A Topological Deep Learning Framework for Neural Spike Decoding

The brain's spatial orientation system uses different neuron ensembles to aid in environment-based navigation. One of the ways brains encode spatial information is through grid cells, layers of decked neurons that overlay to provide environment-based navigation. These neurons fire in ensembles where several neurons fire at once to activate a single grid. We want to capture this firing structure and use it to decode grid cell data. Understanding, representing, and decoding these neural structures require models that encompass higher order connectivity than traditional graph-based models may provide. To that end, in this work, we develop a topological deep learning framework for neural spike train decoding. Our framework combines unsupervised simplicial complex discovery with the power of deep learning via a new architecture we develop herein called a simplicial convolutional recurrent neural network (SCRNN). Simplicial complexes, topological spaces that use not only vertices and edges but also higher-dimensional objects, naturally generalize graphs and capture more than just pairwise relationships. Additionally, this approach does not require prior knowledge of the neural activity beyond spike counts, which removes the need for similarity measurements. The effectiveness and versatility of the SCRNN is demonstrated on head direction data to test its performance and then applied to grid cell datasets with the task to automatically predict trajectories

Molecular configurations and persistence: branched alkanes and additive energies

2022 Spring.Includes bibliographical references.Energy landscapes are high-dimensional functions that encapsulate how certain molecular properties affect the energy of a molecule. Chemists use disconnectivity graphs to find transition paths, the lowest amount of energy needed to transfer from one energy minimum to another. But disconnectivity graphs fail to show not only some lower-dimensional features, such as transition paths with an energy value only slightly higher than the minimum transition path, but also all higher-dimensional features. Sublevelset persistent homology is a tool that can be used to capture other relevant features, including all transition paths. In this paper, we will use sublevelset persistent homology to find the structure of the energy landscapes of branched alkanes: tree-like molecules consisting of only carbons and hydrogens. We derive complete characterizations of the sublevelset persistent homology of the OPLS-UA energy function on two different families of branched alkanes. More generally, we explain how the sublevelset persistent homology of any additive energy landscape can be computed from the individual terms comprising that landscape

Additive energy functions have predictable landscape topologies

Recent work (J. Chem. Phys. 154, 114114) has demonstrated that sublevelset persistent homology provides a compact representation of the complex features of an energy landscape in $3N$-dimensions. This includes information about all transition paths between local minima (connected by critical points of index > 1), and allows for differentiation of energy landscapes that may appear similar when considering only the lowest energy pathways (as tracked by other representations like disconnectivity graphs using index 1 critical points). Using the additive nature of the conformational potential energy landscape of n-alkanes, it became apparent that some topological features --- such as the number of sublevelset persistence bars --- could be proven. This work expands the notion of predictable energy landscape topology to any additive intramolecular energy function, including the number of sublevelset persistent bars as well as the birth and death times of these topological features. This amounts to a rigorous methodology to predict the relative energies of all topological features of the conformational energy landscape in 3N dimensions (without the need for dimensionality reduction). This approach is demonstrated for branched alkanes of varying complexity and connectivity patterns. More generally, this result explains how the sublevelset persistent homology of an additive energy landscape can be computed from the individual terms comprising that landscape

The impact of changes in resolution on the persistent homology of images

Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown