The Spark That Has to Recover

The FitzHugh-Nagumo sketch cycles among spiral waves, a traveling pulse, and a pacemaker-style target pattern. Tap the field and each disturbance either catches, dies, or curls into a rotating arm depending on how much recovery the medium still owes itself.

The FitzHugh-Nagumo model, proposed in the early 1960s as a simplified cousin of Hodgkin-Huxley, removes almost everything. It keeps just two variables: a fast excitation term and a slower recovery term. That stripped-down pair is enough to produce the threshold-and-reset behavior associated with nerve spikes, and once diffusion is added across a grid it also starts to resemble the broader class of excitable media, including the spiral and target-wave patterns that show up in Belousov-Zhabotinsky reaction dishes. The artifact gets a lot of mileage from that reduction by staging the local rule that makes a medium flare, rest, and flare again.

The sketch keeps the model's two timescales visible. Excitation spreads quickly through the voltage field, then the recovery variable follows behind and closes the door for a moment. That delay gives the spirals their shape. A pulse cannot immediately retrace its own path because the medium it just crossed is temporarily refractory. The Wikipedia summary of the model describes the characteristic excursion away from rest before the system relaxes back. The BZ reaction page makes the same point in chemical language, calling excitability the property that lets patterned activity emerge in a medium that would otherwise sit still. The browser version makes both the spark and its forced wait visible.

You can see the model compressed almost to its essentials in gallery/fitzhughnagumo.html:

const dvdt = vi - (vi * vi * vi) / 3.0 - wi + I;
const dwdt = eps * (vi + beta - gamma * wi);

v2[i] = vi + dt * (Dv * lapV + dvdt);
w2[i] = wi + dt * (Dw * lapW + dwdt);

The cubic term makes excitation nonlinear, the recovery term pulls the system back, and the Laplacian spreads each state to neighboring cells. The interface then offers modes that emphasize pattern. Spiral mode shows what happens when a wavefront is broken and leaves a free end. Pulse mode reduces the system to one moving front and one refractory tail. Target mode adds a pacemaker region and lets concentric rings keep reappearing from the same source. I like the restraint here. The piece trusts the equations to be visually legible without extra metaphor.

A two-variable toy model can still offer a serious explanation without claiming biological fidelity. Real tissue has anisotropy, heterogeneity, boundaries, and mess. Even so, this little field shows that pattern depends on recovery as much as stimulation. A medium that could only ignite would saturate into noise. A refractory interval lets it write waves, curls, and rotating traces into space.