How a local stimulus becomes a rotating spiral

Excite one point first

In a stadium wave, a spectator stands, prompts the next person and then sits down. People stay in place while the wave travels. An excitable medium also passes a local state change onward; material does not have to travel the entire distance with the crest.

The Barkley model uses dimensionless excitation $u$ and recovery $v$ . The resting state is $(0,0)$ . A small positive $\varepsilon$ makes excitation fast. Parameters $a,b$ set a local threshold that depends on recovery.

Hold $v$ fixed briefly and omit diffusion. For $0<u<1$ , the sign of $u-(v+b)/a$ sets the reaction direction. Below threshold excitation decreases; above it excitation rises. This is a statement about the fast subsystem. A complete pulse requires the evolving recovery variable too.

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

Only excitation diffuses in this experiment. The variables describe generic excitability; calibration to a particular tissue requires an additional observation model.

Reading: D. Barkley

Why a wave leaves a tail

High $u$ raises $v$ through $\partial_tv=u-v$ . Recovery raises the threshold and eventually helps excitation fall. Excitation can already be low while recovery remains high, making the same new stimulus less effective. This is the refractory period.

A front excites resting material ahead, while its recovering wake resists immediate re-excitation. Two opposing fronts often annihilate because each meets material recently excited by the other. Their collision differs from the linear superposition of small water waves.

Test the sequence by stimulating the same point twice. A short delay may produce no second traveling pulse; after a longer recovery the stimulus works again. A time record reveals this history directly.

A broken front can curl

A straight front has similar surroundings along its length. Near the end of a broken front, curvature, diffusion and recovery are asymmetric. A propagating endpoint can curl sideways and move around a recovering core.

Sustained rotation depends on whether the material ahead has recovered when the tip returns, and whether the curved front can continue propagating. A curved line alone does not guarantee a stable spiral. The experiment evolves an explicit broken-front initial condition.

The default square has side $48$ and no-flux boundaries. Its teaching parameters are $a=0.75,b=0.02,\varepsilon=0.02,D_u=1$ . Grid spacing and time step affect speed and core shape, so quantitative comparisons require a resolution check.

Reading: D. Barkley

Read the relay in a time trace

Leave a probe at a fixed point. Each passing front raises excitation first and recovery afterward. Excitation falls before recovery has returned to its resting value. A sustained spiral repeats this sequence at the same probe.

The spatial field shows propagation; the probe trace shows the order of events at one place. These are different observations. Comparing wave speed with the local recovery time explains when a recently excited point becomes responsive again.

Worked example

One stimulus, two recovery states

With $a=0.75,b=0.02$ and resting $v=0$ , the fast threshold is $0.02/0.75\approx0.0267$ . A local stimulus $u=0.10$ initially drives the reaction upward.

During recovery at $v=0.30$ , the threshold is $0.32/0.75\approx0.427$ . The same $u=0.10$ now decays. This local calculation explains the direction of the refractory response. Whether a stimulus launches a traveling wave also depends on its spatial extent, diffusive losses and full subsequent dynamics.

Try the reasoning

Explain a collision through recovery

Watch the default spiral and compare the timing of excitation and recovery at a fixed point.
Choose opposing fronts. Predict whether they will cross, then observe their meeting region.
Apply two nearby-in-time stimuli at the same point. Repeat after recovery and explain the difference using the local threshold.

Exercises and explanations

Does a right-moving crest imply that every signaling substance moves right at the same speed?
Read the explanation
No. Local state changes can relay a signal. The recovery variable has no diffusion in this model. Material transport must be inferred from the relevant flux.
How does $v$ change when excitation remains close to zero?
Read the explanation
Then $\partial_tv\approx-v$ , so $v(t)\approx v(0)e^{-t}$ . The threshold falls as recovery decays and the medium becomes responsive again.
What could distinguish an excitable wave from a stationary Turing pattern beyond a single photograph?
Read the explanation
Record excitation and recovery at a fixed point, apply paired stimuli at different intervals and observe opposing-front collisions. These time-resolved interventions directly test excitability.

Course sources and calculation scope

The derivations, diagrams and numerical examples are authored for this course. Source locations below distinguish adopted equations, teaching extensions and the one exact published example recalculated here. Interactive fields and noisy observations are model-generated.

A. M. Turing, The Chemical Basis of Morphogenesis (1952), Philosophical Transactions B 237, 37–72
Read §4, pp. 42–43, for the two-cell numerical example and §6 for mode analysis. The reproduced result is early linear growth in §4. No source figure is copied; the Schnakenberg experiments use a different model.
J. Schnakenberg, Simple Chemical Reaction Systems with Limit Cycle Behaviour (1979), Journal of Theoretical Biology 81, 389–400
Original reaction-model reference. The dimensionless irreversible teaching form is specified here; its equilibrium, Jacobian and numerical examples are derived in the course. Parameters are not biological measurements.
D. Barkley, A Model for Fast Computer Simulation of Waves in Excitable Media (1991), Physica D 49, 61–70
Read the two-variable equations in §2 and the discussion of applicability. The experiment solves those continuous equations with its own discretization and teaching initial conditions; source images and rotation frequencies are not reproduced.
E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains (1999), Bulletin of Mathematical Biology 61, 1093–1120
Read §3, equations 9–20, on conservation, transport and growth. The paper mainly studies one-dimensional growth. The two-dimensional uniformly expanding square is a separately derived teaching extension, not a reproduction of its mode-doubling figures.
M. Nishikawa, S. R. Naganathan, F. Jülicher and S. W. Grill, Controlling Contractile Instabilities in the Actomyosin Cortex (2017), eLife 6, e19595
Read Appendix equations 4–9 on density flux, active viscous stress, saturation and force balance. This teaching reduction uses one density, linear turnover and local stress inhibition. It does not fit embryo data or reproduce Figure 2. The source is CC BY.
A. L. Krause, M. A. Ellis and R. A. Van Gorder, Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds (online 2018; issue 2019)
Read equations 4–6 for moving-domain conservation, 14–15 for material coordinates and 44–45 for Schnakenberg reactions and Jacobian. The periodic square and parameters are teaching choices; ellipse geometries and Figures 1–4 are not reproduced.
L. Giomi, Geometry and Topology of Turbulence in Active Nematics, arXiv:1409.1555v2 (2015); Physical Review X 5, 031003
Frontier branch reading on two-dimensional incompressible active nematics, vortex statistics and defects. Those orientation and fluid equations belong to FR22; their results are outside the one-dimensional scalar cortex model.