Strogatz’s Sync

Evidently, as Feedburner suggests, there are quite a few more than the two or three suspected subscribers to this log’s feed. I’ll be amused to see if that number goes down as I post a bit more consistently, but for the moment it’s nice to know there’re ears out there.

I recently got done reading a book called Sync written in ’03 by Steven Strogatz (doctoral advisor to the professor I’m working with at the moment). Disclaimer: if I write about something I read it’s not really meant to be a “review” so much as “random wonderments” post-reading. My hope is that this style keeps the reading somewhat interesting and digestible while maintaining some semblance of coherence.

As an example, one of the first problems he motivates the entire book with is fathoming how exactly the many many tiny pacemaker cells of the heart fire in, you guessed it, sync to contribute to a heart beat:

…Each pacemaker cell is abstracted as an oscillating electrical circuit … a constant input current causes the voltage across the capacitor to charge up, increasing its voltage steadily. Meanwhile, as the voltage rises, the amount of current leaking through the resistor [wired in parallel] increases, so the increase slows down. [By definition,] when the voltage reaches a threshold, the capacitor discharges and the voltage drops to zero.

Next, Peskin idealized the cardiac pacemaker as an enourmous collection of those mathematical oscillators [all equal and able to communicate to one another]. … Specifically, when one oscillator fires, it instantly kicks the voltages of all the others up by a fixed amount.

The effects of the firing are mixed: Oscillators that were close to the threshold are knocked closer to the firing oscillator but those close to baseline are knocked farther out of phase. In other words, a single firing has synchronizing effects for some oscillators and desynchronizing effects for others. The long-term consequences of all these rearrangements are impossible to fathom by common sense alone.

He goes on, with a colleague, to help extend Peskin’s proof that two of them will indeed synchronize in the long-term to an arbitrary number of them (a bit more of a difficult task). Moreover, he emphasizes that this “so the increase slows down” part of the model described above is crucial for synchrony to occur.

The rest of the book extends on that type of oscillation problem quite a bit more. It undertakes a massive pop-sci sampling of examples of synchrony in a wide variety of areas. Sometimes he’d drift a bit off the topic of synchrony, but eventually some thread brought it right back into play. All of it was entertainingly explained, while he reavealing his own personal connection to the work in an anachronistic way. The text is littered with some of the funniest scientific analogies I’ve seen. For instance, his description of a laser begins with:

Imagine you wake up one morning and find yourself on on alien planet, entirely deserted except for a watermellon and step stool beside it. …

He also made it a point to illustrate where we’ve verified certain types of synchrony in practical science (e.g. Josephson arrays, circadian rhythms) while others are less pragmatic to verify (heart and neural studies, for example).

Something that definitely stuck in my mind afterwards is the way the more traditional research of nonlinear dynamical models and their applicability to practical systems evolved into the more current research of complex systems. Discussing some of his earlier work with Art Winfree:

In 1981, nonlinear dynamics had certainly not advanced to the stage where it could predict the behavior of such rotating waves in three dimensions. There was no hope of calculating their evolution in time, their lashing about, their swirling patterns of electrical turbulence. Even if the calculations were possible (assisted by a supercomputer, perhaps), any such attempt would be premature, since one wouldn’t know how to interpret the findings. … So Winfree felt that the first step would learn how to recognize them, to anticipate their features in his mind’s eye; he would worry about their modus operandi later.

This sentiment appearing back in the ’80s for exploring the dynamics of excitable media (with potential applications to an earnest start at modeling the systems-level chemistry of the intenstines, stomach, and heart) also seems to apply to today’s research in complex networks where the pure mathematics and statistics of dynamical null models for systems needs to be developed in tandem with useful methods to probe and explore a system of interest. I can’t help but feeling that a good balance of the two vantage points where both feed each other is a good way to approach this young field.

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