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Traditional Topological Data Analysis (TDA) methods, such as Persistent Homology (PH), rely on distance measures (e.g., cross-correlation, partial correlation, coherence, and partial coherence) that are symmetric by definition. While useful for studying topological patterns in functional brain connectivity, the main limitation of these methods is their inability to capture the directional dynamics - which is crucial for understanding effective brain connectivity. We propose the Causality-Based Topological Ranking (CBTR) method, which integrates Causal Inference (CI) to assess effective brain connectivity with Hodge Decomposition (HD) to rank brain regions based on their mutual influence. Our simulations confirm that the CBTR method accurately and consistently identifies hierarchical structures in multivariate time series data. Moreover, this method effectively identifies brain regions showing the most significant interaction changes with other regions during seizures using electroencephalogram (EEG) data. These results provide novel insights into the brain's hierarchical organization and illuminate the impact of seizures on its dynamics.

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.

Graph neural networks (GNNs) have proven effective in capturing relationships among nodes in a graph. This study introduces a novel perspective by considering a graph as a simplicial complex, encompassing nodes, edges, triangles, and $k$-simplices, enabling the definition of graph-structured data on any $k$-simplices. Our contribution is the Hodge-Laplacian heterogeneous graph attention network (HL-HGAT), designed to learn heterogeneous signal representations across $k$-simplices. The HL-HGAT incorporates three key components: HL convolutional filters (HL-filters), simplicial projection (SP), and simplicial attention pooling (SAP) operators, applied to $k$-simplices. HL-filters leverage the unique topology of $k$-simplices encoded by the Hodge-Laplacian (HL) operator, operating within the spectral domain of the $k$-th HL operator. To address computation challenges, we introduce a polynomial approximation for HL-filters, exhibiting spatial localization properties. Additionally, we propose a pooling operator to coarsen $k$-simplices, combining features through simplicial attention mechanisms of self-attention and cross-attention via transformers and SP operators, capturing topological interconnections across multiple dimensions of simplices. The HL-HGAT is comprehensively evaluated across diverse graph applications, including NP-hard problems, graph multi-label and classification challenges, and graph regression tasks in logistics, computer vision, biology, chemistry, and neuroscience. The results demonstrate the model's efficacy and versatility in handling a wide range of graph-based scenarios.

Topological data analysis (TDA) has become a powerful approach over the last twenty years, mainly due to its ability to capture the shape and the geometry inherent in the data. Persistence homology, which is a particular tool in TDA, has been demonstrated to be successful in analyzing functional brain connectivity. One limitation of standard approaches is that they use arbitrarily chosen threshold values for analyzing connectivity matrices. To overcome this weakness, TDA provides a filtration of the weighted brain network across a range of threshold values. However, current analyses of the topological structure of functional brain connectivity primarily rely on overly simplistic connectivity measures, such as the Pearson orrelation. These measures do not provide information about the specific oscillators that drive dependence within the brain network. Here, we develop a frequency-specific approach that utilizes coherence, a measure of dependence in the spectral domain, to evaluate the functional connectivity of the brain. Our approach, the spectral TDA (STDA), has the ability to capture more nuanced and detailed information about the underlying brain networks. The proposed STDA method leads to a novel topological summary, the spectral landscape, which is a 2D-generalization of the persistence landscape. Using the novel spectral landscape, we analyze the EEG brain connectivity of patients with attention deficit hyperactivity disorder (ADHD) and shed light on the frequency-specific differences in the topology of brain connectivity between the controls and ADHD patients., Comment: 28 pages, 23 figures

We present the unified computational framework for modeling the sulcal patterns of human brain obtained from the magnetic resonance images. The Wasserstein distance is used to align the sulcal patterns nonlinearly. These patterns are topologically different across subjects making the pattern matching a challenge. We work out the mathematical details and develop the gradient descent algorithms for estimating the deformation field. We further quantify the image registration performance. This method is applied in identifying the differences between male and female sulcal patterns., Comment: In press in IEEE ISBI

We introduce PH-STAT, a comprehensive Matlab toolbox designed for performing a wide range of statistical inferences on persistent homology. Persistent homology is a prominent tool in topological data analysis (TDA) that captures the underlying topological features of complex data sets. The toolbox aims to provide users with an accessible and user-friendly interface for analyzing and interpreting topological data. The package is distributed in https://github.com/laplcebeltrami/PH-STAT., Comment: arXiv admin note: text overlap with arXiv:2302.06673

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.

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.

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., Comment: arXiv admin note: text overlap with arXiv:2201.00087

We analyze brain networks by decomposing them into three orthogonal components: gradient, curl, and harmonic flows, through the Hodge decomposition, a technique advantageous for capturing complex topological features. A Wasserstein distance based topological inference is developed to determine the statistical significance of each component. The Hodge decomposition is applied to human brain networks obtained from a resting-state fMRI study. Our results indicate statistically significant differences in the topological features between male and female brain networks., Comment: Will be published in ISBI 2024

Developing reliable methods to discriminate different transient brain states that change over time is a key neuroscientific challenge in brain imaging studies. Topological data analysis (TDA), a novel framework based on algebraic topology, can handle such a challenge. However, existing TDA has been somewhat limited to capturing the static summary of dynamically changing brain networks. We propose a novel dynamic-TDA framework that builds persistent homology over a time series of brain networks. We construct a Wasserstein distance based inference procedure to discriminate between time series of networks. The method is applied to the resting-state functional magnetic resonance images of human brain. We demonstrate that our proposed dynamic-TDA approach can distinctly discriminate between the topological patterns of male and female brain networks. MATLAB code for implementing this method is available at https://github.com/laplcebeltrami/PH-STAT., Comment: In press in journal Foundations of Data Science

Topological data analysis (TDA) approaches are becoming increasingly popular for studying the dependence patterns in multivariate time series data. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes, which can 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. Simulations are crucial for evaluating the performance of proposed TDA methods and testing procedures as well as for creating computation-based confidence intervals. To our knowledge, there are no methods that simulate multivariate time series data with specific and manually imposed 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.

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach which 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. Our aim in this article is to introduce TDA concepts to a statistical audience and provide an approach to analyze multivariate time series data. The application focus will be on multivariate brain signals and brain connectivity networks. Finally, the 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 effects models to capture variations in the topological properties of data collected from multiple subjects

Human brain activity is often measured using the blood-oxygen-level dependent (BOLD) signals obtained through functional magnetic resonance imaging (fMRI). The strength of connectivity between brain regions is then measured as a Pearson correlation matrix. As the number of brain regions increases, the dimension of matrix increases. It becomes extremely cumbersome to even visualize and quantify such weighted complete networks. To remedy the problem, we propose to embed brain networks onto a sphere, which is a Riemannian manifold with constant positive curvature. The Matlab code for the spherical embedding is given in https://github.com/laplcebeltrami/sphericalMDS.

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.

In this paper, we review widely used statistical analysis frameworks for data defined along cortical and subcortical surfaces that have been developed in last two decades. The cerebral cortex has the topology of a 2D highly convoluted sheet. For data obtained along curved non-Euclidean surfaces, traditional statistical analysis and smoothing techniques based on the Euclidean metric structure are inefficient. To increase the signal-to-noise ratio (SNR) and to boost the sensitivity of the analysis, it is necessary to smooth out noisy surface data. However, this requires smoothing data on curved cortical manifolds and assigning smoothing weights based on the geodesic distance along the surface. Thus, many cortical surface data analysis frameworks are differential geometric in nature. The smoothed surface data is then treated as smooth random fields and statistical inferences can be performed within Keith Worsley's random field theory. The methods described in this paper are illustrated with the hippocampus surface data set. Using this case study, we will determine if there is an effect of family income on the growth of hippocampus in children in detail. There are a total of 124 children and 82 of them have repeat magnetic resonance images (MRI) two years later.

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. MATLAB code for the method is available at https://github.com/laplcebeltrami/PH-STAT., Comment: To be published in PLOS Computational Biology

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.

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.

Recent developments in graph theoretic analysis of complex networks have led to deeper understanding of brain networks. Many complex networks show similar macroscopic behaviors despite differences in the microscopic details. Probably two most often observed characteristics of complex networks are scale-free and small-world properties. In this paper, we will explore whether brain networks follow scale-free and small-worldness among other graph theory properties.

This article is based on the first chapter of book Chung (2013), where brain and medical images are introduced. The most widely used brain imaging modalities are magnetic resonance images (MRI), functional-MRI (fMRI) and diffusion tensor images (DTI). A brief introduction to each imaging modality is explained. Further, we explain what kind of curve, volume and surface data that can be extracted from each modality.

Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to use distances and loss functions that recognize topology of data. In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA) and persistent homology that can be used in brain network analysis more effectively. Although there are many recent brain imaging studies that are based on TDA methods, possibly due to the lack of method awareness, TDA has not taken as the mainstream tool in brain imaging field yet. The main purpose of this paper is provide the relevant technical survey of these powerful tools that are immediately applicable to brain network data.

This short paper is based on Chung et al. (2010), where the cosine series representation (CSR) is used in modeling the shape of white matter fiber tracts in diffusion tensor imaging(DTI) and Wang et al. (2018), where the method is used to denoise EEG. The proposed explicit analytic approach offers far superior flexibility in statistical modeling compared to the usual implicit Fourier transform methods such as the discrete cosine transforms often used in signal processing. The MATLAB codes and sample data can be obtained from http://brainimaging.waisman.wisc.edu/~chung/tracts.

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., Comment: 31 pages, 14 figures, 4 tables, code at https://github.com/topolearn

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 then define spectral filters via the LB operator on a graph. We explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters and define an update of the LB operator for pooling in the LBCNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of the ADNI dataset, we showed that the LB-CNN didn't improve 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. 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).

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 was not seen before. In practice, there is often insufficient training data available and augmentation is used to expand the dataset. Even though graph convolutional neural network (graph-CNN) has been widely used in deep learning, there is a lack of augmentation methods to generate data on graphs or surfaces. This study proposes 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 is the same as that of real data. LB-eigDA augments data via the resampling of the LB coefficients. In parallel with LB-eigDA, we introduce a fast augmentation approach, C-pDA, that employs a polynomial approximation of LB spectral filters on surfaces. We design LB spectral bandpass filters by Chebyshev polynomial approximation and resample signals filtered via these filters to generate new data on surfaces. We first validate LB-eigDA and C-pDA via simulated data and demonstrate their use for improving classification accuracy. We then employ the brain images of Alzheimer's Disease Neuroimaging Initiative (ADNI) and extract cortical thickness that is represented on the cortical surface to illustrate the use of the two augmentation methods. We demonstrate that augmented cortical thickness has a similar pattern to real data. Second, we show that C-pDA is much faster than LB-eigDA. Last, we show that C-pDA can improve the AD classification accuracy of graph-CNN.

For random field theory based multiple comparison corrections In brain imaging, it is often necessary to compute the distribution of the supremum of a random field. Unfortunately, computing the distribution of the supremum of the random field is not easy and requires satisfying many distributional assumptions that may not be true in real data. Thus, there is a need to come up with a different framework that does not use the traditional statistical hypothesis testing paradigm that requires to compute p-values. With this as a motivation, we can use a different approach called the logistic regression that does not require computing the p-value and still be able to localize the regions of brain network differences. Unlike other discriminant and classification techniques that tried to classify preselected feature vectors, the method here does not require any preselected feature vectors and performs the classification at each edge level.

There have been many attempts to identify high-dimensional network features via multivariate approaches. Specifically, when the number of voxels or nodes, denoted as p, are substantially larger than the number of images, denoted as n, it produces an under-determined model with infinitely many possible solutions. The small-n large-p problem is often remedied by regularizing the under-determined system with additional sparse penalties. Popular sparse network models include sparse correlations, LASSO, sparse canonical correlations and graphical-LASSO. These popular sparse models require optimizing L1-norm penalties, which has been the major computational bottleneck for solving large-scale problems. Thus, many existing sparse brain network models in brain imaging have been restricted to a few hundreds nodes or less. 2527 MRI features used in a LASSO model for Alzheimer's disease is probably the largest number of features used in any sparse model in the brain imaging literature.

In brain imaging, the image acquisition and processing processes themselves are likely to introduce noise to the images. It is therefore imperative to reduce the noise while preserving the geometric details of the anatomical structures for various applications. Traditionally Gaussian kernel smoothing has been often used in brain image processing and analysis. However, the direct application of Gaussian kernel smoothing tend to cause various numerical issues in irregular domains with boundaries. For example, if one uses large bandwidth in kernel smoothing in a cortical bounded region, the smoothing will blur signals across boundaries. So in kernel smoothing and regression literature, various ad-hoc procedures were introduce to remedy the boundary effect. Diffusion equations have been widely used in brain imaging as a form of noise reduction. The most natural straightforward way to smooth images in irregular domains with boundaries is to formulate the problem as boundary value problems using partial differential equations. Numerous diffusion-based techniques have been developed in image processing. In this paper, we will overview the basics of isotropic diffusion equations and explain how to solve them on regular grids and irregular grids such as graphs., Comment: arXiv admin note: text overlap with arXiv:1710.07849

General linear models (GLM) are often constructed and used in statistical inference at the voxel level in brain imaging. In this paper, we explore the basics of random fields and the multiple comparisons on the random fields, which are necessary to properly threshold statistical maps for the whole image at specific statistical significance level. The multiple comparisons are crucial in determining overall statistical significance in correlated test statistics over the whole brain. In practice, t- or F-statistics in adjacent voxels are correlated. So there is the problem of multiple comparisons, which we have simply neglected up to now. For multiple comparisons that account for spatially correlated test statistics, various methods were proposed: Bonferroni correction, random field theory, false discovery rates and permutation tests. Among them, we will explore the random field approach.

Image acquisition and segmentation are likely to introduce noise. Further image processing such as image registration and parameterization can introduce additional noise. It is thus imperative to reduce noise measurements and boost signal. In order to increase the signal-to-noise ratio (SNR) and smoothness of data required for the subsequent random field theory based statistical inference, some type of smoothing is necessary. Among many image smoothing methods, Gaussian kernel smoothing has emerged as a de facto smoothing technique among brain imaging researchers due to its simplicity in numerical implementation. Gaussian kernel smoothing also increases statistical sensitivity and statistical power as well as Gausianness. Gaussian kernel smoothing can be viewed as weighted averaging of voxel values. Then from the central limit theorem, the weighted average should be more Gaussian.

Recent studies have shown that functional brain brainwork is dynamic even during rest. A common approach to modeling the brain network in whole brain resting-state fMRI is to compute the correlation between anatomical regions via sliding windows. However, the direct use of the sample correlation matrices is not reliable due to the image acquisition, processing noises and the use of discrete windows that often introduce spurious high-frequency fluctuations and the zig-zag pattern in the estimated time-varying correlation measures. To address the problem and obtain more robust correlation estimates, we propose the heat kernel based dynamic correlations. We demonstrate that the proposed heat kernel method can smooth out the unwanted high-frequency fluctuations in correlation estimations and achieve higher accuracy in identifying dynamically changing distinct states. The method is further used in determining if such dynamic state change is genetically heritable using a large-scale twin study. Various methodological challenges for analyzing paired twin dynamic networks are addressed.

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., Comment: Accepted for publication

Background: Recent studies have indicated that functional connectivity is dynamic even during rest. A common approach to modeling the dynamic functional connectivity in whole-brain resting-state fMRI is to compute the correlation between anatomical regions via sliding time windows. However, the direct use of the sample correlation matrices is not reliable due to the image acquisition and processing noises in resting-sate fMRI. New method: To overcome these limitations, we propose a new statistical model that smooths out the noise by exploiting the geometric structure of correlation matrices. The dynamic correlation matrix is modeled as a linear combination of symmetric positive-definite matrices combined with cosine series representation. The resulting smoothed dynamic correlation matrices are clustered into disjoint brain connectivity states using the k-means clustering algorithm. Results: The proposed model preserves the geometric structure of underlying physiological dynamic correlation, eliminates unwanted noise in connectivity and obtains more accurate state spaces. The difference in the estimated dynamic connectivity states between males and females is identified. Comparison with existing methods: We demonstrate that the proposed statistical model has less rapid state changes caused by noise and improves the accuracy in identifying and discriminating different states. Conclusions: We propose a new regression model on dynamically changing correlation matrices that provides better performance over existing windowed correlation and is more reliable for the modeling of dynamic connectivity., Comment: Accepted for publication in Journal of Neuroscience Methods

The permutation test is an often used test procedure in brain imaging. Unfortunately, generating every possible permutation for large-scale brain image datasets such as HCP and ADNI with hundreds images is not practical. Many previous attempts at speeding up the permutation test rely on various approximation strategies such as estimating the tail distribution with known parametric distributions. In this study, we show how to rapidly accelerate the permutation test without any type of approximate strategies by exploiting the underlying algebraic structure of the permutation group. The method is applied to large number of MRIs in two applications: (1) localizing the male and female differences and (2) localizing the regions of high genetic heritability in the sulcal and gyral pattern of the human cortical brain.

Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of $k$-dimensional holes and Betti numbers.The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this study, we propose a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represent the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. We also find the most contributed HHs to the network difference based on the HH dissimilarity. We applied our proposed method to clustering the networks of 4 groups, normal control (NC), stable and progressive mild cognitive impairment (sMCI and pMCI), and Alzheimer's disease (AD). The results showed that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology., Comment: The paper is accepted for publication at the 7th Workshop on Computational Topology in Image Context (CTIC) (http://www.ctic2019.uma.es)

Existing brain network distances are often based on matrix norms. The element-wise differences in the existing matrix norms may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A major disadvantage to element-wise distance calculations is that it could be severely affected even by a small number of extreme edge weights. Thus it is necessary to develop network distances that recognize topology. In this paper, we provide a survey of bottleneck, Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances that are adapted for brain networks, and compare them against matrix-norm based network distances. Bottleneck and GH-distances are often used in persistent homology. However, they were rarely utilized to measure similarity between brain networks. KS-distance is recently introduced to measure the similarity between networks across different filtration values. The performance analysis was conducted using the random network simulations with the ground truths. Using a twin imaging study, which provides biological ground truth, we demonstrate that the KS distance has the ability to determine heritability., Comment: arXiv admin note: text overlap with arXiv:1701.04171

The permutation test is known as the exact test procedure in statistics. However, often it is not exact in practice and only an approximate method since only a small fraction of every possible permutation is generated. Even for a small sample size, it often requires to generate tens of thousands permutations, which can be a serious computational bottleneck. In this paper, we propose a novel combinatorial inference procedure that enumerates all possible permutations combinatorially without any resampling. The proposed method is validated against the standard permutation test in simulation studies with the ground truth. The method is further applied in twin DTI study in determining the genetic contribution of the minimum spanning tree of the structural brain connectivity., Comment: Accepted for publication in MICCAI 2018

A key strength of twin studies arises from the fact that there are two types of twins, monozygotic and dizygotic, that share differing amounts of genetic information. Accurate differentiation of twin types allows efficient inference on genetic influences in a population. However, identification of zygosity is often prone to errors without genotying. In this study, we propose a novel pairwise feature representation to classify the zygosity of twin pairs of resting state functional magnetic resonance images (rs-fMRI). For this, we project an fMRI signal to a set of basis functions and use the projection coefficients as the compact and discriminative feature representation of noisy fMRI. We encode the relationship between twins as the correlation between the new feature representations across brain regions. We employ hill climbing variable selection to identify brain regions that are the most genetically affected. The proposed framework was applied to 208 twin pairs and achieved 94.19% classification accuracy in automatically identifying the zygosity of paired images.

We present the discrete version of heat kernel smoothing on graph data structure. The method is used to smooth data in an irregularly shaped domains in 3D images. New statistical properties are derived. As an application, we show how to filter out data in the lung blood vessel trees obtained from computed tomography. The method can be further used in representing the complex vessel trees parametrically and extracting the skeleton representation of the trees.

We explore the main characteristics of big brain network data that offer unique statistical challenges. The brain networks are biologically expected to be both sparse and hierarchical. Such unique characterizations put specific topological constraints onto statistical approaches and models we can use effectively. We explore the limitations of the current models used in the field and offer alternative approaches and explain new challenges., Comment: 8 pages, 2 figure

This paper surveys various distance measures for networks and graphs that were introduced in persistent homology. The scope of the paper is limited to network distances that were actually used in brain networks but the methods can be easily adapted to any weighted graph in other fields. The network version of Gromov-Hausdorff, bottleneck, kernel distances are introduced. We also introduce a recently developed KS-test like distance based on monotonic topology features such as the zeroth Betti number. Numerous toy examples and the result of applying many different distances to the brain networks of different clinical status and populations are given., Comment: 24 pages, 12 figures

Understanding the modularity of fMRI-derived brain networks or connectomes can inform the study of brain function organization. However, fMRI connectomes additionally involve negative edges, which are not rigorously accounted for by existing approaches to modularity that either ignores or arbitrarily weight these connections. Furthermore, most Q maximization-based modularity algorithms yield variable results with suboptimal reproducibility. Here we present an alternative, reproducible approach that exploits how frequent the BOLD-signal correlation between two nodes is negative. We validated this novel probability-based modularity approach on two independent publicly-available resting-state connectome dataset (the Human Connectome Project and the 1000 Functional Connectomes) and demonstrated that negative correlations alone are sufficient in understanding resting-state modularity. In fact, this approach a) permits a dual formulation, leading to equivalent solutions regardless of whether one considers positive or negative edges; b) is theoretically linked to the Ising model defined on the connectome, thus yielding modularity result that maximizes data likelihood. We additionally were able to detect sex differences in modularity that the most widely utilized methods did not. Results confirmed the superiority of our approach in that: a) correlations with the highest probability of being negative are consistently placed between modules, b) due to the equivalent dual forms, no arbitrary weighting factor is required to balance the influence between negative and positive correlations, unlike existing Q maximization-based modularity approaches. As datasets like HCP become widely available for analysis by the neuroscience community at large, appropriate computational tools to understand the neurobiological information of negative edges in fMRI connectomes are increasingly important.