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Uncoiling the spiral: Maths and hallucinations

Computer generated representations of form constants. The top two images represent a funnel and a spiral as seen after taking LSD, the bottom left image is a honeycomb generated by marijuana, and the bottom right image is a cobweb.

Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. But it’s not just hallucinogenic drugs like LSD, cannabis or mescaline that conjure up these geometric structures. People have reported seeing them in near-death experiences, as a result of disorders like epilepsy and schizophrenia, following sensory deprivation, or even just after applying pressure to the eyeballs. So common are these geometric hallucinations, that in the last century scientists began asking themselves if they couldn’t tell us something fundamental about how our brains are wired up. And it seems that they can.Geometric hallucinations were first studied systematically in the 1920s by the German-American psychologist Heinrich Klüver. Klüver’s interest in visual perception had led him to experiment with peyote, that cactus made famous by Carlos Castaneda, whose psychoactive ingredient mescaline played an important role in the shamanistic rituals of many central American tribes. Mescaline was well-known for inducing striking visual hallucinations. Popping peyote buttons with his assistant in the laboratory, Klüver noticed the repeating geometric shapes in mescaline-induced hallucinations and classified them into four types, which he called form constants: tunnels and funnels, spirals, lattices including honeycombs and triangles, and cobwebs.

In the 1970s the mathematicians Jack D. Cowan and G. Bard Ermentrout used Klüver’s classification to build a theory describing what is going on in our brain when it tricks us into believing that we are seeing geometric patterns. Their theory has since been elaborated by other scientists, including Paul Bressloff, Professor of Mathematical and Computational Neuroscience at the newly established Oxford Centre for Collaborative Applied Mathematics.

How the cortex got its stripes…

In humans and mammals the first area of the visual cortex to process visual information is known as V1. Experimental evidence, for example from fMRI scans, suggests that Klüver’s patterns, too, originate largely in V1, rather than later on in the visual system.

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The first model of pattern formation in animal coats goes back to Alan Turing, better known as the father of modern computer science and Bletchley Park code breaker. Turing was interested in how a spatially homogeneous system, such as a uniform ball of cells making up an animal embryo, can generate a spatially inhomogeneous but static pattern, such as the stripes of a zebra.

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Practical applications of this work include computer vision — computer scientists are already building the inter-connectivity structures that Bressloff and his colleagues played around with into their models, with the aim of teaching computers to detect contours and textures. On a more speculative note, Bressloff’s research may also one day help to restore vision to visually impaired people. “The question here is if you can somehow stimulate part of the visual cortex, [bypassing the eye], and use that to guide a blind person,” says Bressloff. “If one can understand how the cortex is wired up and responds to stimulation, perhaps one would then have a better way of stimulating it in the right way.”