How do connections between neurons store memories and shape the dynamics of neural activity in the brain? How do firing patterns of neurons represent our sensory experiences? The advent of technologies that facilitate simultaneous recordings of large populations of neurons present new opportunities to answer these classical questions of neuroscience. There are mathematical models that are frequently used in network simulations and data analyses that can be employed, but whose mathematical properties are still poorly understood. To guide these efforts, a better understanding of theoretical models of recurrent networks and population codes is essential. This research will focus on two such examples: threshold-linear networks and combinatorial neural codes. The goal is to produce major advances in the mathematical theory of these models, with an eye towards neuroscience applications. Part of the research will involve the analyses of neural activity in the cortex and hippocampus, in collaboration with experimentalists. Despite the focus on neuroscience, the mathematical results have the potential to be sufficiently general so as to be useful in a variety of broader contexts in the biological and social sciences.
A threshold-linear network is a common firing rate model for a recurrent network, with a threshold nonlinearity. These networks generically exhibit multiple stable fixed points, and multistability makes them attractive as models for memory storage and retrieval. Preliminary results have shown that the equilibria possess a rich combinatorial structure, and can be analyzed using ideas from classical distance geometry. The first project will build on this understanding in order to develop a more complete picture of the structure of fixed points and higher-dimensional attractors of these networks. A combinatorial neural code is a collection of binary patterns for a population of neurons. The second project will develop an algebraic classification of combinatorial codes, using the recently developed framework of the neural ring. The neural ring encodes information about a neural code in a manner that makes properties such as receptive field organization most transparent. The resulting methods will be tested and refined using electrophysiological recordings of place cells in the hippocampus. This research will also generate new and interesting problems at the interface of neuroscience with applied algebra, combinatorics, and geometry.