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.

This summary is to try to keep a bigger picture in mind before getting too specific too quickly: a brief survey of a topic of interest in a text. It is specifically in the context of global optimization. I first wrote this post a few months back. Most of meat is obtained from Wales’s Energy Landscapes and Doye and Massen’s “Characterizing the network topology…atomic clusters”. Finding the global minimum and local minima on an energy landscape (a purely theoretical thing determined from pure quantum-mechanical models or hybrid quantum-classical models) gives us cues as to how a set of molecules will structurally stabalize. Practically, this information is invaluable to molecular structure prediction, which helps us better understand how molecules are actually built.

The issue lies in finding the global minimum. I don’t pretend to deeply know this area or detail it here by any means, but I may soon! Finding the global optimum in general is an NP-hard problem in computational complexity. In other words, “there is no known algorithm that is certain to solve the problem within a time that scales as a power of the system size”. So what do we do? Exploit structural properties of a given atomic cluster to drastically improve the running time of the algorithm. (Wales, “Global optimisation”)

This subfield in itself is very active with many different algorithms suited for all sorts of structures. But as far as small worlds go, it seems a basin hopping global optimization approach that could take advantage of the relatively scale-free properties of the landscape. But as implied (Wales, “Small worlds”), we should take caution in considering the cluster we’re applying these algorithms to, how exactly the global optimization search is improved, and whether the small world results apply generally enough to a broader class of molecules or not:

It is not yet clear how general this result may be for other potential energy surfaces [than Lennard-Jones]. For example, the connectivity of the constrained polymer conformations was found to exhibit a Guassian rather than a power law distribution for a noninteracting lattice polymer. For the Lennard-Jones polymer…Doye has found that the connectivity distribution does not really follow a power law, but the scaling is stronger than exponential…If low energy minima are generally highly connected then they should be easier to locate in global optimisation searches…However, the best way to utilise this information more efficiently is not yet clear.

Though it has been a few years since that was written, it seems to still hold true from my cursory understanding. Most of the work that has been done seems to be applied to Lennard-Jones clusters. Needless to say, I’m motivated to dig a little deeper.