New interdisciplinary graduate-level course explores mathematics of the brain
BY DAWN LEVY
On April 3, Stanford's Mathematics Department launches a new course, Math 299, Mathematics of the Brain. Coincidentally, April is "Mathematics Awareness Month," and this year's focus is mathematics and the brain. Teaching Math 299 is Victor Eliashberg, a systems engineer interested in the workings of the brain.
"Understanding the brain as a system requires systems engineering, electrical engineering, computer science, neurobiology, psychology and, most importantly, a broad mathematical background," says Eliashberg, a consulting professor in the Electrical Engineering Department. "Mathematics is the only common point on which to converge the discipline."
The goal of the course is to address the problem of systems integration in brain modeling and help students enter this extremely challenging and promising area of research.
"There is a lot of interesting brain-related research conducted at Stanford and beyond," says Yasha Eliashberg, chair of the Mathematics Department and Victor Eliashberg's brother. "What is missing is an adequate mathematical formalism to efficiently represent and integrate the broad multidisciplinary knowledge. Our department hopes that Math 299 will eventually lead to launching an interdisciplinary program devoted to the brain as an integrated computing system."
The class will meet Tuesdays and Thursdays spring quarter from 2:15 to 3:30 p.m. in Building 380, Room 383N. It is a graduate-level course, but advanced undergraduates from a broad spectrum of majors, from mathematics to biology, are welcome.
The course looks at the math behind the brain's structures and functions. Advances in neuroscience show biological neurons to be much more complex units than those used in traditional brain models. Why does the brain need such complex neurons? "Contrary to popular beliefs, the properties of individual neurons directly affect higher mental functions, such as memory and learning," Victor Eliashberg says. "These 'low-level' properties account for the most mysterious 'high-level' psychological phenomenon—the phenomenon of mental set, or context. The distance between neuroscience and cognitive psychology is much shorter than is generally believed."
In an interview in Yasha Eliashberg's office in Building 380, Victor Eliashberg draws a "Y" to demonstrate the concept of mental set. "What is that?" he asks a visitor. The visitor stabs at a guess: "Um, a fork in a road?" Next to the "Y" Eliashberg writes "ES." "Oh, it's the letter Y," says the visitor. "What about now?" he asks, drawing a new "Y" next to a large bottle. "Now it looks like a glass of champagne," says the visitor.
George W. Zopf Jr. in the early 1960s said the phenomenon of context created a major problem for traditional cognitive modeling computer programs, as they would have to consider a virtually infinite number of possibilities before acting. The human brain knows what to do based on the situation, but a computer often doesn't. It may know how to play chess, but ask it to write a poem, and its programmer will have to write a new set of instructions to get it to execute the new task. "The brain solves this problem by dynamically reconfiguring its knowledge depending on context," Eliashberg says. "Biological neurons have the right resources to accomplish this task. It would take an extremely powerful conventional computer to simulate these resources."
Eliashberg's class promotes a scientific-engineering approach to the problem of information processing in the brain. "Mother Nature is smarter than human systems engineers and mathematicians, so it is important to deal with biologically falsifiable mathematical models rather than just biologically inspired models," he says. "In science, no has more power than yes. Accordingly, to improve a model, it is more important to look for the facts that contradict the model than for the facts that confirm the model."
Eliashberg argues that progress in neuroscience and technology has brought us to the point that it is possible to make a major breakthrough in understanding, reverse engineering and simulating the work of the human brain as an integrated system. "What is needed is a new generation of young researchers combining broad systems engineering, mathematical and biological backgrounds," he says.