Hallucinations, Polar Graphs, Alan Turing, Logarithms and the Leopard

So how does all that go together... it's all about brain chemistry. In a recent article in Plus Magazine (you ought to read it regularly), Marianne Freiberger, one of the Plus Editors gives a really nice capsule history and current status of the research going on about the brain and hallucinations. Amazingly it ties together the entire cast of characters above, and more.. Here is just enough to tease and entice you to go read (study?) the article.

It seems that the hallucinations induced by some drugs and certain illnesses fall into identifiable sets, four of them by one classification scheme, spirals, and funnels (these are often reported by LSD users); honeycombs (from marijuana most often) and different types of lattices like cobwebs or triangles. . Research about what happens in the brain that causes us to see hallucinations has also helped us understand the very mathematical model of how the brain images the world onto the cortex. Amazingly it seems that the imprint on V1 (the part of the cortex where vision is processed first) from the field of vision is achieved by translating a polar graph into Rectangular coordinates. (Do you see a pre-calc project building here?)

The transformation simply turns the polar (r,q) into (log(r), q). (this assumes r is sufficiently large, apparently there is massive complication if you let r get too small. ). So for example, as they point out in the article, a circle in the visual field will plot itself to a straight line in the rectangular coordinate field of V1 (I imagine something like a string of adjacent brain cells being activated). Conversely, a ray emanating from the center of the eye would trace a line perpendicular to that. Now just wait until the next kid asks me why we need to learn to use polar coordinates...B A M...

Now the hallucinations seem to start in the same V1 area of the cortex, so they have been investigating how the illness or drugs can cause the cortex to distort the signals it gets from the eye. It turns out that the theory most helpful, is the theory that Alan Turing, the computer guy, came up with to explain how animals got stripes and spots, like the leopard.. Turing used differential equations to model how two types of agents, an inhibitor and an activator could coexist in an animal and explain the stripes or spots in terms of the rate at which the two chemicals diffuse through the skin. They even have a interactive applet that illustrates the process (Plus is really good about that kind of stuff). Now Turing's model is just the stimulus for a much more complicated approach of the brain behavior of hallucinations, but still a nice story to share with kids.

Speaking of nice stories to share with kids; I close with a great story I heard at a lecture at the Center for Mathematical Sciences at Cambridge from a young lady (sorry I don't have my notes here) who was talking about the Enigma project in Bletchley Park. Alan Turing, of course, was instrumental in the code breaking efforts there and it was while he was there that he laid the foundation for the programmable computer. Now Bletchley was a VERY secret project, and it wasn't made public until years later. She told of a young man who spent the war years there without being able to tell anyone what he was doing. After the war one of his old teachers walked up to him and cursed him for having hid out at a desk job while his friends gave their lives in the war. Turing, it seems, became a problem for the British Security services when their fears that his homosexuality would make him subject to blackmail and therefor a security risk. He eventually was hounded out of public service and could not get work.

Now the side note is that Turing was fascinated with the story of Snow White, and when he was found dead there was an apple laced with arsenic (ok, found out it was cyanide, and apparently they found it in his body, but no one checked the apple??? on his night table beside him, with one bite out of it. There is some question about whether a tortured Turing killed himself, or if he was done away with by the paranoid security agencies. Whichever, the story was passed around by computer geeks down through the years. Then as two young computer nerds were developing a really cool new approach to computing, they decided that they would honor Turing's part in the computer process by symbolizing his death in their logo, an apple with a bite out of it. The story, as she said, is too good not to tell, even if it is totally untrue.