Chad Giusti

Neural manifolds are an indispensable framework for understanding information encoded by activity in neural populations. While some neural manifolds are linear and can be recovered from population activity using standard techniques, many neural manifolds exhibit nonlinear global topology for which such tools can be less effective. Notably, circular and toroidal manifolds describe activity in neural systems across a range of species; common examples include orientation-selective simple cells in primary visual cortex, head-direction cells in thalamic circuits, and grid cells in entorhinal cortex. That such structured information appears in both sensory and deep-brain regions raises a basic question: is the propagation of nonlinear coordinate systems a generic feature of biological neural networks, or must this be learned? If learning is necessary, how does it occur? In this paper, we apply methods from topological data analysis developed to quantitatively measure propagation of such nonlinear manifolds across populations to address these problems. We identify a collection of connectivity and parameter regimes for feed-forward networks in which learning is required, and demonstrate that simple Hebbian spike-timing dependent plasticity reorganizes such networks to correctly propagate circular coordinate systems. We observe during this learning process the emergence of geometrically non-local experimentally observed receptive field types: bimodally-tuned head-direction cells and cells with spatially periodic, band-like receptive fields. These observations provide quantitative support for the hypothesis that simple biologically plausible plasticity mechanisms suffice to induce changes in the structure of neural architectures sufficient to explain the appearance of such features in real neural systems.

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and \emph{in vivo} experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and in vivo experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.

Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces. These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in multi-agent systems, time-varying fMRI scans from neuroscience, and time-dependent scalar fields in hydrodynamics. While persistence diagrams can provide a powerful topological summary of data, the standard space of persistence diagrams lacks the sufficient algebraic and analytic structure required for many theoretical and computational analyses. We enrich the space of persistence diagrams by isometrically embedding it into a Lipschitz-free space, a Banach space built from a universal construction. We utilize the Banach space structure to define bounded variation paths of persistence diagrams, which can be studied using the path signature, a reparametrization-invariant characterization of paths valued in a Banach space. The signature is universal and characteristic which allows us to theoretically characterize measures on the space of paths and motivates its use in the context of kernel methods. However, kernel methods often require a feature map into a Hilbert space, so we introduce the moment map, a stable and injective feature map for static persistence diagrams, and compose it with the discrete path signature, producing a computable feature map into a Hilbert space. Finally, we demonstrate the efficacy of our methods by applying this to a parameter estimation problem for a 3D model of swarming behavior.

A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between homology classes from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing homology classes between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.

A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between cycles from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing cycles between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.

We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.