**A Model of Cardiac Fiber: Problems in Singularly Perturbed Systems**

### Anaheim, California 1985

**JANE CRONIN SCANLON** received her doctorate from the University of Michigan in 1949 under the direction of Erich H. Rothe. After two postdoctoral fellowships, from the Office of Naval Research and the University of Michigan, she worked as a mathematician in the Air Force and for the American Optical Company, and as an instructor at Wheaton College and Stonehill College. In 1957, she moved to the Polytechnic Institute of Brooklyn, and in 1965 took a position as professor at Rutgers University. She became professor emeritus in 1991. She was awarded a Visiting Professorship for Women from the National Science Foundation to spend the 1984-1985 year at the Courant Institute of Mathematical Sciences. At the Joint Mathematics Meetings in Boulder in August 1989, she presented the Pi Mu Epsilon J. Sutherland Frame Lecture,

Scanlon's research has focused on mathematical biology, singular perturbation theory, and nonlinear analysis. She has published more than fifty papers, two research monographs (*Fixed Points and Topological Degree in Nonlinear Analysis* and *Mathematical Aspects of Hodgkin-Huxley Neural Theory*), as well as a textbook (*Differential Equations: Introduction and Qualitative Theory*).

In her Noether Lecture, Scanlon described a system of differential equations that are used to model cardiac Purkinje fibers, which conduct electrical impulses in the heart. These equations, which may be viewed as a singularly perturbed system, were derived by Denis Noble in 1962 by modifying the Hodgkin-Huxley equations in accordance with laboratory data about the Purkinje fiber. Her lecture discussed steps toward two mathematical objectives needed to describe the basic activities of the Purkinje fiber: to establish the existence and stability of a periodic solution, and to develop a theory of entrainment of frequency for the system. Since the time of the lecture, Scanlon has dealt with the first problem by using a geometric approach. "This required considerable time because I had to learn that the methods I first planned to use were ineffective," she notes. "Accepting this disagreeable conclusion required some effort. However, the geometric approach does work, and it seems that it can also be used to study entrainment of frequency in singularly perturbed systems."

Scanlon began her college studies in physics, but she found that she understood the mathematics better than the physics problems to which mathematics was applied. She became more interested in abstract mathematics-- "at that time, the more abstract the better!" she says--but eventually came full circle back to applications when she began studying mathematical problems arising in biological systems. Scanlon says that she was fortunate to be a doctoral student at Michigan, where T. H. Hildebrandt was chair of the mathematics department. "He really cared about mathematical research," she remarks. "He didn't care whether the students were men or women as long as they were interested in mathematics."

A mother of four, Scanlon has maintained her lifelong interest in mathematics because of the satisfaction it brings. "Part of the attraction and fascination of mathematics is the idea of making, you make something that wasn't there before. You start out sometimes with a very vague idea, and you,turn it into something very specific and concrete--if you're lucky!"