Topological Embedding of Human Brain Networks with Applications to Dynamics of Temporal Lobe Epilepsy.

We introduce a novel, data-driven topological data analysis (TDA) approach for embedding brain networks into a lower-dimensional space in quantifying the dynamics of temporal lobe epilepsy (TLE) obtained from resting-state functional magnetic resonance imaging (rs-fMRI). This embedding facilitates the orthogonal projection of 0D and 1D topological features, allowing for the visualization and modeling of the dynamics of functional human brain networks in a resting state. We then quantify the topological disparities between networks to determine the coordinates for embedding. This framework enables us to conduct a coherent statistical inference within the embedded space. Our results indicate that brain network topology in TLE patients exhibits increased rigidity in 0D topology but more rapid flections compared to that of normal controls in 1D topology.

Topological state-space estimation of functional human brain networks.

We introduce an innovative, data-driven topological data analysis (TDA) technique for estimating the state spaces of dynamically changing functional human brain networks at rest. Our method utilizes the Wasserstein distance to measure topological differences, enabling the clustering of brain networks into distinct topological states. This technique outperforms the commonly used k-means clustering in identifying brain network state spaces by effectively incorporating the temporal dynamics of the data without the need for explicit model specification. We further investigate the genetic underpinnings of these topological features using a twin study design, examining the heritability of such state changes. Our findings suggest that the topology of brain networks, particularly in their dynamic state changes, may hold significant hidden genetic information.

Altered topological structure of the brain white matter in maltreated children through topological data analysis.

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging and diffusion tensor imaging. We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children with a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

We employ topological data analysis (TDA) to investigate altered topological structures in the white matter of children who have experienced maltreatment. Persistent homology in TDA is utilized to quantify topological differences between typically developing children and those subjected to maltreatment, using magnetic resonance imaging and diffusion tensor imaging data. The Wasserstein distance is computed between topological features to assess disparities in brain networks. Our findings demonstrate that persistent homology effectively characterizes the altered dynamics of white matter in children who have suffered maltreatment.

Greater white matter degeneration and lower structural connectivity in non-amnestic vs. amnestic Alzheimer's disease.

Introduction: Multimodal evidence indicates Alzheimer's disease (AD) is characterized by early white matter (WM) changes that precede overt cognitive impairment. WM changes have overwhelmingly been investigated in typical, amnestic mild cognitive impairment and AD; fewer studies have addressed WM change in atypical, non-amnestic syndromes. We hypothesized each non-amnestic AD syndrome would exhibit WM differences from amnestic and other non-amnestic syndromes. Materials and methods: Participants included 45 cognitively normal (CN) individuals; 41 amnestic AD patients; and 67 patients with non-amnestic AD syndromes including logopenic-variant primary progressive aphasia (lvPPA, n = 32), posterior cortical atrophy (PCA, n = 17), behavioral variant AD (bvAD, n = 10), and corticobasal syndrome (CBS, n = 8). All had T1-weighted MRI and 30-direction diffusion-weighted imaging (DWI). We performed whole-brain deterministic tractography between 148 cortical and subcortical regions; connection strength was quantified by tractwise mean generalized fractional anisotropy. Regression models assessed effects of group and phenotype as well as associations with grey matter volume. Topological analyses assessed differences in persistent homology (numbers of graph components and cycles). Additionally, we tested associations of topological metrics with global cognition, disease duration, and DWI microstructural metrics. Results: Both amnestic and non-amnestic patients exhibited lower WM connection strength than CN participants in corpus callosum, cingulum, and inferior and superior longitudinal fasciculi. Overall, non-amnestic patients had more WM disease than amnestic patients. LvPPA patients had left-lateralized WM degeneration; PCA patients had reductions in connections to bilateral posterior parietal, occipital, and temporal areas. Topological analysis showed the non-amnestic but not the amnestic group had more connected components than controls, indicating persistently lower connectivity. Longer disease duration and cognitive impairment were associated with more connected components and fewer cycles in individuals' brain graphs. Discussion: We have previously reported syndromic differences in GM degeneration and tau accumulation between AD syndromes; here we find corresponding differences in WM tracts connecting syndrome-specific epicenters. Determining the reasons for selective WM degeneration in non-amnestic AD is a research priority that will require integration of knowledge from neuroimaging, biomarker, autopsy, and functional genetic studies. Furthermore, longitudinal studies to determine the chronology of WM vs. GM degeneration will be key to assessing evidence for WM-mediated tau spread.

Statistical inference for dependence networks in topological data analysis.

Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.

Topological Data Analysis for Multivariate Time Series Data.

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. The aim of this article is to introduce TDA concepts to a statistical audience and provide an approach to analyzing multivariate time series data. The application's focus will be on multivariate brain signals and brain connectivity networks. Finally, this paper concludes with an overview of some open problems and potential application of TDA to modeling directionality in a brain network, as well as the casting of TDA in the context of mixed effect models to capture variations in the topological properties of data collected from multiple subjects.

Unified topological inference for brain networks in temporal lobe epilepsy using the Wasserstein distance.

Persistent homology offers a powerful tool for extracting hidden topological signals from brain networks. It captures the evolution of topological structures across multiple scales, known as filtrations, thereby revealing topological features that persist over these scales. These features are summarized in persistence diagrams, and their dissimilarity is quantified using the Wasserstein distance. However, the Wasserstein distance does not follow a known distribution, posing challenges for the application of existing parametric statistical models. To tackle this issue, we introduce a unified topological inference framework centered on the Wasserstein distance. Our approach has no explicit model and distributional assumptions. The inference is performed in a completely data driven fashion. We apply this method to resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe epilepsy patients collected from two different sites: the University of Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our topological method is robust to variations due to sex and image acquisition, obviating the need to account for these variables as nuisance covariates. We successfully localize the brain regions that contribute the most to topological differences. A MATLAB package used for all analyses in this study is available at https://github.com/laplcebeltrami/PH-STAT.

Altered Topological Structure of the Brain White Matter in Maltreated Children through Topological Data Analysis.

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white-matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI). We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children to a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

Hodge Laplacian of Brain Networks.

The closed loops or cycles in a brain network embeds higher order signal transmission paths, which provide fundamental insights into the functioning of the brain. In this work, we propose an efficient algorithm for systematic identification and modeling of cycles using persistent homology and the Hodge Laplacian. Various statistical inference procedures on cycles are developed. We validate the our methods on simulations and apply to brain networks obtained through the resting state functional magnetic resonance imaging. The computer codes for the Hodge Laplacian are given in https://github.com/laplcebeltrami/hodge.

TOPOLOGICAL LEARNING FOR BRAIN NETWORKS.

This paper proposes a novel topological learning framework that integrates networks of different sizes and topology through persistent homology. Such challenging task is made possible through the introduction of a computationally efficient topological loss. The use of the proposed loss bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations to assess its effectiveness when discriminating networks with different topology. The method is further demonstrated in a twin brain imaging study where we determine if brain networks are genetically heritable. The challenge here is due to the difficulty of overlaying the topologically different functional brain networks obtained from resting-state functional MRI onto the template structural brain network obtained through diffusion MRI.

Topological data analysis of human brain networks through order statistics.

Understanding the common topological characteristics of the human brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological properties of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using the order statistics for analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We found a statistically significant topological difference between the male and female brain networks.

Unified Topological Inference for Brain Networks in Temporal Lobe Epilepsy Using the Wasserstein Distance.

Persistent homology offers a powerful tool for extracting hidden topological signals from brain networks. It captures the evolution of topological structures across multiple scales, known as filtrations, thereby revealing topological features that persist over these scales. These features are summarized in persistence diagrams, and their dissimilarity is quantified using the Wasserstein distance. However, the Wasserstein distance does not follow a known distribution, posing challenges for the application of existing parametric statistical models. To tackle this issue, we introduce a unified topological inference framework centered on the Wasserstein distance. Our approach has no explicit model and distributional assumptions. The inference is performed in a completely data driven fashion. We apply this method to resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe epilepsy patients collected from two different sites: the University of Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our topological method is robust to variations due to sex and image acquisition, obviating the need to account for these variables as nuisance covariates. We successfully localize the brain regions that contribute the most to topological differences. A MATLAB package used for all analyses in this study is available at https://github.com/laplcebeltrami/PH-STAT.

Heterogeneous Graph Convolutional Neural Network via Hodge-Laplacian for Brain Functional Data.

This study proposes a novel heterogeneous graph convolutional neural network (HGCNN) to handle complex brain fMRI data at regional and across-region levels. We introduce a generic formulation of spectral filters on heterogeneous graphs by introducing the k-th Hodge-Laplacian (HL) operator. In particular, we propose Laguerre polynomial approximations of HL spectral filters and prove that their spatial localization on graphs is related to the polynomial order. Furthermore, based on the bijection property of boundary operators on simplex graphs, we introduce a generic topological graph pooling (TGPool) method that can be used at any dimensional simplices. This study designs HL-node, HL-edge, and HL-HGCNN neural networks to learn signal representation at a graph node, edge levels, and both, respectively. Our experiments employ fMRI from the Adolescent Brain Cognitive Development (ABCD; n=7693) to predict general intelligence. Our results demonstrate the advantage of the HL-edge network over the HL-node network when functional brain connectivity is considered as features. The HL-HGCNN outperforms the state-of-the-art graph neural networks (GNNs) approaches, such as GAT, BrainGNN, dGCN, BrainNetCNN, and Hypergraph NN. The functional connectivity features learned from the HL-HGCNN are meaningful in interpreting neural circuits related to general intelligence.

Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering.

This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We define spectral filters via the LB operator on a graph and explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters. We then update the LB operator for pooling in the LB-CNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and Open Access Series of Imaging Studies (OASIS) to demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of two datasets, we showed that the LB-CNN slightly improves classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. The LB-CNN trained via the ADNI dataset can achieve reasonable classification accuracy for the OASIS dataset. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).

Lattice Paths for Persistent Diagrams.

Persistent homology has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. In this paper, we first present a new lattice path representation for persistent diagrams. We then develop a new exact statistical inference procedure for lattice paths via combinatorial enumerations. The lattice path method is applied to the topological characterization of the protein structures of the COVID-19 virus. We demonstrate that there are topological changes during the conformational change of spike proteins.

Fast mesh data augmentation via Chebyshev polynomial of spectral filtering.

Deep neural networks have recently been recognized as one of the powerful learning techniques in computer vision and medical image analysis. Trained deep neural networks need to be generalizable to new data that are not seen before. In practice, there is often insufficient training data available, which can be solved via data augmentation. Nevertheless, there is a lack of augmentation methods to generate data on graphs or surfaces, even though graph convolutional neural network (graph-CNN) has been widely used in deep learning. This study proposed two unbiased augmentation methods, Laplace-Beltrami eigenfunction Data Augmentation (LB-eigDA) and Chebyshev polynomial Data Augmentation (C-pDA), to generate new data on surfaces, whose mean was the same as that of observed data. LB-eigDA augmented data via the resampling of the LB coefficients. In parallel with LB-eigDA, we introduced a fast augmentation approach, C-pDA, that employed a polynomial approximation of LB spectral filters on surfaces. We designed LB spectral bandpass filters by Chebyshev polynomial approximation and resampled signals filtered via these filters in order to generate new data on surfaces. We first validated LB-eigDA and C-pDA via simulated data and demonstrated their use for improving classification accuracy. We then employed brain images of the Alzheimer's Disease Neuroimaging Initiative (ADNI) and extracted cortical thickness that was represented on the cortical surface to illustrate the use of the two augmentation methods. We demonstrated that augmented cortical thickness had a similar pattern to observed data. We also showed that C-pDA was faster than LB-eigDA and can improve the AD classification accuracy of graph-CNN.

Lattice Paths for Persistent Diagrams.

Persistent homology has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. In this paper, we first present a new lattice path representation for persistent diagrams. We then develop a new exact statistical inference procedure for lattice paths via combinatorial enumerations. The lattice path method is applied to the topological characterization of the protein structures of the COVID-19 virus. We demonstrate that there are topological changes during the conformational change of spike proteins.

Fast Polynomial Approximation of Heat Kernel Convolution on Manifolds and Its Application to Brain Sulcal and Gyral Graph Pattern Analysis.

Heat diffusion has been widely used in brain imaging for surface fairing, mesh regularization and cortical data smoothing. Motivated by diffusion wavelets and convolutional neural networks on graphs, we present a new fast and accurate numerical scheme to solve heat diffusion on surface meshes. This is achieved by approximating the heat kernel convolution using high degree orthogonal polynomials in the spectral domain. We also derive the closed-form expression of the spectral decomposition of the Laplace-Beltrami operator and use it to solve heat diffusion on a manifold for the first time. The proposed fast polynomial approximation scheme avoids solving for the eigenfunctions of the Laplace-Beltrami operator, which is computationally costly for large mesh size, and the numerical instability associated with the finite element method based diffusion solvers. The proposed method is applied in localizing the male and female differences in cortical sulcal and gyral graph patterns obtained from MRI in an innovative way. The MATLAB code is available at http://www.stat.wisc.edu/~mchung/chebyshev.