Colloquium

About

Speaker: Dr. Chad Giusti (Oregon State University)

Title: Investigating the Topology of Neural Manifolds

Abstract: The last few decades have seen extraordinary advances in experimental techniques in neuroscience. Work in animal models has moved from single electrodes isolating one or a few neurons to large-scale, long-term recordings of tens of thousands of neurons or more, some spanning multiple brain regions or, in small model organisms such as larval zebrafish, entire brains. These experiments open the door to questions about how populations of neurons encode information en masse, and what neural computation looks like on the macro scale. One fundamental approach to these questions is the study of “neural manifolds,” low-dimensional structures in the space of population activity that are in practice frequently approximated by linear subspaces found via principal component analysis or related methods. Characterizing the relationship between these structures and presented stimuli or observed behavior has proved very effective at revealing which extrinsic variables are encoded by a population and how its activity evolves over time.

However, the brain encodes information that naturally carries non-linear structure. Indeed, two of the most extensively studied brain regions encode inherently circular or toroidal variables. In primary visual cortex, individual neurons are tuned to stimulus orientation. In the hippocampal formation, head-direction cells encode heading and grid cells encode position modulo a lattice. Additionally, dimensionality reduction often discards or obscures fundamental structure in complex data in favor of simple explanatory variables. This, in turn, makes it difficult to rigorously compare activity across populations or to develop basic theory about what computation means at the population scale. Thus, tools that allow for intrinsic analysis of both linear and non-linear structure without dimensionality reduction are desirable in this setting.

Topology provides one lens through which we can address both problems. In this talk, I will discuss how the classical nerve theorem and its relatives provide a natural foundation for a mathematical theory of population-level neural computation. I will then briefly describe recent work with my former postdocs Iris Yoon and Niko Schonsheck on how to track the transmission of topologically structured information across neural populations, and demonstrating that simple spike-timing dependent plasticity is suffificient to allow biological neural networks learn to represent such information.

No background in neuroscience and only minimal exposure to topology (simplicial complexes and a rough notion of homology) will be needed for this talk.