© Ariana Coveney.
My research focuses on developing mathematical models for strongly interacting, complex systems that cannot be analyzed with traditional reductionist techniques. In other words, systems where the whole is greater than the sum of its parts. I particularly focus on the “brain” of the small nematode C. elegans.
Data-driven modeling of biological systems
A key component of my PhD has focused on analysis of calcium imaging data of neurons acting in real time. This network of neurons can be treated as a controlled system using the Dynamic Mode Decomposition with Control (DMDc) framework. [publication coming soon]
Modeling control systems
Dynamic Mode Decomposition with Control (DMDc) is a popular technique for modeling the dynamics of a system with external inputs, aka control signals. I have developed an algorithm with increased accuracy and decreased sensitivity to user-defined parameters: [publication coming soon]
Interpreting models
“All models are wrong…
Interestingly, the way in which they are wrong can be incredibly informative. In particular, most modeling finds a best-fit to data as defined in an L2 sense; this is analogous to positing that the model captures all of the data except for Gaussian random noise. Mathematically, we can search for places in which our models are wrong in a non-Gaussian way, and, using the control signal framework of DMDc, make our models much better.
I am working on developing the theory and some applications of this unsupervised problem, [publication coming soon].
“… but some are useful.”
Machine learning methods are extremely popular at the moment and are increasingly being used with ambiguous datasets for which no “ground truth” is known. Fortunately, they seem to perform very well! Unfortunately, this does not mean that such models have actually captured something about the system that will generalize beyond the specific training data. This issue extends to mathematical modeling more broadly, and another area of my work focuses on which portions of mathematical models, in particular DMDc, are interpretable and how confident we can be. [publication coming soon]