A while ago, I wrote a piece for a series of books on History of Neuroscience in Autobiography [1]. My chapter was called “Beginner Mind”. I choose that title because my research interests have changed so many times that I found myself a beginner over and over, while being able to make use of an ever-growing collection of things at which I was once a beginner.

As an undergrad at Cornell, I majored in math. This was expected since my mother and sister did so before me, and my father was an accountant; I had a math gene. When I pondered grad school in math, I thought I might train to be an algebraist, maybe a topologist, but applied math was barely on my horizon, and being in an entirely different field was in another universe. As it happened, Life had different plans for me.

My graduate training was at UC Berkeley. It was 1959-1963, the bare start of the 60’s and the time of Free Speech Movement. Social justice was a rallying cry, but feminism had not yet emerged, even at Berkeley. As one of the few female graduate students in the very large mathematics department, I felt exposed and “different”. This exposure was both good and bad: it was often highly uncomfortable, but I was also in a position in which anything I did was noticed and could be applauded. My first attempt at working with a thesis advisor did not go well, since he himself was about to change fields and was not very interested in the area (topology) in which I was supposed to be working with him. My second started when Steve Smale, who knew (of) me, started suggesting thesis questions in dynamical systems when he ran into me in the Dept. I had no training in any of them, and finally agreed to try one of them because it was elementary (nothing to learn and no known tools to use on them). He then disappeared on a sabbatical. But he had put together a small group of other Ph.D. students who taught me the field; we all quizzed each other almost every day on what we had learned since the day before. In general, I loved my time at Berkeley, and it gave me useful experience in holding my ground as I wandered into scientific areas in which I was a stranger.

There were many adventures at Berkeley, and I managed to finish a thesis good enough to get me a job as the first woman in a prestigious named instructor position at MIT. There I taught elementary differential equations, which was the first time I learned that subject: As treated in Berkely at that time, dynamical systems was the study of the abstract qualitative properties of ODE’s; one rarely if ever solved one. I had a difficult time at MIT that was only partially the result of the overwhelmingly male (and macho) environment, something that was shared in that era with most other departments of mathematics; I had been struggling for years with personal issues and sensed that the receipt of the Ph.D. would signal the point at which I needed to deal with them. Indeed, I had a choice of excellent possibilities for my first job and chose MIT because I intuited that, at MIT, no one would notice. (I believe I was correct about that.) So, during the two years of that position, I was likely clinically depressed and got very little math done.

That could have been the end of my career, but the work I had done as a graduate student was interesting enough that some schools were willing to overlook my complete lack of productivity since then. One of these was Northeastern University, which is now a well-known research institution, but was then a commuter school with a low ranking. This turned out to be a blessing for me: there was no pressure to do research, and so I was able to look around and figure out what I wanted to do next. Northeastern gave me the space to try to reinvent myself, and I gradually acquired the energy to do so. With a lot of twists and turns (see [1]) that direction turned out to be about pattern formation, focused on a chemical system (Belousov-Zhabotinsky reaction) that oscillated in color when kept stirred up, but formed complex patterns of interacting “target patterns” and spirals when allowed to diffuse without undergoing fluid motion. One of the great strokes of luck in my career was that, when I went back to MIT to ask some technical questions of an expert (Lou Howard), it turned out that he was really interested in the questions I was posing and ready to collaborate on them. We worked together intensely for the next 7 or 8 years, until he left the Boston area. (This shows that “progress” is not always good: in a similar position now, I would have used Google Scholar to try to answer my questions, and not gone looking for a human expert.) The papers became well known and I became known as the “oscillator lady”. During that time, I learned some of the tools of applied math I had never studied, while using my dynamical systems tools to think about patterns.

The next twist in my career came soon after: I was approached by Bard Ermentrout about an issue having to do with a chain of oscillators; he had written a paper on this in an applied math style that the reviewers did not believe, and he wanted some backup. Together, we were able to translate his computations into theorems using dynamical systems methods. At the same time, I had been asked to write a chapter on oscillators for a book on “Central Pattern Generators”, which are networks of neurons that govern periodic motions such as walking, chewing, swimming etc. It took a combination of courage and sheer audacity for me to accept that assignment, since I had never taken a biology course and was hopelessly lost when I first started to learn about the biology of CPGs. But Avis Cohen, who was involved with the book, took me under her wing and sent me cartons of reprints to read. (That was still before Google Scholar). In working with Bard and trying to understand CPGs, I became aware that one prime example of CPGs, swimming in eels, was essentially a chain of oscillators! So, I wrote the CPG chapter (which Avid made me rewrite at least 6 times until a biologist could understand it – before the era of word processors) and a paper with Bard on lamprey locomotion. The latter led to a collaboration with lamprey biologists Karen Sigvardt and Thelma Williams that became well known in the biomath world and included a yearly talk at the Woods Hole course on Methods of Computational Neuroscience as a kind of paradigm for math/bio collaboration. The collaboration with Bard included many papers on interactions of oscillators and was an intense partnership until he told me one day that “he didn’t want to be married anymore; he wanted to be promiscuous”. Of course, mathematical biologists now are almost always “promiscuous” in the sense of working on many projects at once with many collaborators, and that was the right decision for both of us. We stayed friends and collaborators.

The work I had done with Bard was essentially mathematics: one could prove theorems by making judicious assumptions. The next big jump occurred when I started hanging out with Eve Marder, a biologist who worked on lobsters and crabs and described the remarkable complexity of the biological rhythms that these fairly simple creatures were able to produce. I started to try to mathematize the scientific issues but ran into roadblocks: almost every neuron in the small networks she studied was completely different from every other neuron. To understand anything about the outputs of the network, one had to come to terms with the complexity of the underlying biology. Eve was my mentor in the transition from biology inspired math to biology with mathematical underpinnings. Since then, my interests have become more and more biological, with computer simulations taking over from pencil, paper, and theorems.

The work that I have been doing in the last decades has grown more complex and ambitious. At first, with collaborators Roger Traub and Miles Whittington, I tried to understand the how the physiology of underlying various brain rhythms produces the brain dynamics. That gave rise to a long series of papers that is still open ended, as researchers find out more about the layers of biological complexity that support these dynamics. The next step, again open ended, is about the interactions of brain rhythms; such interactions go way beyond entrainment and synchronization; I believe that there are many unproved theorems hidden in that body of data and simulations, but I have been mostly content to be descriptive of the phenomena. What I have been doing most recently is to try to understand, with experimental collaborators, the origin and function of brain rhythms in both normal cognition and pathologies of cognition. This has led me down paths to thinking about, e.g., sensory encoding, attention, working memory, anesthesia, Parkinson’s disease, neural development and its pathologies (e.g., autism), and Alzheimer’s disease. In all these cases, brain rhythms and their interactions are central to brain function, and pathologies of the rhythms have direct and important effects on function. I’m now about 80 years old (and grateful for this time), and still starting on new projects.

I like to tell young researchers that “you are not just what you have previously studied”: the world is wide open to new – and accessible – scientific questions that an applied math background helps one to research. All it takes is curiosity and an occasional willingness to go deep into areas in which one is a beginner.

Reference:
[1] https://www.sfn.org/-/media/SfN/Documents/TheHistoryofNeuroscience/Volume-9/HON_V9Kopell.pdf.