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When Math Meets Nature: Turing Patterns and Form Constants

The views expressed are those of the author and are not necessarily those of Scientific American.


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When I was in fourth grade, my teacher asked us each to write a short “Just So” story in the manner of Rudyard Kipling, author of the classic children’s book Just So Stories. My topic: “How the Mouse Got Its Tail.” I long ago forgot whatever elaborate theory I came up with to explain this rodent feature, but I’m sure it had my classmates on the edge of their seats.

I was reminded of Kipling and my juvenile attempt to emulate him when I interviewed mathematical biologist James Murray for my latest article for Simons Science News on Turing patterns — a mechanism devised in 1952 by Alan Turing to explain such naturally occurring patterns as tiger stripes, leopard spots, the precisely spaced rows of alligator teeth, angelfish stripes, and so forth. One of Kipling’s stories was entitled “How the Leopard Got Its Spots,” and Murray confessed that his interest in the possibility of Turing mechanisms in biological systems was piqued when he read that story to his young daughter and she pressed him to explain how the leopard really got its spots. (His daughter sounds like a natural-born scientist.)

African Leopard in Serengeti, Tanzania. Source: Wikimedia Commons. User:JanErkamp.

From the article:

[Turing] proposed that patterns such as spots form as a result of the interactions between two chemicals that spread throughout a system much like gas atoms in a box do, with one crucial difference. Instead of diffusing evenly like a gas, the chemicals, which Turing called “morphogens,” diffuse at different rates. One serves as an activator to express a unique characteristic, like a tiger’s stripe, and the other acts as an inhibitor, kicking in periodically to shut down the activator’s expression.

To explain Turing’s idea, [Murray] …  imagined a field of dry grass dotted with grasshoppers. If the grass were set on fire at several random points and no moisture were present to inhibit the flames, Murray said, the fires would char the entire field. If this scenario played out like a Turing mechanism, however, the heat from the encroaching flames would cause some of the fleeing grasshoppers to sweat, dampening the grass around them and thereby creating periodic unburned spots in the otherwise burned field.

Turing patterns are a perennial favorite among science writers, especially in light of the 100th anniversary of Turing’s birth last year. Also? Pretty! And narrative angles like why the tiger has stripes play to broad audiences, so editors love them too.

Scientists, on the other hand, have mixed feelings about Turing’s little foray into mathematical biology. Even Murray, who has done seminal work in biological patterning, confessed that he was a little burnt out on Turing after the centenary, pronouncing the mathematician’s contributions to biology rather over-rated. Turing was a mathematician, first and foremost, and his proposed mechanism is (by his own admission) a highly simplified and idealized take on a messy, complicated system.

Nor was he the first to tackle this sort of thing: in 1917, for example, D’Arcy Thompson published On Growth and Form, which also talked about chemical morphogens contributing to periodic patterns. And Boris Belousov independently came up with a closely related model to that proposed by Turing, probably also in the early 1950s, although Belousov struggled to get his work published; it did not appear until 1959, in an obscure journal. Per Murray (in a 2012 paper), Belousov showed “how a group of three reacting chemicals could spontaneously oscillate between a colorless and a yellow solution.”

That doesn’t mean reaction-diffusion and other proposed models for pattern formation can’t be useful: they may lead to breakthroughs later on. This certainly seems to be the case with Turing mechanisms, and the Simons Science News article gives a bit more detail on two recent papers in particular that are generating interest among mathematical biologists: one on how the ridges form on the roof of the mouth in mice, and another on the formation of digit patterns in mouse paws, and why polydactylism may occur.

The latter is particularly intriguing because it might provide additional evidence for the thesis expounded by Neil Shubin (Your Inner Fish), among others, that our hands have an evolutionary antecedent in fish fins. Maria Ros, one of the co-authors, explained that in their experiments, they were able to produce mutated mouse embryos with as many as 14 digits on a paw, looking for all the world like a fan-shaped fin — particularly since they don’t have joints. “We’ve always been interested in why we have five digits and how this is controlled,” she told me. “Our ancestors were polydactyl. In the transition from the fish to tetrapods, from the life in the water to the life on the earth [land, not the planet], there was a transition from the fin to the limb.”

Examples of form constants. Source: Cowan 2002, http://www.scholarpedia.org/article/Models_of_visual_hallucinations

If we really want to get into some interesting speculation, we can think about whether a Turing model can be applied to neurons in the brain, which could be “described mathematically as activators or inhibitors, encouraging or dampening the firing of other, nearby neurons in the brain.” And that could potentially explain why we see certain recurring patterns when we hallucinate.

Back in the 1920s and 1930s, a University of Chicago neurologist named Heinrich Kluever classified hallucination patterns into tidy categories known as form constants: checkerboards, honeycombs, tunnels, spirals and cobwebs.

Over seventy years later, another Chicago researcher, Jack Cowan – who holds dual appointments in mathematics and neurology – set out to reproduce those hallucinatory patterns mathematically, believing they could provide clues to the brain’s circuitry.

While the random fluctuations in brain activity might technically just be “noise,” the brain will take that noise and turn it into a pattern. Since there is no external input when the eyes are closed, that pattern should reflect the architecture of the brain, specifically the functional organization of the visual cortex.

That organization is rather fractal in nature, repeating the same patterns at different size scales. “Like tree branches, the brain recapitulates,” neuroscientist Robin Carhart-Harris told me when we chatted last year for my (forthcoming) book on the science of self. “You are not seeing the cells themselves, but the way they’re organized – as if the brain is revealing itself to itself.”

Cowan found that the predicted patterns from his calculations closely matched what people see when under the influence of LSD, and suspected these patterns might arise from a type of Turing mechanism.

Neurons respond not just to color and brightness in the visual field – the external input – but also to internal interactions with other neurons. Nigel Goldenfeld, a physicist at the University of Illinois, Urbana-Champaign, worked with Cowan on a model for a generic neural network with random connections. They showed that in such a case, the firing of neurons would amplify the Turing effect, making hallucinations more common.

But if our visual cortex actually behaved in this way, it would interfere with our vision. “You don’t want to be enthralled by a hallucinatory spiral when there is a dangerous tiger in front of you,” said Goldenfeld. So he and Cowan speculated that this might be why our brainy architecture is non-random: it confers an evolutionary advantage that limits interactions to stronger short-range connections with nearby neurons. Excited neurons simply follow the familiar uniform diffusion patterns we associate with the behavior of atoms in a gas, and the visual external input from the eyes easily dominates any weaker internal activity.

Sure, it’s highly speculative. That’s part of the excitement of doing cutting-edge science. It will be fascinating to see what biologists and neuroscientists interested in Turing-type mechanisms and biological/brainy patterning come up with next.

References:

Bressloff, Paul C.; Cowan, Jack D.; Golubitsky, Martin; Thomas, Peter J.; Weiner, Matthew C. (2002). “What Geometric Visual Hallucinations Tell Us About the Visual Cortex,” Neural Computation (The MIT Press) 14 (3): 473–491.

Economou, Andrew D. et al. (2012) “Periodic Stripe Formation by a Turing-Mechanism Operating at Growth Zones in the Mammalian Palate,” Nature Genetics 44(3): 348-351.

Economou, Andrew D. and Green, Jeremy B.A. (2013) “Thick and Thin Fingers Point Out Turing Waves,” Genome Biology 14:101.

Ermentrout, G.B. and Cowan, J.D. (1979) “A mathematical theory of visual hallucination patterns,” Biological Cybernetics 34(3): 137-150.

Kondo, Shigeru and Miura, Takashi. (2010) “Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation,” Science 329: 1616.

Murray, James D. (2012) “Why Are There No Three-Headed Monsters? Mathematical Modeling in Biology,” Notices of the AMS 59:6.

Sheth, Rushikesh et al. (2012) “Hox Genes Regulate Digit Patterning by Controlling the Wavelength of a Turing-Type Mechanism,” Science 338: 1476.

Turing, Alan M. (1952) “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 237(641): 37-72.

Jennifer Ouellette About the Author: Jennifer Ouellette is a science writer who loves to indulge her inner geek by finding quirky connections between physics, popular culture, and the world at large. Follow on Twitter @JenLucPiquant.

The views expressed are those of the author and are not necessarily those of Scientific American.





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