This project is focused on applications of computational mathematics to problems arising in neuroscience and systems biology. Beyond utilizing currently available methods in creative ways, the main aim is to develop new computational methods to meet the needs of specific biological applications, particularly in systems biology and neuroscience. In systems biology, the goal is to understand complicated biological processes by studying interactions in a network setting rather than in isolation. While in neuroscience, the goal is to better understand neural connections in the brain, and how the brain interacts with it's external environment. On the systems biology side, the work in the proposed project will help make sense of two types of networks: biochemical reaction networks, deterministic models for molecular interactions, and protein-protein interaction networks, relational data collected, confirmed, and refined over the past decade. On the neuroscience side, the focus will be on neuronal networks, schematics that record connections between neurons, and combinatorial neural codes, a form of discretized cell firing data.
The techniques employed will come from combinatorics and computational algebraic geometry, two subfields with applications in an array of fields including statistics, physics, engineering, and biology. The proposal has three main research components. First, techniques and theory from computational algebraic geometry, such as toric ideals and Groebner bases, will be used to understand and visualize neuroscience data. Second, new algorithms for sampling random graphs with fixed properties will be developed for testing statistical hypotheses about protein-protein interaction and neuronal networks. Third, new algorithms for computing elimination ideals will be developed with the goal of applying these new methods to model selection in biochemical reaction network theory.
