Professor Kormes was concerned. I was excelling in her first-year calculus course at Brooklyn College (1959), but as a pre-med major, I might end up in a career that did not involve mathematics. She thought I might find the area of mathematical biology interesting. She was right, but it would take almost a decade before that path was set.

In my second year of college, my distaste for memorization in comparative vertebrate anatomy prompted me to switch majors. I had been excited by my introductory experimental psychology course, so that was the natural direction. But the style of analysis in the courses in normal and abnormal psychology was beyond my comprehension.

I ended up majoring in chemistry with particular interest in physical chemistry. For graduate studies, I went to the University of Chicago (1963), where Stuart Rice, a former Brooklyn College student, was a young superstar professor. Although I started out doing experiments under Rice’s supervision, destroying several precious cuvettes when lowering them into liquid nitrogen and a fascination with statistical mechanics led to a shift to theoretical studies. My PhD was concerned with theory of atomic motions in liquid argon.

Towards the end of my graduate studies, Jack Cowan was recruited to the University of Chicago to rebuild the Committee on Mathematical Biology, that had been founded by Nicholas Rashevsky. Cowan organized some informal seminars that I attended and found intriguing. But for a postdoc, I decided I would like to go abroad and work with someone who had expertise in chemical physics but who was now working in biology. I found the ideal mentor – Hugh Christopher Longuet-Higgins. He had done superb work in quantum chemistry and in 1967 had set up the Department of Machine Intelligence and Perception in Edinburgh with Donald Michie, an expert in computation, and Richard Gregory, a perceptual scientist. The National Institutes of Health offered a generous fellowship that would allow me to apply methods from statistical physics to the brain. (Yes. I believe my application was that vague!)

Despite Longuet-Higgins injunction (“Reading rots the brain”), I started out by reading some papers on geometric illusions. But actual research was spurred by an unlikely coincidence. On a trip to Syracuse, NY, Longuet-Higgins saw a demonstration from physicist Erich Harth. Start with a blank page and then make a Xerox copy of it. Now take the output and take a copy of it. If you continue iterating, then after about 10 iterations there is quite a stable pattern of random dots. I was fascinated by the spatial distribution of the dots, and harkening back to my study of liquids, I decided to compute the spatial autocorrelation function of the dots. To do this I made a transparency of the dots that I would project on a bull’s eye pattern of circles – placing one dot at the center and counting the number of dots in each ring. By chance, I noticed that when I superposed the transparency on the original in a slightly rotated position, a powerful circular pattern emerged. Longuet-Higgins helped me in preparing an article describing the phenomenon – now known as “Glass patterns”. During the year in Edinburgh, I stayed in touch with Cowan, who offered me a postdoc at Chicago when I was done at Edinburgh.

Figure 2. Glass patterns. A. Dot pattern from starting with a blank page and iterating - always taking a copy of the copy - made in Scotland in early 1969. B. Image A superimposed on itself with a slight rotation.

Before returning to Chicago, I took a touristic trip of several months around Europe with my wife (1969). But I did interrupt the travel for a month long stay in Brussels to visit with Ilya Prigogine’s group to learn about dissipative structures. Although I had intended to work on problems of visual perception in Chicago, I was intrigued by Stuart Kauffman’s recent discoveries about properties of randomly constructed model Boolean genetic networks. I was bothered by the discrete time and discrete states in the Boolean model and developed a way to embed the logic of the Boolean networks in continuous differential equations.

As my postdoctoral fellowship in Chicago was nearing its completion (1972), I started looking for jobs. Although most former students of Stuart Rice were in high demand and were successful in finding tenure track positions in top universities, my fascination with biological questions placed me in a different category. Fortunately, Elliot Montroll, a fine statistical physicist had offered me a postdoctoral position at his recently started Institute for Fundamental Studies in the Department of Physics and Astronomy at the University of Rochester. Montroll’s notion was that people with expertise in statistical mechanics would be able to tackle problems in any area. At Rochester, I carried out studies in vision, genetic networks, pattern formation. I did some teaching and research supervision and my position evolved from Postdoc to Assistant Professor (part-time).

Although I was offered a more permanent position at Rochester, a more intriguing possibility emerged. I had met Michael Mackey at Gordon Conferences in Theoretical Biology. Mackey had a training in Biophysics and was a young professor in the Department of Physiology, McGill University in Montreal. McGill had an interest in encouraging quantitative studies in Physiology and was looking for someone to join Mackey. Even though I had never taken a course in Physiology, I was offered the position (1975). Not only would I get a chance to be next door to Mackey, but I would also be next door to colleagues who were carrying out experiments.

I moved to Montreal. In my first project at McGill, Mackey and I collaborated in formulating delay-differential equations for physiological control systems (1977). One of the equations (now known as the Mackey-Glass equation) displayed chaotic dynamics. Our early paper also proposed the concept that dynamical diseases could be associated with abnormal dynamics arising from bifurcations in physiological control systems.

In my research at McGill, one focus has been experiments and theory that involve the effects of periodic stimulation on biological systems. As the stimulation frequency and amplitude vary, one obtains a variety of rhythms including stable entrainment patterns and occasionally chaotic dynamics. The stability of the rhythms can often be determined theoretically from analysis of fixed points on a low-dimensional return map. As first made clear by V. I. Arnold, there are often universal features concerning the organization of locking zones as stimulation parameters change. We have studied several different systems: periodic inputs to the respiratory rhythm using a mechanical ventilator; periodic electrical stimulation of excitable and oscillatory cardiac tissue; stroboscopic illumination of disks with radial patterns to generate the wagon wheel effect. The precision and beauty of the experimental data mirror the elegance of the theory. I have been delighted that colleagues and students, initially Teresa Trippenbach, Canio Polosa, Alvin Shrier, Michael Guevara and subsequently many others, have been enthusiastic collaborators, both in the design and the interpretation and modeling of experiments. In approaching these diverse systems, we look carefully at the experimental data (often from a small number of preparations or clinical records), develop simplified models, and examine situations in which the simplified models break down to set the course for future research.

Although I have been mainly concerned with basic science questions, I have a continued interest in assessing potential clinical relevance. Prodded by application for a research grant from the Mathematics of Information Technology and Complex Systems National Centre of Excellence in Canada, I contacted colleagues Medtronic, a medical device manufacturer, who pointed to an important practical problem: developing an algorithm to detect atrial fibrillation. This common arrhythmia, characterized by an irregular heartbeat is not fatal, but does have an increased risk of stroke. With Katsumi Tateno, a postdoctoral fellow from Japan, we succeeded in developing an algorithm that was superior to alternatives at the time (2001). The patents that were issued were licensed to Medtronic and devices from Medtronic and now many others are being used for atrial fibrillation detection. Currently, we are excited by our attempts to better understand and model the complex dynamics of premature ventricular complexes (PVCs) in people (see the recent article describing our work by Philip Ball (https://physics.aps.org/articles/v16/2?utm_campaign=weekly&utm_medium=email&utm_source=emailalert). Although cardiologists have identified several potential mechanisms for PVCs, understanding their amazing dynamics is a continuing challenge.

In setting my professional path, I followed my curiosity – from pre-med, to psychology, to chemistry, and then back to medically related questions, always with an interest in complex dynamics and structures. My career trajectory has been shaped more by chance encounters with scientists and discoveries than by a strategic plan for development. I am not sure if this strategy would work now, but it sure has been fun for me.