation speed varies with curvature and local recovery. The tip curls around the refractory region, and the wave can rotate repeatedly around a core.

Arthur Winfree’s 1972 observations of spiral chemical activity made this geometry famous in BZ media. Barkley’s model abstracts the same generic phenomenon. The spiral is not programmed as a rotating object. It is a self-maintained wavefront whose tip continually encounters a spatial gradient of readiness.

Target waves and pacemakers

Repeated activation at one location emits concentric waves. In real BZ experiments, spontaneous or heterogeneous pacemaker sites can produce target patterns. In AL.Floys, target-like waves arise from the chosen initial seed or from repeated user excitation. The rings are a record of trigger timing and refractory recovery.

The distance between rings depends on wave speed and excitation period. If a new pulse is launched too soon, the nearby medium remains refractory and the pulse may fail. This gives the visitor a direct way to experience a recovery timescale.

Spiral breakup and wave turbulence

Under some parameters, a rotating wave remains coherent. Under others, the tip meanders, fronts become unstable or spirals fragment into many interacting wavelets. This transition is not random decoration. It reflects the relation between activation timescale, recovery, diffusion, curvature and numerical stepping.

The resulting “turbulence” is a state of proliferating defects and wavefront interactions in an excitable medium, not fluid turbulence governed by Navier–Stokes equations.

One family, many physical realisations

Excitable-wave mathematics appears in chemical reactions, cardiac electrical activity, neural tissue, calcium waves, ecological invasion fronts and other systems. Similar geometry does not mean identical mechanisms. The Barkley model is useful because it strips the phenomenon to activation, recovery and diffusion.

This is also why Excitable Waves is more truthful than BZ Reaction. The public sphere can mention BZ chemistry as the most visually resonant historical example while allowing the model’s conceptual reach to remain broader.

Boris Belousov discovered chemical oscillation in a bromate-based reaction in the 1950s. Anatol Zhabotinsky developed the experimental system and demonstrated spatial waves in unstirred media during the 1960s. The reaction challenged the then-common intuition that a well-mixed chemical system should move monotonically toward equilibrium rather than oscillate visibly.

Field, Körös and Noyes provided a detailed kinetic mechanism in 1972. Field and Noyes then reduced the mechanism to the Oregonator, a compact model that retained important oscillatory behaviour of the chemical reaction.

Arthur Winfree’s 1972 paper “Spiral Waves of Chemical Activity” showed that broken chemical wavefronts can curl into persistent spirals and argued that crucial aspects of their behaviour are geometrical. Spiral waves subsequently became a unifying object across chemical and biological excitable media.

Dwight Barkley’s 1991 model made the study computationally efficient. Starting from two reaction–diffusion variables, Barkley constructed a minimal system whose resolution could be traded against speed while preserving qualitative wave dynamics. That model—not the BZ chemical mechanism—is the direct technical identity of the sphere.

The accurate lineage is:

Belousov and Zhabotinsky’s oscillating chemistry → FKN/Oregonator chemical mechanisms → Winfree’s spiral-wave geometry → Barkley’s generic excitable-medium model → AL.Floys interactive field

Aspect	Current implementation
Entities / field	Two continuous scalar arrays, activator u and recovery/inhibitor v, on a square grid.
Local kinetics	Barkley cubic activator equation and linear recovery toward u.
Spatial coupling	Five-point discrete Laplacian with separate diffusion coefficients for u and v.
Initial conditions	Curated seed morphologies including broken fronts, target-wave sources, collisions and distributed blooms.
Boundary conditions	Periodic wrap.
Numerical method	Explicit Euler-style stepping with configurable dt and multiple internal steps per rendered frame; fields are clamped to a finite range.
Randomness	Seeded morphology variation; the field evolution itself is deterministic for fixed state and parameters.
Rendering interpretation	Palette and glow map combinations of u and v to colour. They do not show a specified chemical indicator concentration.
Public simple controls	Seed style, steps per frame, colour palette, glow and brush radius.
Public tools	Activator/Spawn, Inhibitor/Brush and Erase.

Title hierarchy (resolved)

The pack description still says “Belousov–Zhabotinsky-style excitable medium (Barkley model).” The public atlas title is Excitable Waves, so visitors meet the model class first and encounter BZ chemistry as historical inspiration in the Field Note and this essay—not as a claim that the code implements the full chemical mechanism.

Public-surface weakness: the main conceptual parameters are hidden

The simple ecospheres.ai surface exposes stepsPerFrame, which primarily controls how much model time advances per displayed frame, while hiding a, b and epsilon, the parameters that define excitability and recovery. A visitor can make the waves run faster but cannot directly explore why a medium is quiescent, excitable or unstable.

A better simple surface would replace or supplement Steps per frame with one carefully designed conceptual control, for example:

• Excitability, mapped through a validated path in (a,b) space; or
• Recovery time, mapped primarily to epsilon with stable presets.

Until such a control exists, the public guide should use presets and interventions to teach excitability rather than pretending that frame-step count is a physical parameter.

Threshold response — small inputs fade, sufficient inputs launch waves

In a calm medium, compare a weak activator touch with a stronger one. The nonlinearity appears as a qualitative difference, not merely a brighter version of the same response.

Refractory wake — the field remembers recent activation

Behind a bright front lies a region that cannot immediately support another wave. Follow the whole activation–recovery cycle rather than only the leading edge.

Collision annihilation — waves consume one another’s readiness

When two fronts meet, they disappear instead of passing through. The event exposes the refractory mechanism.

Target waves — timing becomes spacing

Concentric rings emitted from one source encode the interval between excitations. Closely spaced rings may fail if the medium has not recovered.

Spiral formation — a broken boundary becomes a rotating source

Watch an open wave tip curl around its refractory wake. The centre is not a mechanical rotor; it is a continually renewed geometrical defect.

Core and curvature — not all parts of a front move equally

Strongly curved regions propagate differently from broad fronts. The spiral core therefore has a characteristic size and rotation period.

Wave turbulence — coherent spirals can fragment

In more unstable presets, identify individual wave tips and watch them multiply or annihilate. The disorder is organised around moving defects.

First 30 seconds

1. Load Calm Excitable Medium. Use a brief Activator pulse in an empty region and watch the complete cycle: front, wake, recovery.
2. Trigger the same location again immediately. If the second pulse fails or weakens, wait and try a third time after recovery.
3. Load Target Waves and follow one ring until it collides with another.
4. Load Spiral Waves and focus on the tip rather than the full colour field. The tip’s motion is the organising centre.

Three experiments

Experiment	Question	Do	Watch for	Why it matters
Threshold and recovery	What makes the medium excitable rather than merely diffusive?	Calm Excitable Medium; apply a weak Activator pulse, then a stronger one; repeat quickly and then after a delay.	Failed subthreshold response, travelling wave, refractory failure, restored response.	Makes threshold and recovery directly observable.
Collision	Why do two waves annihilate instead of cross?	Load a collision-compatible seed or create two separated activator pulses.	Fronts expand, meet and vanish at the contact line.	Demonstrates that each wave leaves temporarily unavailable medium behind.
Making a spiral	How can rotation arise without a rotating object?	Start from a broad wave; use Inhibitor to cut a gap through one part of the front.	Open ends curl, one may die and another may settle into rotation.	Reveals the spiral as a broken front organised by curvature and refractoriness.

Parameter	What it really controls	Increase it	Decrease it	What to watch
a	Shape of the activator nullcline and effective excitability.	Can alter threshold, wave speed and spiral stability.	Can move the medium toward quiescence or another instability.	Regime changes rather than a simple linear effect.
b	Threshold offset coupled to recovery.	Typically makes excitation harder in this parameterisation.	Typically makes activation easier.	Whether a pulse propagates or dies.
epsilon	Fast/slow timescale separation.	Slower activator excursion relative to recovery.	Faster activation and sharper separation.	Wave width, recovery and spiral stability.
Diffusion u	Spread of activation.	Faster/broader front propagation.	Slower or failed spatial transmission.	Wave speed and front thickness.
Diffusion v	Spread of recovery/inhibition.	More spatially coupled refractory state.	More local recovery.	Core size and interaction between fronts.
dt	Numerical integration increment.	Faster model advance but reduced stability margin.	Slower, safer stepping.	Numerical instability versus real regime change.
Steps per frame	Number of internal updates per rendered frame.	Faster visible evolution and higher computation.	Slower evolution.	Pacing, not excitability.
Seed style	Initial wave geometry.	Categorical.	Categorical.	Which wave defects are introduced at reset.
Warm-up	Early hidden/progressive development.	More mature startup waves.	More visible formation transient.	How the world is staged.
Brush radius	Size of local activator/inhibitor intervention.	Broader wave source or cut.	More local defect.	Critical nucleus and spiral creation.
Palette / glow	Rendering only.	Stronger visual emphasis.	Flatter appearance.	Never interpret as changed chemical kinetics.

Calm Excitable Medium

The best conceptual baseline. It isolates threshold, propagation and recovery without immediately overwhelming the viewer with many wave tips.

Target Waves

Makes repeated source timing and ring collision legible. Use it to connect temporal period with spatial wavelength.

Spiral Waves

The iconic regime. Focus on tip geometry, core size and refractory wake rather than treating the spiral as a spinning image.

Turbulent Oscillation

A high-defect regime in which fronts fragment and wave tips proliferate. The public name would be more precise as Wave Turbulence, because the visible state is not simply a global oscillator.

Dense Chemical Bloom

A visually dense seeded field. The preset should be renamed Dense Wave Field or Dense Excitations so that it does not imply a simulated chemical species or literal bloom.

Recommended learning order:

Calm Excitable Medium → Target Waves → Spiral Waves → Wave Turbulence

Activator / Spawn

Raises u locally and initiates excitation when the perturbation is sufficiently large and the medium is ready. It is the cleanest tool for threshold experiments.

Inhibitor / Brush

Raises v locally, suppressing activation and creating a refractory-like obstacle. Cutting a moving front with inhibitor is the most instructive way to generate an open tip.

Erase

Reduces both fields in the selected region. It does not necessarily restore the exact resting state in the same way as a model-aware reset, so it should be framed as field suppression rather than deletion of an object.

This is not a numerical BZ reaction vessel. There is no bromate, malonic acid, catalyst oxidation state, bromide control, Field–Körös–Noyes reaction mechanism or Oregonator chemistry.

The colours are not calibrated chemical concentrations. The “activator” and “inhibitor” labels name abstract dynamical roles.

The same model is not a faithful heart or brain simulation merely because excitable waves occur in cardiac and neural tissue. Those systems contain ion channels, cell geometry, anisotropy, conduction pathways and biological heterogeneity that are absent here.

A rotating spiral is not a rotating material object. It is a propagating wavefront organised around a phase singularity or refractory core.

“Wave turbulence” is not fluid turbulence. It is irregular creation, motion and annihilation of excitable wavefronts and tips.

Excitable Waves contributes propagating form with memory of recent activity. Turing Patterns creates relatively persistent concentration domains. Flow transports material and vorticity. Excitable Waves instead transmits an event: each location briefly activates, becomes unavailable and recovers. That local temporal cycle is converted into expanding rings, annihilating fronts and rotating spirals across space.

Foundational and primary sources

1. Dwight Barkley, “A model for fast computer simulation of waves in excitable media”, Physica D 49, 61–70 (1991). DOI: 10.1016/0167-2789(91)90194-E.
2. Arthur T. Winfree, “Spiral Waves of Chemical Activity”, Science 175, 634–636 (1972). DOI: 10.1126/science.175.4022.634.
3. Richard J. Field, Endre Körös and Richard M. Noyes, “Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate–cerium–malonic acid system”, Journal of the American Chemical Society 94, 8649–8664 (1972). DOI: 10.1021/ja00780a001.
4. Richard J. Field and Richard M. Noyes, “Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction”, Journal of Chemical Physics 60, 1877–1884 (1974). DOI: 10.1063/1.1681288.
5. Dwight Barkley, “Linear stability analysis of rotating spiral waves in excitable media”, Physical Review Letters 68, 2090–2093 (1992). DOI: 10.1103/PhysRevLett.68.2090.

Editorially important distinction

BZ chemistry is a foundational experimental example. The Barkley equations are a generic phenomenological excitable-medium model. The public page should explain their relation without calling the model a BZ reaction.