FR21 · Excitation and recovery

Excitable waves and spirals

Why does a wave curl into a spiral, and why do two waves disappear when they meet? A broken wave front can curl around its recovering wake and keep exciting fresh material. Colliding fronts meet regions that are still recovering, so they usually die out.

Touch a recovering sheet to launch a wave. Follow a spiral, collide two fronts and watch the material become excitable again.

Model assumptions

The dimensionless Barkley model represents excitation and recovery on a square with reflecting boundaries. It has no calibrated cell chemistry, membrane voltage or three-dimensional tissue geometry. Colors show computed fields.

Cyan: excitationAmber: recovery

Time

6.00

Excited area

0.100

Probe excitation

0.500

Probe recovery

0.300

A point becomes excitable again

Both curves come from the white probe on the sheet. Cyan rises quickly; amber takes longer to fall.

Watch the fronts meet

Every row stores the actual excitation along the center line. Position increases rightward, time downward. The latest row is at the bottom.

Start with the essentials

Focus question: Why does a wave curl into a spiral, and why do two waves disappear when they meet?
One-sentence intuition: An excitation wave carries a temporary change of state. The recovering material behind each front explains both spiral rotation and annihilation on collision.

Core mathematical model

Excitation spreads and recovery follows

\partial_t u=D\nabla^2u+\varepsilon^{-1}u(1-u)\left(u-\frac{v+b}{a}\right),\qquad\partial_t v=u-v

Both fields and time are dimensionless. Excitation is the fast field; the recovery field changes locally and raises the threshold. The boundary reflects the diffusing field.

The resting threshold

u_{\mathrm{th}}=\frac{v+b}{a}

With recovery held fixed, an excitation above the threshold initially grows. Recovery rises after excitation, so a second stimulus can fail.

Common difficulties

A wave does not carry the material across the sheet

Typical misconceptionThe bright patch is a lump of substance moving through the medium.

Better mental modelThe color indicates a local state. Each patch rises and falls as its neighbors trigger it. Only the fast field has a diffusive flux in this model.

Spirals and stationary patterns need different tests

Typical misconceptionAny reaction–diffusion pattern is a stationary Turing instability.

Better mental modelHere a finite stimulus leaves a recovering trail in an otherwise stable medium. Follow the wave and recovery time, rather than a growing infinitesimal stationary mode.

Run the experiment

01
Predict the collision
Select Two fronts collide. Follow the slanted bands in the saved center-line record.
What to observe: The incoming fronts meet and end. Each would need to enter material that the other front has just left recovering.
02
Stimulate a recovering region
Select A half-recovered sheet. Drag the same short stroke above and below the midline.
What to observe: The fresh region supports a growing front. A sufficiently recovered region can be excited again later.

Excitable waves and spirals

A point becomes excitable again

Watch the fronts meet