Evaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System

Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We comprehensively and rigorously assess its performance in simulated neural recordings of the brain's spatial representation system. Grid, head direction, and conjunctive cell populations each span low-dimensional topological structures embedded in high-dimensional neural activity space. We evaluate the ability for persistent cohomology to discover these structures for different dataset dimensions, variations in spatial tuning, and forms of noise. We quantify its ability to decode simulated animal trajectories contained within these topological structures. We also identify regimes under which mixtures of populations form product topologies that can be detected. Our results reveal how dataset parameters affect the success of topological discovery and suggest principles for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.

Generalized penalty for circular coordinate representation

<p style='text-indent:20px;'>Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high-dimensional applications. To do that, we use a generalized penalty function instead of an <inline-formula><tex-math id="M1">\begin{document}$ L_{2} $\end{document}</tex-math></inline-formula> penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analyses to support our claim that circular coordinates with generalized penalty will detect the change in high-dimensional datasets under different sampling schemes while preserving the topological structures.</p>

State space discovery in spatial representation circuits with persistent cohomology

Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We explore the application of persistent cohomology to the brain’s spatial representation system. We simulate populations of grid cells, head direction cells, and conjunctive cells, each of which span low-dimensional topological structures embedded in high-dimensional neural activity space. We evaluate the ability for persistent cohomology to discover these structures and demonstrate its robustness to various forms of noise. We identify regimes under which mixtures of populations form product topologies can be detected. Our results suggest guidelines for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.

The Comparison of Diffeomorphic Images Based on the Construction of Persistent Homology

An object shape analysis is a problem that is related to such areas as geometry, topology, image processing and machine learning. For analyzing the form, the deformation between the source and terminal form of the object is estimated. The most used form analysis model is the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model. The LDDMM model can be supplemented with functional non-geometric information about objects (volume, color, formation time). The paper considers algorithms for constructing sets of barcodes for comparing diffeomorphic images, which are real values taken by persistent homology. A distinctive feature of the use of persistent homology with respect to methods of algebraic topology is to obtain more information about the shape of the object. An important direction of the application of persistent homology is the study invariants of big data. A method based on persistent cohomology is proposed that combines persistent homology technologies with embedded non-geometric information presented as functions of simplicial complexes. The proposed structure of extended barcodes using cohomology increases the effectiveness of persistent homology methods. A modification of the Wasserstein method for finding the distance between images by introducing non-geometric information was proposed. The possibility of the formation of barcodes of images invariant to transformations of rotation, shift and similarity is considered.