Self-organizing, art, and mathematical mutants

Deciphering the principles of self-organizing systems is often at the heart of new ideas in biology, including neurobiology. A complex, self-organizing system contains a large

number of elements that have predictable, local interactions with each other, but these local interactions create global properties that cannot be predicted from even the most well-understood local events. This is why mechanical models of these systems fail. Flocks and swarms illustrate this kind of configuration. I’ve written before about how bees and ants optimize foraging routes, and how their optimization solutions have contributed to problem solving strategies in computer science. I’ve also posted about physicists’ observation that birds in a flock followed the lead of precisely their seven closest neighbors, regardless of the density of the flock. The rippling effect that this had on the flock resembled the physics of magnetism. In other words the birds aligned with their neighbors the way the electron spin of particles aligns as metals become magnetized.

The collective behavior of self-organized biological systems, like ant and bee colonies, has come to be called swarm intelligence. The complexity of the group’s behavior is one that could never be managed by an individual member. Nor is it directed by any individual members. Schools of fish, flocks of birds, and herds of animals, all display swarm intelligence, where the actions of individual members of are determined by an inherent set of rules, and these actions transform the behavior of the group. To many observers, this kind of collective behavior indicates that members are assembling, distributing, and processing information. This focus on the role played by information has inspired a good deal of multidisciplinary study.

While doing a little research on swarm intelligence, I came upon an article published in 2006 in the Bulletin of Mathematical Biology, and written by an immunologist whose research on the immune system involves the properties of self-organization. The author is Irun R. Cohen and the title of the article is Informational Landscapes in Art, Science, and Evolution. Cohen uses a multimedia work of art called Listening Post as “a prototypic example for considering the creative role of informational landscapes in the processes that beget evolution and science.”  The work Cohen evaluates is one that relies on self-organizing principles.

There is a trail of thoughts that lead to Cohen’s argument. These include the provocative idea that new meaning can be created from what is often thought of as interference or ‘noise’ in a signal. In other words, new meaning can be created from unstructured, random, or meaningless signals or representations. One of the keys to this possibility comes from a principle of self-organization whose formulation Cohen attributes to a colleague – biophysicist and philosopher Henri Atlan. In the context of a discussion of evolution and natural selection, Cohen summarizes Atlan’s view.

Atlan’s argument goes like this: Existing information first generates surplus copies of itself, which happens regularly in reproducing biological systems. The surplus copies can then safely undergo mutations, and so create modified (new), added information without destroying the untouched copies of the old information. The system thus becomes enriched; it now contains the new information along with the old information. Indeed, it appears that the complexity of vertebrate evolution was preceded and made possible by a seminal duplication of the ancestral genome…

..Information, in other words, feeds back on itself in a positive way; a great amount of information, through its variation, leads to even more information. And as information varies it increases, and so does complexity.

This, I think, is a beautiful idea. Cohen then shows us how the artwork Listening Post shares the features of an organism, and he explains its two sides – a visual-auditory display designed by an artist whose content is driven by an algorithm developed by a mathematician. But this is how it behaves:

The algorithm randomly samples, in real time, the many thousands of chats, bulletin boards, and bits of message that flow dynamically through the cyberspace of the Internet. This simultaneous me ́lange of signals, in the aggregate, is meaningless noise. The algorithm, by seeking key words and patterns of activity, artfully exploits this raw information to construct patterns of light, sound, and words that please human minds. The substrate of information flowing over the Internet is in constant flux so the patterns presented by Listening Post are unpredictable at the fine microscopic scale; but at the macroscopic scale of sensible experience, Listening Post is manifestly pleasing…Listening Post transforms the Internet’s massive informational landscape into a comprehensible miniature. Two attributes of Listening Post illustrate our theme: the work feeds on information designed for other purposes and it survives by engaging our minds.

Cohen develops precise definitions of information, signal, noise, and meaning. These are necessary to the clarity of his broad parallels, like this one addressing the informational structure of the cell and the internet:

In place of electromagnetic codes generated by computer networks, the information flowing within and through the cell—life’s subunit—is encoded in molecules. But the informational structure of both networks, cell and Internet, is similar: Each molecule in a cell, like a chat box signal, emerges from a specific origin, bears an address, and carries a message. Our problem is that the cell’s molecules are not addressed to our minds, so we don’t understand them. The mind exists on a different scale than does the cell; the mind and the cell live in different informational landscapes. We are unable to directly see molecular information; we have to trans- late the cell’s molecules and processes into abstract representations: words, numbers, and pictures…The informational landscape of the cell-organism-species-society is like the informational landscape of the Internet; viewed in the aggregate it is incomprehensible noise.

In his quest to understand, Cohen argues that computers can help. But specifications and the ordering of data are not enough. Understanding is not simply representation. Understanding is primarily an interaction with information. In general we understand complex information by transforming it into a meaningful signal, whether it be meaningful images, symbols, sounds, actions, etc… And this is what Listening Post does.

Part of the point of this article is to explain a research strategy Cohen and colleagues have developed, called Reactive Animation, that they hope will bring them closer to this ideal. He sees it as a synthesis of biology and information science, “between mind and computer.” He and his colleagues have developed a way to record and catalog complex scientific data and “have the data themselves construct representations that stimulate human minds productively.” The effort sounds rich and promising. But what I find most intriguing about these information driven, self-organizing ideas is the view that information breeds information, generating diversity and greater complexity. And productive offspring would be information that breeds information captivating to the human mind. I couldn’t help but consider that mathematics itself is just such a system.

Ideas have crept into mathematics that can look like aberrant variations of other things. The imaginary unit, for example, defined as the square root of negative one, is one such mutation. There is no number which when multiplied by itself will produce -1. It looks like a mistake. This odd use of number symbols showed up when 16th century Italian mathematicians employed a quirky algebraic trick to find the solutions to some cubic equations. The scheme involved the use of numbers that included the square roots of negative numbers. While it was agreed that these roots had no meaning, they nonetheless made it possible to extract the real number solutions that were sought. But these anomalies held the attention of mathematicians through the 18th century. Gauss was one among many who defended their value and explored their meaning. Eventually they produced an entirely new set of numbers – the complex numbers – made from a real number and some multiple of the imaginary unit. These numbers were not only acceptable, they produce beautiful results in mathematics and are extraordinarily useful in physics and engineering. The complex number found a home on the complex plane and produced the branch of mathematics called complex analysis. The system has certainly been enriched.

This may be one of the easiest parallels to draw, but there are others like Weierstrass’ monster function that is now understood by chaos theory and its relationship to fractals. Or, infinitesimals, once a hindrance to the acceptance of the calculus and now the foundation of non-standard analysis. Or the clarifications in geometry provided by the oxymoron: the point at infinity.   

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