Topology and coding in neural networks
Wednesday, April 16, 2014 - 1:30pm - 2:30pm
Natural and artificial neural networks are traditionally studied using classical methods from dynamical systems and linear algebra. However, there are many problems of interest for which topological tools can offer new perspectives. Here, we survey our recent work on the relationship between the structure of a network and that of the data it encodes: first, using classical tools from combinatorial topology, we show that a simple one-layer feed-forward network can encode any prescribed simplicial complex, but that a large class of non-convex codes cannot be encoded; second, using recent notions and computer tools for computing persistent homology, we discover signatures of geometric structure in the correlations of activity among neurons in the hippocampal network in rats. In the latter project, we frame persistent homology of clique complexes as a tool for the study of equivalence classes of real symmetric matrices under the action of the group of monotone increasing functions, building on work of M. Kahle describing Betti curves of such classes to interpret the results. Time permitting, we will also touch on work in progress regarding detecting convexity of codes. This is in various parts joint work with C. Curto, V. Itskov and W. Kronholm. No background in biology or networks is assumed!