Science After Sunclipse

In the late 1970s, Cowan began to suspect that the culprit was not mescaline or LSD itself, but rather the visual cortex at the back of the brain, and in particular the section known as the primary visual cortex, V1. (In technical terms, this means focusing on topological rather than hodological hallucinations, studying those which might arise from a specific part of the brain rather than from malfunctions in the connections among multiple regions.) From magnetic-resonance imaging studies, we know that V1 becomes active when a subject is asked to examine closely an image presented to the eyes. Neuroscientists have discovered a great deal about V1; we know, for example, that its cells are organized into columns, perpendicular to the cortical surface, which are about a millimeter wide (the exact size varies among different mammal species). However, Cowan realized, to a decent first approximation, the visual cortex could be treated mathematically as a uniform surface, essentially a flat plane.

HOW THE CORTEX GETS ITS SPOTS

What patterns of activity would arise when this patch of brain-stuff is perturbed by some outside influence? The answer lies in V1's symmetry: no matter where you stand over it, it looks roughly the same. Shift (or translate) it left or right, and it appears unchanged. To a mathematician, this approximate "translational symmetry" carries a direct implication: the natural patterns or "eigenmodes" of neural activity moving across the cortex will be plane waves.

Imagine a wave whose crest is a straight line, rippling placidly across a tank of water. The situation in V1 is much the same, except that instead of the displacement of water, the quantity of interest is the neural activity at a given point. In physics jargon, the set of all points and their associated neural firing rates constitutes a field, and the variation over time of that field is tractable via the tools of field theory.

Oddly enough, this thread of the research stretches back to Alan Turing. After he won the War for England by breaking the Enigma cipher and reading Hitler's mail, Turing grew interested in biology. In particular, he tried to explain how periodic patterns — from the centipede's segments to the zebra's stripes — could arise from the flow of chemicals during embryonic development. Turing described, in the last paper he published, what he called "The chemical basis of morphogenesis": he imagined two substances, an activator and an inhibitor, which spread throughout the developing organism. The activator would, for example, up the production of the pigment melanin, while the inhibitor would decrease it. Turing found that if one chemical spread more rapidly than the other, then the smooth placidity of the uniform expanse of cells could break down, and a pattern of stripes would arise: rows of dark cells with lots of melanin separated by strips of light cells with none.

Now, clever as this idea was, it isn't the way organisms make segments. The way repeating vertebrae get made, for example, is through a different, but equally seductive mechanism. However, one of the odd things about mathematics is that it can have uses beyond the place where it was first invented. In the study of the brain, we also hear about "activation" and "inhibition": individual nerve cells can be stimulated to greater levels of activity by certain inputs, and reined back by other inputs. Turing's basic scheme for studying how a placid sheet of cells can suffer a breakdown of its stability and see a repeating pattern arise turns out to be just the thing for getting a handle on how patterns of activity can shape themselves in the cortex.

HOW THE EYES TALK

The relationship between the eyes and the visual cortex is an interesting one, with significant ramifications. It is not the case that, if a square is projected onto the retina, the cortical columns in V1 will "light up" in a square pattern, like the image from a camera being reproduced on a television screen. Instead, a more sophisticated "mapping" exists between our field of view and our primary visual cortex, a relationship known as a complex logarithmic map. A complex logarithm turns circles into straight lines, for example, so that if we stare at a light circle on a dark background, a straight line of neural activity will cut across V1. (If two lines meet at an angle, a complex logarithmic map will turn them into two other lines meeting at the same relative angle. This is a type of conformal map, which preserves angles locally but distorts shapes at larger scales.)

"Spiral" form constant provoked by LSD,
from Bresloff, Cowan et al. (2002)The logical question is then, what images seen by the eyes correspond to the eigenmodes characteristic of the perturbed V1? What would you have to see for the ordinary activity of V1 to look like the patterns seen when the brain is perturbed by a migraine or a drug?

The answer is that the eigenmodes of this simple model correspond to two of Klüver's form constants.

Now, the visual cortex is not a perfectly uniform, completely symmetric surface. For starters, the cortex can recognize orientation: patches of cortex have a sense of directionality, such that a particular small piece of cortex will become active when its region of the visual field contains a feature tilted at a certain angle. (In the macaque, for example, these "iso-orientation patches" are about 0.7 mm across, the width of a mechanical pencil point.) When this feature is added to the cortex model, the other two form constants emerge from the equations. Shapes seen in hallucinations arise naturally from the symmetries of V1.

THE BRAIN OF THE SPHERICAL COW

I had the good fortune to see Jack Cowan explain this research in person, although on that day, the emphasis was on other ways the model could be tested — using EEG measurements and so forth — rather than on hallucinations provoked by the 1960s. For that, the reader can watch the video from California.

A complex-systems expert who sat next to me in the audience opined that Cowan had "averaged out the neuroscience" in the first few minutes of his presentation: this model of the cortex omits almost all of the detailed biochemical knowledge we have on neurons. This is at once the blessing and the bane of such research.

Physicists like to joke that their field studies "spherical cows", entities so far abstracted from the real world that they lose the relevant details. Studying vast collections of neurons using statistical field theory is rather like dissecting the brain of a spherical cow: If a "toy model" exhibits behaviour which resembles that of the real, living system, then we can provisionally neglect the intricacies of detail. Features in EEG traces or visual hallucinations which occur on top of the toy model's predictions might then be due to those biological peculiarities.

FURTHER READING

Those who wish to brave the mathematical details can read the literature:

• P. C. Bresloff et al., "What Geometric Visual Hallucinations Tell Us about the Visual Cortex" Neural Computation 14 (2002):473–491. Also available here.
• M. Buice and J. Cowan, "Field-theoretic approach to fluctuation effects in neural networks" Physical Review E. (29 May 2007).

For a broader perspective, try the review article

• G. Deco et al., "The Dynamic Brain: From Spiking Neurons to Neural Masses and Cortical Fields" PLoS Computational Biology 4, 8 (2008).

This is very much a field under development. To get the flavour of current research, see, e.g., the preliminary report of Daniel Fraiman et al., "Ising-like dynamics in large-scale functional brain networks" (2008), arXiv:0811.3721.

AFTERWORD

The back story to this post is a little funny:

I wrote the first version of it last September, when the features editor of New Scientist showed up and said, essentially, "Well why don't you write a popular science story, Mr. Critic-Pants." Given that Greg Egan and I had been complaining pretty vocally, he had every reason to be nettled; reining in my instinct for snarky replies, I took the next block of free time I had and wrote the above post.

At that, it's shorter than I'd thought it would be: 1600 words instead of 2400 or thereabouts.

I had it almost finished when the folks at the n-Category Café advised, "No Man but a blockhead ever did anything involving being peppered with buckshot, except for money," so I stuck it on the drafts pile while I tried to figure out what to do with it, and then I got busy with other things.

Came the day when I got Pharyngulated and I realized I should put up something for people coming by, something perhaps of more general interest than my forthcoming essay on the Dirac Equation. I looked on ye olde draft pile, saw this item and realized that it read fine. I tweaked the beginning a little and pushed it out into the 'tubes. This post is a slightly edited version of that earlier draft.

Now, I'll have to figure out how I should handle the next installments. This area of research touches upon several ideas which have wide applicability: mean-field theories, phase transitions, critical points, certain properties of networks, etc. My plan at the moment is to explore each of those general topics and connect them to this specific example, rather than to give a blow-by-blow exposition of a technical paper.

Finding that sort of basic exposition online can be harder than one might expect. . . .