Wednesday, December 10, 2008

Pattern formation phase experiments

Today I played around with trying to understand where the patterns come from in simple oscillators. In the following picture, the center region is exactly 180 phase shifted relative to the outside (circular "boundaries" as always). Note the cool reconnection events about 1/3 and 2/3 the way up from the bottom (t=0).

The interesting thing here is that the boundaries begin to oscillator faster than the surrounding regions. The center goes through 6 cycles in the time it takes the boundary to go through 7. At 7 edge-cycles versus 6 center-cycles there's a disconnection event where the two regions become disjoint and then reconnect one cycle later. These discontinuities are where interesting patterns emerge.

Why should it be that the boundary oscillates faster than the center? This is a bit counter-intuitive. Imagine two oscillators sitting next to each other and diffusing some of their energy into each other. Consider the moment when the first oscillator is at its maximum value and the second is at its minimum. At this moment the first oscillator is dumping a lot of its material into its neighbor. In other words, right when the second should be at its minimum value it is instead being "pullled forward" by the incoming flux. Conversely, by dumping flux into its neighbor, the first never quite makes it to maximum value and thus sort of short-cuts it way to the downward part of the cycle. Half a wavelength later the reverse is true. Thus, both oscillators act to pull the other one ahead and thus they both run a little faster as their amplitudes are reduced.

As noted before, work on coupled oscillators is as old as Huygens 1665 paper. Here's a more recent synthetic biological investigation from Garcia-Ojalvo, Elowitz, and Strogatz. What I haven't found yet (probably because I haven't looked hard yet) is a paper showing the spatial dynamics of such coupled oscillators as demonstrated here.

So what happens when the two regions are not started exactly 180 out of phase? Yet another interesting instability forms. Here's the same thing at 170 degrees:

This time the boundary between the two regions begins to wobble around as the two sides compete for control of the boundary space. This instability also creates interesting disconnection / reconnection events around 7 cycles. And what if we symmetry break the size of the two areas? Here's 180 degree separation with the center region being a bit smaller than the outer:

Now you see the unstable edge oscillation like above case after the perturbations travel all the way around and end up interacting with the center during the second reconnect event. Clearly such patterns are all reminiscent of diffraction scattering and other sorts of complicated spatial pattern-forming phenomena where waves are bouncing around inside of closed spaces -- I find all such phenomena hard to intuit and these examples are no different. Where things get fun IMHO is to see how noise plus such simple oscillators generates interesting formations as the ones I posted a few days ago.

Several people have asked me what is the relationship is between these simulations I'm showing here and cellular automata? I argue that these systems are analog, memoryless versions of CAs. While CAs are very logically simple, they aren't nearly as hardware simple as the systems I'm working on here. For example, Wolfram's lovely illustration of all 256 binary 1D CA rules are simple rules, but their implementation presupposes both memory and an a priori defined lattice that includes left/right differentiation. However, as Wolfram points out on page 424 of NKS, the symmetric 1D rules do generate interesting short-term random patterns when initialzied with random state so these are a good binary model for the analog systems pre-supposed here.

Meanwhile, my friend Erik Winfree's lab has very cleverly built DNA crystaline structures that have do define a lattice and thus can implement the Turing comple rules at molecular scales. But on the scale of complexity, I'd argue that these amorphous analog systems are "simpler" in the sense that I can more easily imagine them evolving from interacting amplifiers that would have independent precursor functionality and without imposing a lattice. Erik might disagree, but anyway, it's this idea of evolving-interacting-amplifiers that I'm going work on as I continue this.

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