**Oscillators and Networks of
Them: Which Differences
Make a Difference**

### Baltimore, Maryland 1992

**NANCY KOPELL** received her AB from Cornell University in 1963 and her PhD from the University of California, Berkeley in 1967. She held a Moore Instructorship at the Massachusetts Institute of Technology for two years, and then went to Northeastern University in 1969. Since 1986, she has been a professor at Boston University. She held visiting positions at the Centre National de la Recherche Scientifique in France (1970), MIT (1975, 1976-1977), and the California Institute of Technology (1976). She received Guggenheim and Sloan Fellowships, and she was an Invited Speaker at the International Congress of Mathematicians in 1983. In 1990, she was a Plenary Speaker at two meetings of the Society for Industrial and Applied Mathematics. That same year, she received one of the celebrated "genius awards," a MacArthur Foundation Fellowship. In the last few years, she has given the Volmer Fries Memorial Lecture at Rensselaer Polytechnic Institute, the Mark Kac Memorial Lectures at Los Alamos National Laboratories, and the 1993 University Lecture at Boston University.

Kopell uses and develops methods of dynamical systems to attack problems of applied mathematics. She is especially interested in questions involving self-organization in physical and biological systems. With L. N. Howard, she wrote a series of papers on pattern formation in oscillating chemical systems. More recently, in work with G. B. Ermentrout, she has developed mathematics appropriate to analyzing neural networks that govern rhythmic motor activities in animals, such as walking, swimming, and breathing. Such systems are, roughly speaking, large collections of units, each of which is an oscillator or a close mathematical relative of an oscillator. The aim of the mathematics is to help sort out which properties of the units and their interactions have implications for the emergent properties of the networks. The techniques Kopell uses include extensions of invariant manifold theory, averaging theory, and geometric methods for singularly perturbed equations. She has brought together an active group of physiologists and mathematicians who collaborate on these problems. In addition, Kopell has been working on geometric techniques in dynamical systems.

"The overall problem that we're trying to approach is to get a better understanding of the relationship between structure and function," says Kopell. "That means we're trying to understand why these networks are constructed the way they are in order to serve the functi6ns that they serve." Kopell's work has shown that mathematics can provide powerful tools for attacking such problems. In particular, her work with two biologists, Karen Sigvardt and Thelma Williams, led to striking results in understanding the neural processes of swimming in lampreys, which are primitive vertebrates. "It was a leap of faith to intuit that mathematics actually could end up giving biologists insight into these networks that they work so closely with," Kopell remarks. It has been "tremendously exciting, to get talented biologists to take it seriously."