The morphological diversity of neurons supports the complex information-processing capabilities of biological neuronal networks. A major challenge in neuroscience has been to reliably describe neuronal shapes with universal morphometrics that generalize across cell types and species. Inspired by algebraic topology, we have developed a topological descriptor of trees that couples the topology of their complex arborization with their geometric structure, retaining more information than traditional morphometrics. The topological morphology descriptor (TMD) has proved to be very powerful in separating neurons into well-defined groups on morphological grounds. The TMD algorithm led to the discovery of two distinct morphological classes of pyramidal cells in the human cortex that also have distinct functional roles, suggesting the existence of a direct link between the anatomy and the function of neurons. The TMD-based classification also led to the objective and robust morphological clustering of rodent cortical neurons. Recently we proved that the TMD of neuronal morphologies is also essential for the computational generation (i.e., synthesis) of dendritic morphologies. Our results demonstrate that a topology-based synthesis algorithm can reproduce both morphological and electrical properties of reconstructed biological rodent cortical dendrites and generalizes well to a wide variety of different dendritic shapes. Therefore it is suitable for the generation of unique neuronal morphologies to populate the digital reconstruction of large-scale, physiologically realistic networks.

How complex, natural signals are represented in the activity (spiking) patterns of neural populations is not well understood. For this talk, I will describe data from a series of experiments that examine the spatiotemporal pattern of song-evoked spiking in populations of simultaneously recorded neurons in the secondary auditory cortices of European starlings (Sturnus vulgaris). Single neurons in these regions display composite receptive fields that incorporate large numbers (a dozen or more) orthogonal features matched to the acoustics of species typical song. Considered independently, the spiking response of a given neuron at a given point in time is therefore ambiguous with respect to the stimulus. Applied topology provides a promising tool to resolve this ambiguity and capture invariant structure in the spiking coactivity in arbitrarily large neural populations. I will show that the topology of the population spike train carries stimulus-specific structure that is not reducible to that of individual neurons. I then introduce a topology-based similarity measure for population coactivity that is sensitive to invariant stimulus structure and show that this measure captures invariant neural representations tied to the learned relationships between natural vocalizations.

This talk introduces a method for characterizing the dynamics of
time-evolving data within the framework of topological data analysis
(TDA), specifically through the lens of persistent homology. Popular
instances of time-evolving data include flocking or swarming behaviors
in animals, and social networks in the human sphere. A natural
mathematical model for such collective behaviors is that of a dynamic
metric space. In this talk I will describe how to extend the well-known
Vietoris-Rips filtration for metric spaces to the setting of dynamic
metric spaces. Also, we extend a celebrated stability theorem on
persistent homology for metric spaces to multiparameter persistent
homology for dynamic metric spaces. In order to address this stability
property, we extend the notion of Gromov-Hausdorff distance between
metric spaces to dynamic metric spaces. This talk will not require any
prior knowledge of TDA. This talk is based on joint work with Facundo
Memoli and Nate Clause.

Even in the absence of external stimuli, neural activity is both highly dynamic and organized across multiple spatiotemporal scales. The continuous evolution of brain activity patterns during rest is believed to help maintain a rich repertoire of possible functional configurations that relate to typical and atypical cognitive phenomena. Whether these transitions or "explorations" follow some underlying arrangement or instead lack a predictable ordered plan remains to be determined. Here, using a precision individual connectomics dataset, we aimed at revealing the rules that govern transitions in brain activity at rest. The dataset includes individually defined parcellations and ~5 hours of resting state functional Magnetic Resonance Imaging (fMRI) data for each participant – both of which allowed us to examine the topology and dynamics of at-rest whole-brain configurations in an unprecedented detail. We hypothesized that by revealing and characterizing the overall landscape of whole-brain configurations we could interpret the rules (if any) that govern transitions in brain activity at rest. To generate the landscape of whole- brain configurations we used Topological Data Analysis (TDA)-based Mapper approach that could reveal shape of the underlying dataset as a graph (a.k.a. shape graph). We observed a rich topographic landscape in which the transition of activity from one canonical brain network to the next involved a large, shared attractor-like basin, or "transition state", where all networks were represented equally prior to entering distinct network configurations. The intermediate transition state and traversal through it seemed to provide the underlying structure for the continuous evolution of brain activity patterns at rest. In addition, differences in the manifold architecture were more consistent within than between subjects, providing evidence that this approach contains potential utility for precision medicine approaches.

Functional magnetic resonance imaging (fMRI) measures brain activity by detecting changes in blood oxygenation levels (BOLD signals). Relating these activity measurements to behavioral and cognitive measures is a topic of great interest in neuroscience. The functional architecture of the brain can be described as a dynamic system where components interact in flexible ways, constrained by physical connections between regions. Using correlation, either in time or in space, as an abstraction of functional connectivity, we perform topological data analysis of resting-state fMRI data.  We find that temporal vs. spatial functional connectivity can encode different aspects of cognition and personality. Topological analyses using persistent homology show that persistence barcodes are significantly correlated to individual differences in cognition and personality, with high reproducibility. Topological data analyses, including approaches to model connectivity in the time domain, are promising tools for representing high-level aspects of cognition, development, and neuropathology.

This talk is a summary of joint works with Sourabh Palande, Keri L. Anderson, Jeffrey S. Anderson, Archit Rathore, Brandon Zielinski, and Tom Fletcher.

We propose a mathematical formalism for neural information networks endowed with assignments of resources (computational or metabolic or informational), suitable for describing assignments of concurrent or distributed computing architectures and associated binary codes, governed by a categorical form of the Hopfield network dynamics, and measures of informational complexity in the form of a cohomological version of integrated information.

The classical theory of nonlinear dynamics exhibits wonderfully rich and exotic structures. I will argue that as we move to an era of data driven dynamics it offers too many riches. Stated differently, if we only have finite data at our disposal we need a simpler theory of dynamics. I will present such a theory based on combinatorics and algebraic topology. After describing the theory, I will apply it to the analysis of dynamics of networks.

The brain represents the perceived world via the activity of individual neurons, or groups of neurons. There is an increasing body of evidence that neural activity in a number of sensory systems is organized on low-dimensional manifolds. Understanding the neural representations (a.k.a. the neural code) thus requires methods for inferring the structure of the underlying stimulus space, as well as natural decoding mechanisms that takes advantage of this structure.   

Neural representations are constrained by receptive field properties of individual neurons as well as the underlying neural network. It is therefore essential to utilize these constraints in any meaningful analysis of  the underlying space.  In my talk, I will describe two different methods, based on computational topology and differential geometry that take advantage of the receptive field properties to infer the dimension of (non-linear) neural representations  as well as a geometry-based learning algorithm, that can be re-interpreted as an output of a neural network.  I will illustrate the first method by inferring basic features of the neural representations in the mouse olfactory bulb.

The stability of persistence diagrams is among the most important results in applied and computational topology but most results are with respect to the bottleneck distance between diagrams. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this talk I will discuss new stability results with respect to the p-Wasserstein distance between persistence diagrams. The main result is stability of persistence diagrams between different functions on the same finite simplicial complex in terms of the p-norm of the functions. This has applications to image analysis, persistence homology transforms and Vietoris-Rips complexes. This is joint work with Primoz Skraba.