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We interviewed Nancy Kopell, the winner of SIAM’s 2013 Jürgen Moser Lecture prize for her lifetime contributions in applied dynamical systems, about mathematics, her career, current frontiers in mathematical neuroscience, and more. Nancy is a professor at Boston University in the Department of Mathematics & Statistics, is Director of their Cognitive Rhythms Collaborative, and is Co-Director of the Center for Computational Neuroscience & Neural Technology. Nancy has won numerous awards and made pioneering contributions to dynamical systems, mathematical neuroscience, and mathematical biology more generally. Take a look at her website and her Society for Neuroscience biography for more details.

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*DSWeb*: As with many people in applied mathematics, your Ph.D. dissertation was in theoretical mathematics. Please describe this transition in your research. Was it gradual? What were the most challenging parts of this change? Why did you decide on mathematical neuroscience in particular?

*Nancy:* After I received my Ph.D, I entered both a prestigious position at MIT and a deep depression. I’ve written about this in a recent autobiographical chapter. I was able in those years to start to make that transition, basically by doing a lot of reading and thinking about what might be good questions. After finding an area (involving pattern formation in chemistry, something I started knowing nothing about), I had the good luck to attract Lou Howard of MIT as a collaborator (see the chapter for more details). By this time I was at Northeastern University. The next few years contained my real postdoc training, working with Lou on a series of papers that taught me new methodology while using my dynamical systems background.

The challenges were enormous: I didn’t have applied math skills, and I didn’t have much background in chemistry. Furthermore, the area I wanted to explore, self-organization in chemical systems, didn’t exist as a field; I contacted Lou Howard because he was an expert in fluid mechanics, a field that did study self-organization, even though the chemical patterns did not involve fluid motion, only diffusion. But I did have a good nose for a good scientific question, and a sense that the dynamical systems point of view would be useful. And my nose is still in good shape!

The biggest transition in my career was from pure mathematics to a combination of pure and applied math, not to speak of computational methods. The kinds of questions one asks can be very different, and the standards of success even more so. Much of applied math is very pure, in the sense that strict assumptions are made so that theorems can be proved. Though I still continue to prove theorems (or rather, my students do), many of the questions I want to address are too messy to yield to theorems, and what I’m looking for are insights and predictions, and the creation of a framework in which to think about questions that start out not well-posed.

It took me a long time until I started working on neuroscience, and then I was chosen, rather than choosing: I had gotten a reputation for work with oscillators, and a colleague in neuroscience asked me to write something about oscillators in neuroscience. As usual, I didn’t know the first thing, so I read a lot to try to understand what were the scientific issues. Along the way, I (with Bard Ermentrout) managed to solve a problem about some oscillators associated with motion (swimming in eel-like animals), which started a career in a new direction. More details are in the chapter mentioned above.

*DSWeb*: Also going back to your early work, what was it like being a student of Steve Smale? Did this shape your own student supervision style?

*Nancy:* I was part of Steve’s first cohort of students. Steve was not my first thesis advisor, but I was floundering after having received honors in my Qualifying Exam. I was also one of the very few women in a class of hundreds at UC Berkeley, so I did not fade into the background. I became Steve’s student almost by accident: when I saw him in the hall, he would suggest questions, none of which I knew anything about. After a few such attempts, he suggested something that did not require any background (and was very hard for that reason), and I agreed to work on it; in the end, I disproved his conjecture. This was likely my first experience in starting from nothing to do a scientific problem, something I continued throughout my career. Indeed, that autobiographical chapter is called “Beginner Mind”. Steve was away for much of when I was working, but his other students (Mike Shub and Jacob Palis) took me in and basically taught me the field. We formed a close-knit group, and the experience of that group has definitely shaped how I work with students and postdocs, recreating a scientific community.

*DSWeb*: Do you have advice for young mathematicians and young scientists? Do you have additional advice for young women scientists, or is this the same advice?

*Nancy:* One piece of advice: whatever you studied does not define you. You must dig deep in whatever you are doing, but it is good to also have peripheral vision, and look around to see what else is in the world. Some more advice: having good questions is at least as important as having technique, and I always tell my students that questions come first. (I ask over and over: “What is the question to which this is the answer?” )

I’m not sure that I have special advice for women. Both men and women often find the early years of a scientific career to be emotionally difficult, with lack of confidence in their own abilities and the knowledge that they will be expected to “produce”. Nowadays, this is even harder than when I was starting out, with more intense competition. I think it is good for them to find friends to talk out issues and become aware that what they are experiencing is not shameful or even unusual. It is especially important to talk about possible career directions, since the competition for professorial research slots has become so much more intense. Some of the people who receive Ph.D.s may be happier working in industry or teaching in a liberal arts school, even if they like research. I talk with students and postdocs about their lives and what makes them happy, and whether they think they can deal with the particular kinds of pressure that academia imposes. It is said that more women than men are sensitive to such sociological (not intellectual!) issues, but I don’t have a clear sense of whether this is true any more.

*DSWeb*: In your Moser Lecture, you made a point of bringing up the importance of mechanisms (and considering dynamics) and not just relying on "omics". We saw that you have subsequently written an article about this. In the few years since your talk, what progress do you see in this direction? What do you think still needs to be done better? What are the exciting opportunities for researchers here?

*Nancy:* Oh my! This is a big question! Let's start by saying that progress is as much a question about the sociology of science as about science itself. The progress has to involve physiologists as well as mathematicians, and I see hope in the way my work is received by that community and the growing number of interested collaborators. The science of dynamics of the brain (“dynomics”) is still at a very early stage; it is much more complex than “genomics”, large parts of which can now be automated. I’m now taking on large issues, such as rule-based decisions, the nature of predictions, active sensing, and other questions about normal cognition, as well as continuing with anesthesia, Parkinson’s disease, and other pathological states. For the questions that interest me, the modeling has many challenges, including working with large and diverse networks of neurons representing interactions of different regions of brain. This has led a number of people in my group to work on issues of efficient coding and computation of large and complex networks, and to develop a new platform (“DynaSim”) for creating large simulations and sharing parts of them. On an abstract level, there is big set of open questions concerning interaction of dynamical systems and statistics. For example, almost all neural simulations involve deterministic systems with some randomness thrown in. But it is not at all understood how the nature of the randomness may affect the qualitative properties of the outcomes.

For questions of dynamical systems, the biggest overall question for me is to understand how the dynamical properties of a target network affects its response to a temporally structured input. My Moser lecture was mostly about work at the beginning of this program. [See the next question for more.] More generally, I see the overlap of dynamical systems and neuroscience as wide open scientifically: almost everywhere one looks there are interesting questions. A caveat is that there are barriers to entry: one needs to learn a lot of neuroscience and some computer science, and the relevant parts are learned best by experience rather than classes.

*DSWeb*: What about your own recent work in this direction? What are some of your results since you gave your Moser Lecture? What new directions have you been thinking about?

*Nancy:* In the last few years, the mathematical thrust of my work has been about target networks with multiple time scales receiving temporally structured inputs with several time scales. This happens a lot in the brain since the intrinsic currents in cells provide a wide range of time scales, and an individual region often gets rhythmic inputs at multiple frequencies. To my surprise, longer time scales in the target can aid the ability of the target to phase lock to faster time scales. In related work, I’ve also been investigating interactions among different layers of a network, which is mathematically similar to interaction of different regions. One major theme of that work is “resonance” of the target, which turns out to be a more complex idea than resonance in standard linear systems. With my group, I’ve been exploring various notions of resonance, and how these resonances help gate inputs having temporal structure to networks in which excitatory cells have strong feedback inhibition from inhibitory cells in the same network.

*DSWeb*: Outside of your research area, what do you think are the most exciting areas of current (and future) work in dynamical systems and applied mathematics?

*Nancy:* There is so much to follow in neuroscience that even my peripheral vision does not allow me to answer this with confidence. Even within neuroscience, there are very different kinds of models (less biophysical) that are addressed to trying to understand major cognitive issues. These can involve dynamical systems as well.

*DSWeb*: Do you have any further thoughts on the role of computing versus pen-and-paper calculations versus writing in your work? How has this changed over the years? How, if at all, do you think should mathematical sciences curricula change based on these experiences?

*Nancy:* I never used a pen — too many mistakes! But pencil, paper, and eraser (and what I once described under “Methods” in an early neuroscience poster, the method of thinking) were the main tools of my work up until just before the current century**. **I was definitely part of the “pure” branch of applied math: the science inspired me to formulate fairly precise questions that could be answered by theorems. This methodology is still a substantial part of mathematical neuroscience. But one thing that has changed is what is now considered convincing: In the early days of computers, one needed proofs to back up simulations; now simulations are required to back up proofs! This makes sense since the proofs work under very restrictive conditions, and simulations can be used to show that the phenomena in question still exist even if the hypotheses are relaxed.

In current applied math, including neuroscience, numerical computation has become central. When I first started including simulations in the work of my group, I took it for granted that students could easily pick up those tools. How naïve! Now I look for significant background in scientific computing in any student I take on, and think this should be a part of the training of any applied mathematician. Data have become more complex and data sets more overwhelming in size, and learning how to approach such data sets is now far more critical than in my earlier career. Students need to know sophisticated analysis methods, which can be specific to the subject area. For neuroscience, texts on such subjects are only now appearing. As in experimental labs, the tools change as much as the questions, and the former can constrain the latter; how fast a computation can be made and how efficiently alterations can be made in code partly determine what questions can be addressed.

*DSWeb*: Over the course of your career, how has the meaning (and expectations?) of "applied mathematics" evolved? What changes have occurred both in terms of the types of questions that are asked and the types of techniques that are used?

*Nancy:* I think applied math became more ambitious after computers became available and powerful. The questions that could be asked became more closely allied with the central themes of the relevant areas of science. BC (before computers), much more of applied math was addressed to mathematical questions inspired by the science. Communities of applied mathematicians were more organized by their tools (bifurcation theory, geometric singular perturbation, asymptotic analysis, data analysis, …) and less by their questions. For example, the math bio crew had an ongoing yearly conference that would include ecology, genetics, immunology, physiology, and any kinds of mathematics that might be relevant to any of the above — all in the same conference. This became less attractive as applied-math people started to go instead to conferences focused on their scientific interests. Now much of computational science is integrated into the scientific fields that they serve, rather than being a different, if related, field. This is leading to some growing pains: it is much less easy to evaluate computational work than theorems within math departments, and less easy for the scientific fields to judge their contributions versus those of more traditional experiments. The idea that math (even applied math) is highly allied with “certainty” is strained, and it is more necessary to articulate what are the goals of any given piece of work and whether they have (more or less) been accomplished. The traditional experimental fields are more used to this uncertainty.

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