One pair of equations models the traveling and spiral waves of the Belousov-Zhabotinsky reaction and of heart muscle, where spiral waves are a leading explanation for fibrillation.
The Gray-Scott patterns of the previous essay come from an activator and a faster-diffusing inhibitor. Make the inhibitor slow instead and the pair describes an excitable medium: each part rests until pushed past threshold, fires, then stays refractory until it recovers.
The Belousov-Zhabotinsky (BZ) reaction carries such waves in a thin unstirred layer (Zaikin and Zhabotinsky 1970), and a broken wave curls into spirals (Winfree 1972). Spiral waves also turn in heart muscle, where they are a leading explanation for ventricular tachycardia and fibrillation (Davidenko et al. 1992). This essay builds one model of both and measures what a wave needs to come back around.
The FitzHugh-Nagumo equations (FitzHugh 1961; Nagumo, Arimoto and Yoshizawa 1962) are one of the simplest models. An activator \(u\) drives firing; an inhibitor \(v\) shuts it down, slowly, at a rate set by the small number \(\epsilon\):
\[ \begin{aligned} \frac{du}{dt} &= u - \frac{u^3}{3} - v + I, \\ \frac{dv}{dt} &= \epsilon\,(u + a - b\,v). \end{aligned} \]
Every figure uses \(b = 0.8\). In the BZ reaction's Oregonator model (Field and Noyes 1974), built on the Field-Koros-Noyes mechanism (1972), the fast species is bromous acid and the slow one the oxidized catalyst, cerium(IV), which releases the bromide that halts autocatalysis. In heart muscle the pair stand for membrane voltage and a lumped recovery variable. With a steady drive \(I = 0.5\) (Figure 1), an unstable fixed point makes the cell tick; a stable one leaves it resting until kicked.
Figure 1. Phase plane of a single excitable cell. Left, the trajectory in \((u, v)\); the dashed curves are the nullclines \(du/dt = 0\) and \(dv/dt = 0\). Right, \(u\) (solid) and \(v\) (dashed) against time. Raising \(a\) slides the fixed point left along the \(u\)-nullcline. On the middle branch it is unstable and the cell oscillates; just past the left knee (about \(a = 0.84\) to \(0.87\), depending on \(\epsilon\)) it turns stable and the cell rests. Kick adds 0.5 to \(u\): a resting cell near the knee answers with a full spike, while at \(a = 1.3\) the same kick dies away.
Stirred, the solution (bromate and malonic acid in sulfuric acid, with cerium) ticks between yellow cerium(IV) and colorless cerium(III) (Field, Koros and Noyes 1972). Unstirred, each point of a thin film is an FHN unit coupled to its neighbors by diffusion, \(D\,\nabla^2 u\). With no drive (\(I = 0\)) and \(a = 0.5\) each point rests until a neighbor fires, so a wave rolls out from any stimulus; below about \(a = 0.46\) the whole dish flashes by itself. Colliding fronts die in each other's refractory trails; water waves would pass through.
Click anywhere to start a circular wave. Click two spots to watch them collide.
Figure 2. BZ dish (simulated). A 200 by 200 grid of FitzHugh-Nagumo cells with \(D = 0.5\), \(I = 0\), no-flux edges: orange where the activator has spiked, dark purple in the refractory wake, deep blue at rest. The two opening waves meet, annihilate, and leave the dish quiet. Larger \(a\) makes the waves slower and narrower; below \(a = 0.46\) the whole dish fires at once.
Heart muscle is excitable in the same way. At threshold a myocyte's sodium channels open, the voltage spikes and holds on a plateau while calcium enters, and potassium current brings it back down. The sodium channels stay inactivated until the membrane has repolarized, and until then the cell is refractory (Klabunde, Cardiovascular Physiology Concepts).
Figure 3 paces a row of FHN cells from one end. Started from rest, it follows every stimulus at intervals down to 438 steps. At 436 every second stimulus lands on refractory tissue beside the electrode and dies: 2:1 block.
Figure 3. A paced cable (simulated). 200 cells with the same equations and parameters as the dish (\(a = 0.5\), \(\epsilon = 0.015\)), no-flux ends. Every pacing interval the five leftmost cells are set to \(u = 2\) (amber). "Got through" counts the last ten stimuli that excited cell 10; the middle-cell readout gives the latest action potential's duration (APD, steps with \(u > 0.3\)), the rest before it (DI), and its speed over the next 20 cells. Reset restarts from rest at the current interval. "Poke both ends" sends two pulses that meet and annihilate.
Figure 4 sweeps the interval down and back up. A beat after a short rest, or diastolic interval (DI), is briefer and slower: after 273 steps it lasts 137 steps and moves 0.060 cells per step, against 236 and 0.15 after a long rest. Cardiologists call the shortening restitution, and it lets the cable keep up: sped up gradually, it follows every stimulus down to 420 steps, yet once in 2:1 it stays there until 440. In between, the response depends on history.
Figure 4. Rate-dependent block and restitution, measured. The Figure 3 cable paced from 900 steps down to 150 and back, 10 steps at a time, 16 stimuli at each interval, never reset. Left: share of the last 8 stimuli that reached the middle cell, shortening (violet) and lengthening (cyan); amber marks the intervals where the two disagree. Right: every beat seen at the middle cell in those halves, its APD and its speed against the DI before it. Click the left panel, or focus it and use the arrow keys, to set Figure 3's interval.
A pulse that returns to tissue it has already excited needs no pacemaker. That is re-entry, simplest on a ring (Figure 5). One stimulus launches two pulses that annihilate on the far side; make one side refractory and a single pulse circulates indefinitely at the ring's own rhythm. Shrink the ring and the pulse runs into its own recovering tail, slows, and laps sooner: 993 steps on 150 cells, about 330 on 12. On 10 it dies.
Figure 5. Re-entry on a ring (simulated). The Figure 3 cable with its ends joined. One-way start excites six cells and makes the 20 behind them refractory; two-way start excites six cells only. The slider resizes the ring one cell at a time, removing or copying its most recovered cell, so a circulating pulse survives the change. Right: lap time against length, measured by shrinking a ring from 150 cells one cell at a time (three laps each) until the pulse dies; the dashed line is the lap time at the speed of an isolated pulse, 0.15 cells per step (one lap of a 300-cell ring). From 132 to 150 cells the pulse runs slightly faster than that, by up to 2%. The dot is the ring on the left.
A spiral is re-entry without a hole, circulating around its own tip (Figure 6). Here it turns every 470 steps, a little slower than the fastest pacing Figure 4's cable follows. Davidenko et al. (1992) filmed spirals in isolated heart muscle turning about every 180 ms, three to five times a normal heart rate; re-entry that outpaces the heart's pacemaker is widely held to cause the most dangerous arrhythmias (Fenton et al. 2002). Gray, Pertsov and Jalife (1998) traced fibrillation to spiral tips at a few sites, forming and vanishing.
Click (or focus the sheet and press Enter) to deposit a stimulus. Switch to "Quench" to erase activity locally.
Figure 6. Spiral formation and defibrillation (simulated). A 250 by 250 FHN sheet with no-flux edges and the dish's equations. "Break wavefront" cuts a planar wave so its free end curls into one spiral; "Seed spiral pair" plants a broken front with two free ends, which curl into two counter-rotating spirals. "S1-S2 protocol" runs the protocol below with the slider's delay. "Many spirals" scatters short broken fronts; FHN spirals at these settings do not break up, so the count holds steady, while richer cardiac models show several ways for spirals to break (Fenton et al. 2002). "Defibrillate" excites every cell at once. The readouts count phase singularities (spiral tips) and give the median firing period of four probe cells.
The S1-S2 protocol makes a spiral on purpose: a planar wave (S1), then a stimulus behind it (S2) while its tail recovers (Figure 7). Too early, S2 dies on refractory tissue; too late, its wave spreads into recovered tissue with no free end. In between, it can spread only where the tissue has recovered, so its upper end is free and curls.
Figure 7. The vulnerable window, from real runs. Each bar is one run of the Figure 6 sheet: S1 excites the four leftmost columns, S2 excites the lower half for \(x < 80\) after the delay shown. Grey: S2 dies (fewer than 2,500 more excited cells than S1 alone has, 400 steps after S2), up to 450 steps. Blue: S2 launches a wave but no spiral tip remains 1,000 steps after S2, from 960. Red: a spiral tip remains, from 480 to 930. The snapshots show the sheet 500 steps after S2 in the selected run of each kind. Rerun uses Figure 6's sliders: at \(\epsilon = 0.020\) the window moves earlier, to 360 to 840 steps. Click a bar, or focus the chart and use the arrow keys, to run that delay in Figure 6.
Winfree's When Time Breaks Down (1987) treated cardiac re-entry as a problem in topology. A defibrillator shock depolarizes much of the heart at once; Zipes and colleagues (1975) found in dogs that fibrillation stopped when a critical mass of the ventricles was depolarized. In the model every cell fires and recovers together, no wave is left with tissue to enter, and the sheet is quiet within about 250 steps.
The same waves gather Dictyostelium amoebae (cyclic AMP, concentric and spiral; Tomchik and Devreotes 1981) and spiral through frog oocytes (calcium; Lechleiter et al. 1991). The driven cell of Figure 1, though, fires on its own. The next essay couples many such oscillators, each with its own frequency, and asks when they fall into step.