Ran Levi: Topological analysis of neural networks

ABSTRACT: A standard way to schematically represent networks in general, and neural network in particular, is as a graph. Depending on context, graphs representing networks can be directed or undirected, and in some cases carry labels or weights on their vertices and edges. With any graph one can associate a variety of combinatorial and topological objects, as well as certain algebraic invariants. The problem in using such ideas in neuroscience has been that obtaining connectivity data from a biological brain is very hard and expensive with existing techniques. Hence only very small connectivity patterns are understood, and extracting meaningful topological structures is not likely. Artificial neural networks, created by probabilistic rules can give much larger graphs, but they only provide a partial solution, as with existing knowledge their connectivity and functionality is not capable of reliably reproducing biological neuronal networks.

In this talk I will survey an on-going collaborative project where we apply topological techniques and ideas to the study of the brain. The project was motivated by the creation of a biologically accurate digital reconstruction of a small part of the cortex of a young rat by the Blue Brain Project. This digital model provided a way of extracting very accurate structural and functional information. Hence it allows us to extract large directed structural connectivity graphs that are a good approximation to the connectivity in a biological tissue. We have also developed ways of extracting time series of graphs corresponding to the reaction of the system to a variety of stimuli. These directed graphs can be studied using a combination of novel ideas and basic algebraic topology. I will describe some of our methods and the results obtained by applying them to the Blue Brain reconstruction. In the second part of the talk I will also discuss purely theoretical advances inspired by neuroscience.