Oscillators with different natural rhythms, from metronomes to fireflies, fall into step once their coupling passes a threshold. Kuramoto's model predicts that threshold from the distribution of their rhythms.
The excitable cell of the previous essay oscillates on its own under a steady drive. Here many oscillators with different natural rhythms are coupled: when do they fall into step, and how sharply? Metronomes on a board that can roll freely pull each other into step through the board's motion, with no leader (Pantaleone 2002).
Yoshiki Kuramoto's 1975 model captures this with $N$ oscillators, each with its own natural frequency $\omega_i$ and each pulled toward the others:
\[ \begin{aligned} \frac{d\theta_i}{dt} &= \omega_i + \frac{K}{N}\sum_{j=1}^{N} \sin(\theta_j - \theta_i) \\ &= \omega_i + K r \sin(\psi - \theta_i), \end{aligned} \]
where $r e^{i\psi} = \frac{1}{N}\sum_j e^{i\theta_j}$ is the average of the phases as points on a circle. In the second form (Strogatz 2000) each oscillator feels only the mean phase $\psi$, pulled with strength $Kr$.
At $K = 0$ the oscillators scatter around the circle; $r$, the arrow's length, runs from 0 (incoherent) to 1 (in step).
Figure 1. 50 oscillators colored by natural frequency, drawn from a Lorentzian with half-width $\gamma = 0.5$ (so $K_c = 1$). Raise the coupling and the middle colors gather into a cluster while the extreme ones keep lapping it.
Which ones join? An oscillator's lag $\phi = \theta - \psi$ behind the mean phase, which turns at rate $\Omega$, obeys $d\phi/dt = \omega - \Omega - Kr\sin\phi$. If $|\omega - \Omega| \le Kr$ this has a fixed point and the oscillator locks; otherwise it slips, at average rate $\Omega \pm \sqrt{(\omega - \Omega)^2 - (Kr)^2}$.
Figure 2. 200 oscillators (one fixed Lorentzian draw, $\gamma = 0.5$) run for 200 time units from the same start at each $K$. Each dot is one oscillator: natural frequency across, average rate over the last 100 time units up. Locked ones share the group's rate $\Omega$; drifting ones fall between that plateau and the dashed diagonal where an uncoupled oscillator would sit. The amber lines are $\Omega \pm Kr$ with $r$ measured in the run, and the gray curve is the slip formula. At $K = 1.5$ the lines enclose exactly the 150 that lock, and at every $K$ from 1 to 3 the count inside them matches the count locked. The plateau sits at $\Omega = -0.10$, not 0, because 200 draws do not average to the distribution's center. Below $K_c$ the rule stops working: $r$ flickers, there is no steady group to lock to, and at $K = 0.5$ only 5 lock where the lines would admit 13.
For a symmetric, single-peaked frequency density $g$, Kuramoto's self-consistency argument puts the threshold at $K_c = 2/(\pi g(0))$ (Kuramoto 1975, 1984; Strogatz 2000). For a Lorentzian of half-width $\gamma$ that is $K_c = 2\gamma$, and above it $r = \sqrt{1 - K_c/K}$, for infinitely many oscillators. Figure 3 measures three finite sizes; the $K$ you pick there also drives Figure 1.
Drag across the chart to set $K$
Figure 3. Measured while you read: at each $K$, 20 runs of 50 oscillators, 8 of 200 and 4 of 1,000, each with fresh Lorentzian frequencies ($\gamma = 0.5$) and random phases, run for 100 time units with $r$ averaged over the second half; bars span one standard deviation across runs. Gray is the infinite-$N$ curve $\sqrt{1 - K_c/K}$. At $K = 0$ the floor falls from 0.13 to 0.06 to 0.03, the length of an average of $N$ random unit vectors, $\sqrt{\pi/4N}$. Multiply by $\sqrt N$ and below $K_c$ the three sizes fall roughly onto one curve; the dotted line there is $\sqrt{\pi/4}\,/\sqrt{1 - K/K_c}$, that random average amplified as the coupling nears threshold. Above $K_c$ the sizes agree with the theory to a few hundredths.
For Lorentzian frequencies Ott and Antonsen (2008) found that the infinite system obeys one equation exactly, \[ \frac{dr}{dt} = -\gamma r + \frac{K}{2}\, r\,(1 - r^2), \] when the phases start as a wrapped Lorentzian bump, the same for every frequency. Below $K_c = 2\gamma$ coherence decays at rate $\gamma - K/2$; above it the fixed point is Kuramoto's curve again.
Figure 4. Three populations of 100, 1,000 and 10,000 oscillators ($\gamma = 0.5$) start from phases drawn from that family with the chosen $r$; gray is the one-line equation solved from the same start. Each change of a slider or New draw starts fresh frequencies and phases. On the first draw at $K = 1.5$ the 10,000 never stray more than 0.03 from the curve, while 1,000 and 100 stray by 0.2 to 0.3. Their gaps are largest during the rise, where a slightly different start shifts the whole curve in time, and afterwards they jitter around the fixed point.
The threshold depends only on the density at the center: $K_c$ is 1 for the Lorentzian ($\gamma = 0.5$), $2\sigma\sqrt{2/\pi} \approx 0.80$ for the Gaussian ($\sigma = 0.5$) and $6/\pi \approx 1.91$ for the uniform density on $-1.5$ to $1.5$. Above it the rest of the shape matters. Requiring the locked oscillators, at $\sin\phi = \omega/Kr$, to reproduce $r$ gives Kuramoto's condition $r = Kr\int_{-\pi/2}^{\pi/2}\cos^2\phi\; g(Kr\sin\phi)\,d\phi$ (Strogatz 2000). A uniform density has no tails to recruit: everyone locks once anyone can, and $r$ jumps from 0 to $\pi/4$ (Pazó 2005).
Click a point to load its distribution and coupling into Figure 6
Figure 5. $r$ against $K/K_c$ for 200 oscillators per distribution, with frequencies at evenly spaced quantiles of the density so there is no sampling noise. Each point is one run of 150 time units from random phases (fixed seed), $r$ averaged over the second half, with a bar from its lowest to highest value there. The uniform population stays below 0.1 up to $K_c$ and is locked at 0.82 by $1.02\,K_c$, with one restless point between; the theory's jump to $\pi/4 = 0.785$ is the vertical part of its faint curve. The Gaussian climbs faster than the Lorentzian (0.66 against 0.40 at $1.2\,K_c$, both on their curves) because it has fewer stragglers in its tails.
A bimodal distribution has no single threshold. Over a window of coupling two groups lock separately, one per peak, and lap each other, so $r$ beats: the standing-wave state that Martens et al. (2009) solved for two Lorentzian peaks. Stronger coupling merges the groups.
Figure 6. The same model with four frequency distributions, 60 oscillators each (a fresh random draw every time you pick one). The bimodal one has peaks at $\pm 1$ with standard deviation 0.25; with 60 oscillators its two groups appear from about $K = 0.8$ and merge between $K = 2$ and $2.2$. Try $K \approx 1.5$: two clusters of opposite colors turn in opposite directions and $r$ swings through a wide range; at $K = 3$ they are one cluster and the range collapses to a single value.
Male Pteroptyx malaccae fireflies gathered in trees in Thailand flash in synchrony about once every 560 ms (Buck and Buck 1968). A standard cartoon of how flashes can align is Mirollo and Strogatz's (1990) pulse coupling: each oscillator charges toward a threshold, fires and resets, and each flash raises the others' charge by $\varepsilon$, firing any it pushes past the threshold. With a charge that rises fast and then slowly, a pulse advances a nearly charged oscillator further in phase than a freshly reset one, so phases bunch up. Below, each firefly sees only its eight neighbors.
Click on the grid (or focus it and press Enter) to perturb a region
Figure 7. 1,200 pulse-coupled fireflies on a 40 by 30 grid, natural rates spread by about 8 percent. Each dot flashes when its charge reaches the threshold and pushes its eight neighbors' charges up by $\varepsilon$. The strip below traces the order parameter of all their phases over the last 60 simulated seconds: low while the field is patchy, near 1 when the grid flashes together. Counting a start as together when $r$ averages above 0.9 from 30 to 60 s, 0 of 20 starts get there at $\varepsilon = 0.01$, 11 at 0.02 and 19 at 0.03. With the straight charge $r$ averages 0.10 to 0.12 even at $\varepsilon = 0.1$.
In a ring each drummer hears only its two neighbors. Besides unison, a ring can lock into a twisted state, a traveling wave whose phase winds a whole number of turns around it (Wiley, Strogatz and Girvan 2006); about one run in five at the default settings ends twisted.
Click an oscillator to knock it out of phase
Figure 8. Twelve ring-coupled oscillators, natural tempos drawn uniformly within the tempo-spread value either side of 2. The state readout compares each drummer's tempo over the last 5 seconds: "locked" when all agree to within 0.02 rad/s, with the winding number when the phases twist around the ring; when the tempos are too spread for the coupling it counts the runs of neighbors that share a tempo. Locking can take a minute or more. The recovery timer reports when order returns above 0.8 after a perturbation; a kick that leaves the ring twisted never gets back above 0.8. Recovery is faster at higher coupling: in node runs of three kicked drummers the median time back above 0.8 fell from about 1.9 s at coupling 0.5 to 0.14 s at 2. With sound on, each flash is a click pitched by ring position.
London's Millennium Bridge swayed on its opening day, 10 June 2000, as pedestrians fell into step with its sideways motion and made it larger. Strogatz and colleagues (2005) modeled the crowd as synchronizing oscillators with the bridge as the shared board, and estimated the damping a crowded footbridge needs to stay below threshold; it reopened in 2002 with dampers.