From the galactic scale of the cosmos in the night sky to the orbits and rotations of the planets in our solar system, from the machines that influence our lives daily to the cells in our brain that enable us to perceive the world: it’s mathematics all the way down.

In the beginning of his education, Tobias Timofeyev — a doctoral student in UVM’s College of Engineering and Mathematical Sciences — was interested in pure mathematics at Bard College, a private liberal arts college in Annondale-on-Hudson, New York, just under 200 miles away from Burlington.

“I played cello, I also did some musical ensembles, but my main focus was just pure mathematics,” Timofeyev said. But something changed. “I decided I wanted to engage with the world a little bit more, which led me to my current program, applied math, at UVM.”

The vast field of mathematics can be broken down into two distinct but overlapping areas: pure and applied mathematics. Simply, pure math is math done for its own sake, while applied math is the “practical” utilization of math in the real world. But that distinction isn’t always so simple. Modern encryption methods, a bedrock of safety on the internet, is based on number theory, the branch of  pure mathematics almost entirely devoted to the study of integers and arithmetic function.

At UVM, Timofeyev — who, along with several other graduate students, was recently awarded a $147,000 (over three years) National Science Foundation (NSF) Graduate Research Fellowship — is taking applied math to the cellular level, using mathematical biology to study and model the neuronal connections in the human brain.

For some, the concept of “mathematical biology” may seem contradictory. But, as Timofeyev explained, it’s just an extension of applied mathematics.

“There’s a beauty in mathematics in the applied spaces,” Timofeyev said. “You can look at a system, a small piece of the body, and extract key components from that system, abstracting into a mathematical model to understand the dynamics that are going on.”

The branch of mathematical biology applies to more than just the human body. Honeybees build hexagonal beehives, utilizing as little material as possible. Ferns with distinct growth patterns can be described geometrically with fractals, which can then be simulated with computer modeling to predict growth patterns. Timofeyev’s research, however, is largely concerned with the brain.

“There is this model of neuronal dynamics with two populations of neurons, one excitatory and one inhibitory,” Timofeyev said. “They have this interaction that leads to an oscillatory voltage behavior.”

While someone reads this article, millions of neurons send messages throughout their brain to recollect definitions of words and pattern recognition in sentences to form complete thoughts. These neurons take sensory information from this article’s images and translate them nearly instantaneously into discernable visual data in their brains.

The majority of neurons in the brain are excitatory, activating the electrical current that run’s through everyone’s brains. Inhibitory neurons restrict that flow and make other neurons less likely to send signals of their own.

“I’m looking at models of regions in the brain and understanding how the fibers connecting them should impact of the voltage of other regions,” Timofeyev said. “You can build a whole network of these different regions and ask yourself, ‘if this part is going up and down like this and that part is going up and down like that, how do those regions interact with each other?’”

A common approach to modelling voltage and neuronal activity in the brain might lead to  computer visualization, utilizing super computers — like the Vermont Advanced Computing Core — to build and demonstrate different scenarios for a simulated brain. Timofeyev, however, is extracting models from complex mathematical equations.

“There are these differential equations — equations that incorporate variables that are changing over time — that can describe brain activity,” Timofeyev said. “I think of a computer outputting a three dimensional model of the brain with wires running through everything, showing the oscillations of the voltages between those two types of neurons, but it’s just equations. The computer needs to have those equations to do what you’re asking it to do.”

While the implications of modelling the brain and formulating complex equations to better understand complex neuronal voltages may be obvious — improved brain modelling can help better understand how different medical treatments can affect the brain — Timofeyev said these complex differential equations could be put to use in another applied setting: power grids.

“The same equations that I’m using to understand the brain, a modified version of those can used to understand power grids. The power we get from a wall socket is an alternating current, so you can already understand it’s oscillating that originates from a power station,” Timofeyev said. “But how does that distribution work? How much power can you push through these cables? How resistant is this system to damage? You can study the network structure and wonder if one connection breaks, will that be too much for the rest to handle.”

Like a population of  excitatory  and inhibitory neurons, a hydroelectric turbine in a dam may interact with other generators or loads in the network in a myriad of ways. This interaction can be described and modelled with complex differential equations like those on which Timofeyev is working.

Before Timofeyev comes back to UVM in the fall, he’s spending his summer as an intern for the National Renewable Energy Lab under the NSF’s mathematical sciences program, researching power flow equations.

“These equations describe the states of variables such as voltage magnitude and phase across a power grid,” Timofeyev said. “The application is the power grids we have in cities and countries. Better understanding these equations and their characteristics, allows fewer resources to be spent modeling, and maybe also better optimization of how we use the grid.”

This is the second of four stories on UVM graduate students awarded prestigious National Science Foundation Graduate Research Fellowships.