That was one of the things that drove me to physics, because physicists are very proud of being able to derive things from first principles, and everything logically fits together. A lot of people have a wrong impression of math because of a bad high school class or a bad experience with math early on. That happened to me with biology.

As a grad student, I finally sat in on a neuroscience course, and I started learning more and more. And then I made the decision in my fourth year of grad school that when I finished my Ph.D., I was going to switch to neuroscience. I wasn’t planning to abandon my math and physics background, but I decided to apply it to neuroscience rather than string theory.

Toward the end of my fourth year, there was a professor, Ken Harris, then at Rutgers University, who was spamming math departments across the country. He sent an email to everyone in the Duke University math department about how he was trying to recruit people with math and physics backgrounds to his neuroscience lab. I went to meet Harris and he offered me a job as a postdoc, essentially on the spot. After getting my Ph.D., I spent three years in his lab, doing mostly data analysis and traditional computational neuroscience — learning a completely new set of tools and a lot of neuroscience.

Do you have any regrets?

No, not really. Early on when I made this transition, the only thing that I had regrets about was that I didn’t know how much math I would get to use. I plunged straight in. I was like, “I’m going to learn neuroscience, I’m not going to worry about applying specific mathematical tools, I’m just going to be open-minded and immerse myself.” It was almost like an exchange program, when you go to another country and live with a family and immerse yourself in the new culture and language. That’s what I did with neuroscience. I just went full in and didn’t try to force my own background onto it.

There were times when I thought, “OK, I’m never going to do real math again. I’m never going to prove another theorem, because all I’m doing is data analysis and computational work, and it’s interesting, but it’s not really math.” And so I did have that feeling at times, with some regret.

But then it was kind of amazing. After a few years, I started having my own ideas. I started asking questions that were meaningful to neuroscientists, and having ideas about how I could tackle some neuroscience problems that were quite mathematical — bringing in tools even from areas of math like topology and commutative algebra and combinatorics that were not traditionally being used. That was very exciting. I got to use much more sophisticated math but still address questions that were meaningful to the neuroscience.

In what direction is your research heading currently?

One research direction I’ve gone into, and the reason I was invited to the Janelia conference, is related to the larger project of trying to understand the relationship between a network’s connectivity structure and its dynamics — the dynamics being the patterns of neural activity we observe in neural recordings.

For instance, you may have some set of neurons that fire in a regular sequence. You could be recording, say, 10 neurons, and you could see over and over again the sequence of: 3, 5, 6, 7, 9 … 3, 5, 6, 7, 9 … and it repeats. The question would be: Why are you seeing this pattern of activity? What does it tell you about the underlying connections between the neurons, and what do the connections tell you about the activity pattern? How does the structure of the network affect the more ephemeral dynamics? How are these patterns of activity being used to encode and transmit information in the brain?

And you developed your “network songs” to help explain this interplay?

Oh, you found the network songs! That was kind of a gimmick; I used the songs in talks a few years ago and people loved them. The idea was to show the rhythmic activity that these networks generate. Sometimes it’s hard to see that just by looking at plots.

So I set the activity to music: You have individual neurons and their firing rates. Their activity levels go up and down, and as a collective population, they generate rhythms. I did something very simple: I assigned a piano key note to each neuron, and then I used the neuron’s firing rate to modulate the amplitude. When it’s high-firing, that note is playing loudly, and when it’s low-firing, it’s not playing at all. It’s another way of representing the very same thing that’s on the plot, just a solution to a differential equation. But you can hear it, and somehow the ear picks up recurring patterns better than the eye. It’s really just a way of turning the same information into sound so that the rhythmicity can be more apparent. It makes the point nicely that these networks are very rhythmic.

Ultimately, those dynamics are what we believe is leading to behavior, and leading to perceptual experiences, memory recall and so on. And the work that I’m most excited about right now is on a very nice model of a neural network, the combinatorial threshold-linear networks model. My collaborator Katherine Morrison and I have been playing around with the model and proving theorems as well — we can actually prove theorems that allow us to tie the structure of the graph to important features of the dynamics. That’s been very exciting.

What does it mean to be proving theorems about the brain, so to speak, in this context?

Of course, you still have to do experiments. Proving something about your abstract model is not proof that you are right about the actual science. But it gives you insights into things that can then be checked in a more traditional, scientific-method-type way.

In a sense, a mathematical model is a simplified model in the same way that a worm, like Caenorhabditis elegans, is a model organism: There is a lot of Nobel Prize work in neuroscience that was done on worms, very small creatures with very simple nervous systems, because these simplified models really do give insight into the human brain. In a similar way, we can prove theorems in simple mathematical models and gain real insight into how the model is working. That insight then gives us things to look for in the real system, the real brain.

When you can prove a theorem, it basically captures infinitely many examples that you could never individually check on your own — even with the most powerful computers. And often, the essence of what is needed for a proof to work gives insight into the key features that make the phenomenon tick. You could discover something in simulations by playing around with the parameters, and then make the guess, “I think this feature is what’s causing the phenomenon to happen.” But if you can really prove it, even if the proof requires a simpler setting, then you can nail down the key features that lead the phenomenon to work. Being able to do real math in relation to a model like this helps a lot in gaining insight into the mechanisms driving the phenomenon.

For example, there are certain periodic patterns of activity that we see in real neural circuits, and we always wonder, “OK, what’s causing these rhythms of neural activity to emerge in the network?” And in the context of a simple model one can actually prove, to some extent, which features are causing that pattern to emerge.

This gives you something to look for. If I have my connectomics data, if I have some structural information about the network, I can check: Do I see the same structural pattern in the network that is giving rise to this phenomenon in my model? So the mathematics gives a way of generating hypotheses.

The one bad thing about the traditional scientific method is that it doesn’t really tell you how to generate hypotheses in the first place. When you get to complex systems, there is this combinatorial explosion of possible hypotheses, and you cannot check them all. Having good models, beautiful models, is a really nice way of generating hypotheses in a controlled and rigorous fashion.

Overall, what do you find most fascinating about neuroscience?

The field is evolving very rapidly right now. New types of neurons are discovered all the time, and new coding properties of neurons, too. We’re just now figuring out how networks are structured in the brain.

I remember hearing about the 1960s as this great age for particle physics: A new particle was being discovered every week, and theorists were busily trying to figure out how to organize the particles, and what kind of mathematical structures should be involved, and what kinds of symmetries were present, and so on. There were lots of interesting connections to mathematics. That’s the sort of exciting thing I wanted to be a part of.

When I got into neuroscience, it was kind of like that. Instead of a new particle discovered every week, it was a new type of neuron, or new rules for how neurons are connected, or new experiments showing interesting and surprising results. It felt like — and still feels like — a very exciting playground for a theorist.