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Pitt mathematician tracks origin of hallucinations

Monday, August 02, 1999

By Byron Spice, Science Editor, Post-Gazette

Mention the word "mathematics" and many people respond by closing their eyes. It is thus a happy coincidence that this article about mathematics is actually best understood if you close your eyes for awhile.

By closing your eyes and gently pressing your fingers against your eyelids, you will eventually experience a phenomenon called pressure phosphenes -- patterns of light that appear in geometric forms, such as exploding stars, spirals, honeycombs, cobwebs and tunnels.

These personal light shows can also be triggered by flickering lights, by the onset of migraine headaches and, infamously, by hallucinatory drugs such as LSD. And they all can be explained, in part, by mathematics.

"Instability is the key thing here," said G. Bard Ermentrout, a University of Pittsburgh mathematician who specializes in the mathematical modeling of neurologic phenomena. Physiology and the way that images are mapped onto the brain's visual cortex play large roles in hallucinatory patterns, to be sure, but mathematics helps explain how instabilities in the brain arise and contribute to these patterns.

Whether it's by pressure on the retina or by drugs that scramble signals in the brain, the effect is the same: Stimulation to the brain area responsible for vision is cut off. Without the normal flow of signals, instabilities arise, Ermentrout said, and it's possible to devise equations that explain how those instabilities can generate various patterns.

Ermentrout's interest in visual hallucinations dates back 20 years, when he was working on his doctorate degree at the University of Chicago. He had studied number theory as an undergraduate, but in graduate school became interested in what was then a new discipline, mathematical biology.

"I developed a theory of mescaline hallucinations," said Ermentrout, 45, pausing during an interview in his Thackeray Hall office to pull down a 1980 textbook, "Mathematical Biology" by James D. Murray, that includes his findings. "When you've been around this field for more than 10 years," he added with a laugh, "everything you've written suddenly is a classic."

The basic concept of the theory, however, has remained unchanged for two decades. Images that strike the tightly packed nerve cells of the retina, he explained, are mapped by the eye as a series of concentric circles. But, as animal experiments have shown, when these images are transferred to the brain's visual cortex, the geometry changes to lines that form a grid.

The only exception to this mapping scheme, he added, are images from the fovea, the depression at the center of the retina responsible for the eye's sharpest vision.

Pressure on the eyeball inhibits signals from the retina. The lack of signals, however, has the effect of making the brain's cortex more excited. "Neurons really want to fire when they're hooked up," Ermentrout said. To counteract this natural tendency to fire, he explained, inhibitory cells slightly outnumber excitatory cells in the cortex.

But when the retina is no longer sending signals to these excitatory and inhibitory cells, the brain is on its own and the natural tendency of neurons to fire takes over. Rather than responding to stimuli from the eye, neurons in the visual cortex become unstable and begin firing spontaneously.

"That's where the math comes in," Ermentrout said. As a neuron fires randomly, a phenomenon called lateral inhibition kicks in. The stimulated neuron has the effect of stimulating the neurons immediately surrounding it on the grid of the visual cortex and suppressing the neurons just beyond.

When these patterns of alternating stimulation and inhibition in the visual cortex are projected back into the coordinates of the retina, the result is the hallucinatory patterns seen as phosphenes. Adjacent areas of stimulated and inhibited neurons become alternating black and white patterns. The orientation of these stimulated and inhibited areas in the cortex affects the images seen. Areas that follow up and down lines in the cortex translate into bull's-eye or tunnel-shaped retinal images; alternating side to side lines appear as starbursts; a checkerboard pattern appears as a honeycomb.

In addition to pressing on the eyelids, these instabilities can be triggered by sensory deprivation chambers or by the onset of migraine headaches.

"I get migraines without the visual patterns," Ermentrout groused, "which really sucks."

In addition to lack of stimuli, instabilities can be driven by a specific type of stimuli. What are called "flicker phosphenes" are triggered by pulses of light -- a strobe light, for instance, or a contraption once sold under the name of the LSD Flight Simulator. This latter device is a mask that fits over the eyes; blowing into a tube causes a disk in the mask to turn, allowing flickers of light to reach the eye.

The flicker phosphenes appear, Ermentrout said, when light pulses reach about 20 per second -- twice as fast as some common brain processes. Just as sound waves at a resonant frequency can shatter crystal or winds of a certain velocity once caused the Tacoma Narrows Bridge in Washington state to ripple and break apart, light pulses at this critical frequency can cause instabilities.

LSD and mescaline work by yet another mechanism -- interfering with brain chemistry. LSD trails, for instance, are a phenomenon in which an object moving rapidly across the field of vision appears as a continuous image or a series of frozen images. This occurs because LSD slows the reaction of inhibitory cells, Ermentrout suggested last month during an address at Computation Neuroscience '99, a scientific meeting at the Pittsburgh Hilton and Towers.

A slow-moving object doesn't leave LSD trails because the brain has time to turn off one image of the object before it overlaps the next. A fast-moving object, however, overwhelms the impaired brain.

"This is just fun stuff," Ermentrout said, though he has collaborated in the past with psychiatrists on studies of LSD effects. But what was once the fledgling field of mathematical biology has, with the addition of computers, become a major new tool.

As evidenced by the hundreds of researchers attending last month's conference here, computation neuroscience has become a hot field, helping neuroscientists develop models and test ideas about how the brain works.

"Computers can handle lots of data and create models," said Dennis Glanzman, chief of theoretical and computational neuroscience research at the National Institute of Mental Health. "The models are extremely valuable." Researchers, for instance, can see the effects of different sodium ion concentrations in brain cells by using the computer models. The same experiments in humans could be fatal.

Computers also make it easier to translate experimental findings into a mathematically grounded theory, Glanzman and Ermentrout agreed. "Computers make it so easy," said Ermentrout, who is an adjunct faculty member in neurobiology in addition to being a professor of mathematics. "Everybody wants a theory now," he added in mock exasperation.

His own research has included collaborations with Nancy Kopell, a Boston University mathematician and a 1990 recipient of a MacArthur Foundation "genius grant." That work, based in part on studies of lampreys and the intestinal tract, analyzes how the neural system synchronizes rhythmic motor activity, such as walking, swimming and breathing.

"It's biologically interesting," Ermentrout said, "but it also has some interesting math."

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