Fireflies

I saw your comment: "The cells need not have identical individual frequencies to reach sychronization."

This seems counter-intuitive to me.

It does make (intuitive) sense, though, because the only time that makes a significant difference is when a lot of fireflies flash around one firefly simultaneously, in which case the firefly wouldn't decide to flash twice in a row but instead interpret it as a singular flash. And if a firefly sees another one flash when it's about to, it would naturally want to synchronize its flashing with the other.

So what this line does (which is in the section of code where a flash alters the fireflies around it) is that after advancing the other fireflies clocks, if any are over 1 then it puts them in the right state to flash on the next time they're evaluated. If you mod it, then you would have the logic:

"if I'm a firefly and see my neighbour flash while just about to flashing myself, don't flash but reset my clock to part way through the cycle"

rather than

"if I'm a firefly and see my neighbour flash while just about to flashing myself, flash as well"

Edit - setting it to one ensures that each neighbour can't keep adding to the clock, advancing it to a random point in the cycle.

Some of these processes use physical latency to time the events: if the fireflies were roughly of the same maturity, they would be of similar size and thus have clocks of similar duration.

I think this is it. There could be some missing detail (e.g. a firefly is unable to "see" when it flashes) which keeps them from nudging their clocks once already in sync

But the algorithm described by the author has a bigger problem: It wouldn't actually sync the fireflies!

If EVERYONE nudged their clocks forward, then an out-of-sync firefly would be nudged the same as an in-sync firefly, and synchronization wouldn't increase over time!

Your rule avoids this problem:

> If you see a neighbour nearby flash, when you are not flashing, nudge your clock forward by a bit.

If you look in "show clocks" mode, it looks like this is exactly what is implemented.

Specifically, an ODE is given for firefly $i$'s phase angle $ \theta_i$

$$\dot\theta_i = \omega_i + K / N \sum_{j=1}^N \sin(\theta_j - \theta_i)$$

This gets interesting because the flies natural frequencies $\omega_i$ are also assumed to be randomly distributed. So you don't get perfect phase-synchronization--flies with fast natural frequencies lead the pack as it loops around the phase ring, and flies with slow natural frequencies are dragged along at the back. Relative to the mean phase, your excess phase approaches a smooth increasing function of your natural frequency. But, for high enough values of $K$, you do get frequency-synchronization--everyone oscillates at the average frequency.

(For low values of $K$, or too-large natural frequencies, "rogue oscillators" emerge in a SNIPER bifurcation. They zoom around the phase ring at a different frequency, briefly slowing as they pass through the cloud of synchronized oscillators. Also applies for too-slow rogues.)

In the video and the OP's simulation, it looked the natural frequencies were all pretty similar, if not the same. However, there was a second addition that is not in the original Kuramoto model (but is in most subsequent models): rather than observing all other flies, only observe nearest neighbors. This can be added by putting a symmetric boolean adjacency matrix $A_{i,j}$ right before the $\sin$ in the previous equation.

This has the effect of making excess phase a smooth function of not only the natural frequency, but also some structural feature imposed by the network. In random networks like Erdos-Renyi, this feature is the node degree, but in the video it looks like it might be the long axis of the bush (so, like, one of the eigenvectors of the graph Laplacian).

(In general, coupled oscillators, such as circadian gene clocks, show this smooth dependence of excess phase on per-unit heterogeneities, which is the topic for the first half of my PhD dissertation. The second half is figuring out what the heterogeneities are when you only have recordings of the dynamics to go by.)

Your objection is just that the model is too simple, but everyone knows linear regression predicts quite a lot, quadratic polynomials estimate physical motion very well, etc.

On the other hand, I don't think the authors make the claims about faithfully and objectively measuring reality that you say they do. You're building up a straw man there.