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Mathematics Colloquium: Soft Clustering Decoding of Neural Codes
4:10 PM; Neill Hall 5W
Abstract: Methods based on Rate Distortion theory have been successfully used to cluster stimuli and neural responses in order to study neural codes at a level of detail supported by the amount of available data. They approximate the joint stimulus-response distribution by soft-clustering paired stimulus-response observations into smaller reproductions of the stimulus and response spaces. An optimal soft clustering is found by maximizing an information-theoretic cost function subject to both equality and inequality constraints, in hundreds to thousands of dimensions. The method of annealing has been used to solve the corresponding high dimensional non-linear optimization problems. The annealing solutions undergo a series of bifurcations in order to reach the optimum. We study that system using equivariant bifurcation theory. The optimal models found by distortion methods have symmetries: any soft clustering can lead to another equivalent model simply by permuting the labels of the classes. These symmetries are described by SN, the algebraic group of all permutations on N symbols. The symmetry of the bifurcating solutions is dictated by the subgroup structure of SN. In this presentation we describe these symmetry breaking bifurcations in detail, and indicate some of the consequences stemming from the form of the bifurcations.