Friday, September 18, 2009

Fractals, brains, Pi

A fellow-traveler on SomaSimple found this blogpost, Fractal Thoughts on Fractal Brains by David Pinkus, an interesting sequence of thoughts spurred by an open access article, Broadband Criticality of Human Brain Network Synchronization.

The word "fractal" always conjures up in my mind an image of the Norwegian coastline, then another little tidbit that always has stuck with me, about pi,
"Pi also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river (Stølum 1996, Singh 1997)."

From the blogpost:
"The design, results and context for this study are very sophisticated, and the implications are quite abstract. So I’m going to do my best to be clear. First the context: Many natural systems exhibit fractal organization and behavior. A fractal is a branchlike structure. Think of a tree: (1) Trees have many more small branches than large ones. This characteristic is also sometimes called a “power-law” or “inverse power law” or a “1/f” organization. Each of these terms means that there are exponentially more small branches compared to big ones. (2) Trees are “self-similar,” meaning that small branching patterns resemble larger ones. This characteristic is also sometimes called “scale invariance” or “scale free” because no matter the size you are looking at, the general branching shape is the same. (3) The complexity of tree branching patterns can be quantified. Fractals are called “fractals” because they exist in fractional dimensions. A line fits perfectly in one-dimension. A plane (like a piece of paper) fits in two-dimensions. Fractals fit in between a line and a plane (or in the real world between two and three dimensions). More simply, because they are so complex, with huge numbers of tini tiny branches, trees never quite reach three dimensions. If you put them in a box, there will always be some space left over.

You may quickly recognize that many other natural structures besides trees are fractals: Neurons, rivers, the respiratory system, the circulatory system, geological fault lines, snow-flakes, and so on."

As if fractal form doesn't sound complex enough, there is fractal behaviour, apparently:

"Natural systems also produce fractal behavior over time or in dynamics. Earthquakes are a common example. There are many more small earthquakes than large ones (which is nice by the way). Other examples include the size of extinction events in animal species, numbers of academic publications (a few researchers do huge amounts of work and the rest of us do just a little), numbers of hits to web-sites, wait times in stop-and-go traffic, and word usage in literature (i.e., zipf’s law)."

Fractals as verbs as well as nouns. I've often thought of the brain as more verb than noun.

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