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.

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.

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.

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.

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.

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.

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 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.

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.

Nociceptor Neurons Magnify Host Responses to Aggravate Periodontitis

Periodontitis is a highly prevalent chronic inflammatory disease that progressively destroys the structures supporting teeth, leading to tooth loss. Periodontal tissue is innervated by abundant pain-sensing primary afferents expressing neuropeptides and transient receptor potential vanilloid 1 (TRPV1). However, the roles of nociceptive nerves in periodontitis and bone destruction are controversial. The placement of ligature around the maxillary second molar or the oral inoculation of pathogenic bacteria induced alveolar bone destruction in mice. Chemical ablation of nociceptive neurons in the trigeminal ganglia achieved by intraganglionic injection of resiniferatoxin decreased bone loss in mouse models of experimental periodontitis. Consistently, ablation of nociceptive neurons decreased the number of osteoclasts in alveolar bone under periodontitis. The roles of nociceptors were also determined by the functional inhibition of TRPV1-expressing trigeminal afferents using an inhibitory designer receptor exclusively activated by designer drugs (DREADD) receptor. Noninvasive chemogenetic functional silencing of TRPV1-expressing trigeminal afferents not only decreased induction but also reduced the progression of bone loss in periodontitis. The infiltration of leukocytes and neutrophils to the periodontium increased at the site of ligature, which was accompanied by increased amount of proinflammatory cytokines, such as receptor activator of nuclear factor κΒ ligand, tumor necrosis factor, and interleukin 1β. The extents of increase in immune cell infiltration and cytokines were significantly lower in mice with nociceptor ablation. In contrast, the ablation of nociceptors did not alter the periodontal microbiome under the conditions of control and periodontitis. Altogether, these results indicate that TRPV1-expressing afferents increase bone destruction in periodontitis by promoting hyperactive host responses in the periodontium. We suggest that specific targeting of neuroimmune and neuroskeletal regulation can offer promising therapeutic targets for periodontitis supplementing conventional treatments.

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).

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.

Cardiovascular benefits outweigh risks in patients with atrial fibrillation in REDUCE-IT (Reduction of Cardiovascular Events with Icosapent Ethyl-Intervention Trial)

Background/Introduction

REDUCE-IT, a multinational, double-blind trial, randomized 8179 statin-treated patients with controlled low density lipoprotein cholesterol, elevated triglycerides, and cardiovascular (CV) risk, to icosapent ethyl (IPE) 4 grams/day or placebo. IPE reduced the primary (CV death, myocardial infarction [MI], stroke, coronary revascularization, hospitalization for unstable angina) and key secondary (CV death, MI, stroke) endpoints 25% and 26%, respectively (each p<0.0001), and individual components including stroke (28%), MI (31%), cardiac arrest (48%), and sudden cardiac death (31%) (all p≤0.01). With IPE, bleeding was greater (11.8% vs 9.9%; p=0.006), serious bleeding trended higher (2.7% vs 2.1%; p=0.06), and atrial fibrillation/flutter (AF/F) hospitalization endpoints increased (3.1% vs 2.1%; p=0.004).

Purpose

To evaluate the effects of IPE on the risk of CV events and safety measures in patients by either history of AF/F or in-study occurrence of positively adjudicated AF/F hospitalization.

Methods

Conduct post hoc efficacy and safety subgroup analyses of patients with or without either baseline history of AF/F or in-study adjudicated AF/F hospitalization, including hospitalization for ≥24 hours; AF/F not meeting endpoint criteria were reported as adverse events.

Results

Patients with (n=751; 9.2%) AF/F history at baseline (vs without; n=7428; 90.8%) (Figure 1), or those with (n=211; 2.6%) positively adjudicated in-study AF/F hospitalization endpoints (vs without; n=7968; 97.4%) (Figure 2), had higher event rates of primary, key secondary, and fatal or nonfatal stroke endpoints, but relative risk reductions with IPE were not significantly different (all interaction p-values [pint]=ns). Similar reductions were observed with IPE across the prespecified endpoint testing hierarchy in patients with or without AF/F history or in-study hospitalization endpoints. Patients with baseline AF/F history had similar relative risk for in-study occurrence of AF/F hospitalization with IPE versus placebo (pint=0.21) but had greater absolute risk (12.5% vs 6.3%, IPE vs placebo) vs patients without baseline AF/F history (2.2% vs 1.6%, IPE vs placebo); i.e., recurrent AF/F in those with a prior history of AF/F was more prevalent than de novo AF/F. Serious bleeding trended higher regardless of AF/F history or in-study AF/F hospitalization endpoints (all pint=ns); absolute risk of serious bleeding was greater in patients with AF/F history at baseline (7.3% vs 6.0%) vs those without a baseline history of AF/F (2.3% vs 1.7%), and serious bleeding also trended higher in patients with in-study AF/F hospitalization (8.7% vs 6.0%) vs without (2.5% vs 2.0%) [all IPE vs placebo].

Conclusion

REDUCE-IT patients with AF/F history or in-study AF/F hospitalization endpoints had greater CV risk, but similar relative risk reduction in primary, key secondary, and fatal or nonfatal stroke endpoints with IPE.

Funding Acknowledgement

Type of funding sources: Private company. Main funding source(s): Amarin Pharma, Inc.

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.

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.

Neural Pathways of Craniofacial Muscle Pain: Implications for Novel Treatments

Craniofacial muscle pain is highly prevalent in temporomandibular disorders but is difficult to treat. Enhanced understanding of neurobiology unique to craniofacial muscle pain should lead to the development of novel mechanism-based treatments. Herein, we review recent studies to summarize neural pathways of craniofacial muscle pain. Nociceptive afferents in craniofacial muscles are predominantly peptidergic afferents enriched with TRPV1. Signals from peripheral glutamate receptors converge onto TRPV1, leading to mechanical hyperalgesia. Further studies are needed to clarify whether hyperalgesic priming in nonpeptidergic afferents or repeated acid injections also affect craniofacial muscle pain. Within trigeminal ganglia, afferents innervating craniofacial muscles interact with surrounding satellite glia, which enhances the sensitivity of the inflamed neurons as well as nearby uninjured afferents, resulting in hyperalgesia and ectopic pain originating from adjacent orofacial tissues. Craniofacial muscle afferents project to a wide area within the trigeminal nucleus complex, and central sensitization of medullary dorsal horn neurons is a critical factor in muscle hyperalgesia related to ectopic pain and emotional stress. Second-order neurons project rostrally to pathways associated with affective pain, such as parabrachial nucleus and medial thalamic nucleus, as well as sensory-discriminative pain, such as ventral posteromedial thalamic nuclei. Abnormal endogenous pain modulation can also contribute to chronic muscle pain. Descending serotonergic circuits from the rostral ventromedial medulla facilitate activation of second-order neurons in the trigeminal nucleus complex, which leads to the maintenance of mechanical hyperalgesia of inflamed masseter muscle. Patients with temporomandibular disorders exhibit altered brain networks in widespread cortical and subcortical regions. Recent development of methods for neural circuit manipulation allows silencing of specific hyperactive neural circuits. Chemogenetic silencing of TRPV1-expressing afferents or rostral ventromedial medulla neurons attenuates hyperalgesia during masseter inflammation. It is likely, therefore, that further delineation of neural circuits mediating craniofacial muscle hyperalgesia potentially enhances treatment of chronic muscle pain conditions.

Statistical model for dynamically-changing correlation matrices with application to brain connectivity

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.

Heritability of nested hierarchical structural brain network

When a brain network is constructed by an existing parcellation method, the topological structure of the network changes depending on the scale of the parcellation. To avoid the scale dependency, we propose to construct a nested hierarchical structural brain network by subdividing the existing parcellation hierarchically. The method is applied in diffusion tensor imaging study of 111 twins in characterizing the topology of the brain network. The genetic contribution of the whole brain structural connectivity is determined and shown to be robustly present over different network scales.

STATISTICAL INFERENCE ON THE NUMBER OF CYCLES IN BRAIN NETWORKS

A cycle in a graph is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. While the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, enumerating cycles in the network is not easy and often requires brute force enumerations. In this study, we present a new scalable algorithm for enumerating the number of cycles in the network. We show that the number of cycles is monotonically decreasing with respect to the filtration values during graph filtration. We further develop a new statistical inference framework for determining the significance of the number of cycles. The methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the functional human brain network.

Exact topological inference of the resting-state brain networks in twins

A cycle in a brain network is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. Whereas the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, it is unclear how to perform statistical inference on the number of cycles in the brain network. In this study, we present a new statistical inference framework for determining the significance of the number of cycles through the Kolmogorov-Smirnov (KS) distance, which was recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using the random network simulations with ground truths. By using a twin imaging study, which provides biological ground truth, the methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the resting-state functional connectivity in 217 twins obtained from the Human Connectome Project. The MATLAB codes as well as the connectivity matrices used in generating results are provided at http://www.stat.wisc.edu/∼mchung/TDA.

In this paper, we propose a new topological distance based on the Kolmogorov-Smirnov (KS) distance that is adapted for brain networks, and compare them against other topological network distances including the Gromov-Hausdorff (GH) distances. KS-distance is recently introduced to measure the similarity between networks across different filtration values by using the zeroth Betti number, which measures the number of connected components. In this paper, we show how to extend the method to the first Betti number, which measures the number of cycles. The performance analysis was conducted using random network simulations with ground truths. Using a twin imaging study, which provides biological ground truth (of network differences), we demonstrate that the KS distances on the zeroth and first Betti numbers have the ability to determine heritability.

Altered dynamic electroencephalography connectome phase-space features of emotion regulation in social anxiety

Emotion regulation deficits are commonly observed in social anxiety disorder (SAD). We used manifold-learning to learn the phase-space connectome manifold of EEG brain dynamics in twenty SAD participants and twenty healthy controls. The purpose of the present study was to utilize manifold-learning to understand EEG brain dynamics associated with emotion regulation processes. Our emotion regulation task (ERT) contains three conditions: Neutral, Maintain and Reappraise. For all conditions and subjects, EEG connectivity data was converted into series of temporally-consecutive connectomes and aggregated to yield this phase-space manifold. As manifold geodesic distances encode intrinsic geometry, we visualized this space using its geodesic-informed minimum spanning tree and compared neurophysiological dynamics across conditions and groups using the corresponding trajectory length. Results showed that SAD participants had significantly longer trajectory lengths during Neutral and Maintain. Further, trajectory lengths during Reappraise were significantly associated with the habitual use of reappraisal strategies, while Maintain trajectory lengths were significantly associated with the negative affective state during Maintain. In sum, an unsupervised connectome manifold-learning approach can reveal emotion regulation associated phase-space features of brain dynamics.

Discrete Heat Kernel Smoothing in Irregular Image Domains

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 of heat kernel smoothing are derived. As an application, we show how to filter out noisy data in the lung blood vessel trees obtained from computed tomography. The method can be further used in representing the complex vessel trees parametrically as a linear combination of basis functions and extracting the skeleton representation of the trees.

Statistical Challenges of Big Brain Network Data

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.

Topological Data Analysis of Single-Trial Electroencephalographic Signals

Epilepsy is a neurological disorder that can negatively affect the visual, audial and motor functions of the human brain. Statistical analysis of neurophysiological recordings, such as electroencephalogram (EEG), facilitates the understanding and diagnosis of epileptic seizures. Standard statistical methods, however, do not account for topological features embedded in EEG signals. In the current study, we propose a persistent homology (PH) procedure to analyze single-trial EEG signals. The procedure denoises signals with a weighted Fourier series (WFS), and tests for topological difference between the denoised signals with a permutation test based on their PH features persistence landscapes (PL). Simulation studies show that the test effectively identifies topological difference and invariance between two signals. In an application to a single-trial multichannel seizure EEG dataset, our proposed PH procedure was able to identify the left temporal region to consistently show topological invariance, suggesting that the PH features of the Fourier decomposition during seizure is similar to the process before seizure. This finding is important because it could not be identified from a mere visual inspection of the EEG data and was in fact missed by earlier analyses of the same dataset.

Connectivity in fMRI: Blind Spots and Breakthroughs

In recent years, driven by scientific and clinical concerns, there has been an increased interest in the analysis of functional brain networks. The goal of these analyses is to better understand how brain regions interact, how this depends upon experimental conditions and behavioral measures and how anomalies (disease) can be recognized. In this paper, we provide, first, a brief review of some of the main existing methods of functional brain network analysis. But rather than compare them, as a traditional review would do, instead, we draw attention to their significant limitations and blind spots. Then, second, relevant experts, sketch a number of emerging methods, which can break through these limitations. In particular we discuss five such methods. The first two, stochastic block models and exponential random graph models, provide an inferential basis for network analysis lacking in the exploratory graph analysis methods. The other three addresses: network comparison via persistent homology, time-varying connectivity that distinguishes sample fluctuations from neural fluctuations, and network system identification that draws inferential strength from temporal autocorrelation.

A Novel Registration-Based Semiautomatic Mandible Segmentation Pipeline Using Computed Tomography Images to Study Mandibular Development

OBJECTIVE

We present a registration-based semiautomatic mandible segmentation (SAMS) pipeline designed to process a large number of computed tomography studies to segment 3-dimensional mandibles.

METHOD

The pipeline consists of a manual preprocessing step, an automatic segmentation step, and a final manual postprocessing step. The automatic portion uses a nonlinear diffeomorphic method to register each preprocessed input computed tomography test scan on 54 reference templates, ranging in age from birth to 19 years. This creates 54 segmentations, which are then combined into a single composite mandible.

RESULTS

This pipeline was assessed using 20 mandibles from computed tomography studies with ages 1 to 19 years, segmented using both SAMS-processing and manual segmentation. Comparisons between the SAMS-processed and manually-segmented mandibles revealed 97% similarity agreement with comparable volumes. The resulting 3-dimensional mandibles were further enhanced with manual postprocessing in specific regions.

CONCLUSIONS

Findings are indicative of a robust pipeline that reduces manual segmentation time by 75% and increases the feasibility of large-scale mandibular growth studies.

ABNORMAL HOLE DETECTION IN BRAIN CONNECTIVITY BY KERNEL DENSITY OF PERSISTENCE DIAGRAM AND HODGE LAPLACIAN

Community and rich-club detection are a well-known method to extract functionally specialized subnetwork in brain connectivity analysis. They find densely connected subregions with large modularity or high degree in brain connectivity studies. However, densely connected nodes are not the only representation of network shape. In this study, we propose a new method to extract abnormal holes, which are another representation of network shape. While densely connected component characterizes network's efficiency, abnormal holes characterize inefficiency. The proposed method differs from the existing hole detection in two respects. One is to use Hodge Laplacian to obtain a harmonic hole in the linear combination of edges, rather than a subset of edges. The other is to use the kernel density estimation of persistence diagram of random networks to determine the significance of a hole, rather than using the persistence of a hole. We applied the proposed method to find the abnormality of metabolic connectivity in the FDG PET data of ADNI. We found that, as AD severely progressed, the brain network had more abnormal holes. The localized holes showed how inefficient the structure of brain network became as the disease progressed.

Topological Properties of the Structural Brain Network in Autism via ϵ-Neighbor Method

OBJECTIVE

Topological characteristics of the brain can be analyzed using structural brain networks constructed by diffusion tensor imaging (DTI). When a brain network is constructed by the existing parcellation method, the structure of the network changes depending on the scale of parcellation and arbitrary thresholding. To overcome these issues, we propose to construct brain networks using the improved $\varepsilon $-neighbor construction, which is a parcellation free network construction technique.

METHODS

We acquired DTI from 14 control subjects and 15 subjects with autism. We examined the differences in topological properties of the brain networks constructed using the proposed method and the existing parcellation between the two groups.

RESULTS

As the number of nodes increased, the connectedness of the network decreased in the parcellation method. However, for brain networks constructed using the proposed method, connectedness remained at a high level even with an increase in the number of nodes. We found significant differences in several topological properties of brain networks constructed using the proposed method, whereas topological properties were not significantly different for the parcellation method.

CONCLUSION

The brain networks constructed using the proposed method are considered as more realistic than a parcellation method with respect to the stability of connectedness. We found that subjects with autism showed the abnormal characteristics in the brain networks. These results demonstrate that the proposed method may provide new insights to analysis in the structural brain network.

SIGNIFICANCE

We proposed the novel brain network construction method to overcome the shortcoming in the existing parcellation method.

The significance of negative correlations in brain connectivity

Understanding the modularity of functional magnetic resonance imaging (fMRI)-derived brain networks or "connectomes" can inform the study of brain function organization. However, fMRI connectomes additionally involve negative edges, which may not be optimally accounted for by existing approaches to modularity that variably threshold, binarize, or arbitrarily weight these connections. Consequently, many existing Q maximization-based modularity algorithms yield variable modular structures. Here, we present an alternative complementary approach that exploits how frequent the blood-oxygen-level-dependent (BOLD) signal correlation between two nodes is negative. We validated this novel probability-based modularity approach on two independent publicly-available resting-state connectome data sets (the Human Connectome Project [HCP] and the 1,000 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. Additionally, we were able to detect novel and consistent sex differences in modularity in both data sets. As data sets like HCP become widely available for analysis by the neuroscience community at large, alternative and perhaps more advantageous computational tools to understand the neurobiological information of negative edges in fMRI connectomes are increasingly important.

Topological Network Analysis of Electroencephalographic Power Maps

Meditation practice as a non-pharmacological intervention to provide health related benefits has generated much neuroscientific interest in its effects on brain activity. Electroencephalogram (EEG), an imaging modality known for its inexpensive procedure and excellent temporal resolution, is often utilized to investigate the neuroplastic effects of meditation under various experimental conditions. In these studies, EEG signals are routinely mapped on a topographic layout of channels to visualize variations in spectral powers within certain frequency ranges. Topological data analysis (TDA) of the topographic power maps modeled as graphs can provide different insight to EEG signals than standard statistical methods. A highly effective TDA technique is persistent homology, which reveals topological characteristics of a power map by tracking feature changes throughout a filtration process on the graph structure of the map. In this paper, we propose a novel inference procedure based on filtrations induced by sublevel sets of the power maps of high-density EEG signals. We apply the pipeline to simulated and real data, where we compare the persistent homological features of topographic maps of spectral powers in high-frequency bands of EEG signals recorded on long-term meditators and meditation-naive practitioners.

Topological Distances Between Brain Networks

Many existing brain network distances are based on matrix norms. The element-wise differences may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to develop network distances that recognize topology. In this paper, we introduce Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances. GH-distance is often used in persistent homology based brain network models. The superior performance of KS-distance is contrasted against matrix norms and GH-distance in random network simulations with the ground truths. The KS-distance is then applied in characterizing the multimodal MRI and DTI study of maltreated children.

Integrative Structural Brain Network Analysis in Diffusion Tensor Imaging

In diffusion tensor imaging, structural connectivity between brain regions is often measured by the number of white matter fiber tracts connecting them. Other features such as the length of tracts or fractional anisotropy (FA) are also used in measuring the strength of connectivity. In this study, we investigated the effects of incorporating the number of tracts, the tract length, and FA values into the connectivity model. Using various node-degree-based graph theory features, the three connectivity models are compared. The methods are applied in characterizing structural networks between normal controls and maltreated children, who experienced maltreatment while living in postinstitutional settings before being adopted by families in the United States.

Exact Topological Inference for Paired Brain Networks via Persistent Homology

We present a novel framework for characterizing paired brain networks using techniques in hyper-networks, sparse learning and persistent homology. The framework is general enough for dealing with any type of paired images such as twins, multimodal and longitudinal images. The exact nonparametric statistical inference procedure is derived on testing monotonic graph theory features that do not rely on time consuming permutation tests. The proposed method computes the exact probability in quadratic time while the permutation tests require exponential time. As illustrations, we apply the method to simulated networks and a twin fMRI study. In case of the latter, we determine the statistical significance of the heritability index of the large-scale reward network where every voxel is a network node.

Characterizing mandibular growth using three-dimensional imaging techniques and anatomic landmarks

OBJECTIVE

To provide quantitative data on the multi-planar growth of the mandible, this study derived accurate linear and angular mandible measurements using landmarks on three dimensional (3D) mandible models. This novel method was used to quantify 3D mandibular growth and characterize the emergence of sexual dimorphism.

DESIGN

Cross-sectional and longitudinal imaging data were obtained from a retrospective computed tomography (CT) database for 51 typically developing individuals between the ages of one and nineteen years. The software Analyze was used to generate 104 3DCT mandible models. Eleven landmarks placed on the models defined six linear measurements (lateral condyle, gonion, and endomolare width, ramus and mental depth, and mandible length) and three angular measurements (gonion, gnathion, and lingual). A fourth degree polynomial fit quantified growth trends, its derivative quantified growth rates, and a composite growth model determined growth types (neural/cranial and somatic/skeletal). Sex differences were assessed in four age cohorts, each spanning five years, to determine the ontogenetic pattern producing sexual dimorphism of the adult mandible.

RESULTS

Mandibular growth trends and growth rates were non-uniform. In general, structures in the horizontal plane displayed predominantly neural/cranial growth types, whereas structures in the vertical plane had somatic/skeletal growth types. Significant prepubertal sex differences in the inferior aspect of the mandible dissipated when growth in males began to outpace that of females at eight to ten years of age, but sexual dimorphism re-emerged during and after puberty.

CONCLUSIONS

This 3D analysis of mandibular growth provides preliminary normative developmental data for clinical assessment and craniofacial growth studies.

Integrated multimodal network approach to PET and MRI based on multidimensional persistent homology

Finding underlying relationships among multiple imaging modalities in a coherent fashion is one of the challenging problems in multimodal analysis. In this study, we propose a novel approach based on multidimensional persistence. In the extension of the previous threshold-free method of persistent homology, we visualize and discriminate the topological change of integrated brain networks by varying not only threshold but also mixing ratio between two different imaging modalities. The multidimensional persistence is implemented by a new bimodal integration method called 1D projection. When the mixing ratio is predefined, it constructs an integrated edge weight matrix by projecting two different connectivity information onto the one dimensional shared space. We applied the proposed methods to PET and MRI data from 23 attention deficit hyperactivity disorder (ADHD) children, 21 autism spectrum disorder (ASD), and 10 pediatric control subjects. From the results, we found that the brain networks of ASD, ADHD children and controls differ, with ASD and ADHD showing asymmetrical changes of connected structures between metabolic and morphological connectivities. The difference of connected structure between ASD and the controls was mainly observed in the metabolic connectivity. However, ADHD showed the maximum difference when two connectivity information were integrated with the ratio 0.6. These results provide a multidimensional homological understanding of disease-related PET and MRI networks that disclose the network association with ASD and ADHD. Hum Brain Mapp 38:1387-1402, 2017. © 2016 Wiley Periodicals, Inc.

Degree-based statistic and center persistency for brain connectivity analysis

Brain connectivity analyses have been widely performed to investigate the organization and functioning of the brain, or to observe changes in neurological or psychiatric conditions. However, connectivity analysis inevitably introduces the problem of mass-univariate hypothesis testing. Although, several cluster-wise correction methods have been suggested to address this problem and shown to provide high sensitivity, these approaches fundamentally have two drawbacks: the lack of spatial specificity (localization power) and the arbitrariness of an initial cluster-forming threshold. In this study, we propose a novel method, degree-based statistic (DBS), performing cluster-wise inference. DBS is designed to overcome the above-mentioned two shortcomings. From a network perspective, a few brain regions are of critical importance and considered to play pivotal roles in network integration. Regarding this notion, DBS defines a cluster as a set of edges of which one ending node is shared. This definition enables the efficient detection of clusters and their center nodes. Furthermore, a new measure of a cluster, center persistency (CP) was introduced. The efficiency of DBS with a known "ground truth" simulation was demonstrated. Then they applied DBS to two experimental datasets and showed that DBS successfully detects the persistent clusters. In conclusion, by adopting a graph theoretical concept of degrees and borrowing the concept of persistence from algebraic topology, DBS could sensitively identify clusters with centric nodes that would play pivotal roles in an effect of interest. DBS is potentially widely applicable to variable cognitive or clinical situations and allows us to obtain statistically reliable and easily interpretable results. Hum Brain Mapp 38:165-181, 2017. © 2016 Wiley Periodicals, Inc.

Persistent Homology in Sparse Regression and Its Application to Brain Morphometry

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in drastically speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.

Statistical Inference Models for Image Datasets with Systematic Variations

Statistical analysis of longitudinal or cross sectional brain imaging data to identify effects of neurodegenerative diseases is a fundamental task in various studies in neuroscience. However, when there are systematic variations in the images due to parameter changes such as changes in the scanner protocol, hardware changes, or when combining data from multi-site studies, the statistical analysis becomes problematic. Motivated by this scenario, the goal of this paper is to develop a unified statistical solution to the problem of systematic variations in statistical image analysis. Based in part on recent literature in harmonic analysis on diffusion maps, we propose an algorithm which compares operators that are resilient to the systematic variations. These operators are derived from the empirical measurements of the image data and provide an efficient surrogate to capturing the actual changes across images. We also establish a connection between our method to the design of wavelets in non-Euclidean space. To evaluate the proposed ideas, we present various experimental results on detecting changes in simulations as well as show how the method offers improved statistical power in the analysis of real longitudinal PIB-PET imaging data acquired from participants at risk for Alzheimer's disease (AD).

Multi-resolution statistical analysis of brain connectivity graphs in preclinical Alzheimer's disease

There is significant interest, both from basic and applied research perspectives, in understanding how structural/functional connectivity changes can explain behavioral symptoms and predict decline in neurodegenerative diseases such as Alzheimer's disease (AD). The first step in most such analyses is to encode the connectivity information as a graph; then, one may perform statistical inference on various 'global' graph theoretic summary measures (e.g., modularity, graph diameter) and/or at the level of individual edges (or connections). For AD in particular, clear differences in connectivity at the dementia stage of the disease (relative to healthy controls) have been identified. Despite such findings, AD-related connectivity changes in preclinical disease remain poorly characterized. Such preclinical datasets are typically smaller and group differences are weaker. In this paper, we propose a new multi-resolution method for performing statistical analysis of connectivity networks/graphs derived from neuroimaging data. At the high level, the method occupies the middle ground between the two contrasts - that is, to analyze global graph summary measures (global) or connectivity strengths or correlations for individual edges similar to voxel based analysis (local). Instead, our strategy derives a Wavelet representation at each primitive (connection edge) which captures the graph context at multiple resolutions. We provide extensive empirical evidence of how this framework offers improved statistical power by analyzing two distinct AD datasets. Here, connectivity is derived from diffusion tensor magnetic resonance images by running a tractography routine. We first present results showing significant connectivity differences between AD patients and controls that were not evident using standard approaches. Later, we show results on populations that are not diagnosed with AD but have a positive family history risk of AD where our algorithm helps in identifying potentially subtle differences between patient groups. We also give an easy to deploy open source implementation of the algorithm for use within studies of connectivity in AD and other neurodegenerative disorders.

4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: a holistic treatment of multiple disconnected anatomical structures

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind HyperSPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; HyperSPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM's burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures.

TOPOLOGICAL EPILEPSY SEIZURE DETECTION IN ELECTROENCEPHALOGRAPHIC SIGNALS

We propose a seizure detection method for electroencephalographic (EEG) epilepsy data based on a novel multi-scale topological technique called persistent homology (PH). Among several PH descriptors, persistence landscape (PL) possesses many desirable properties for rigorous statistical inference. By building PLs on EEG epilepsy signals smoothed by a weighted Fourier series (WFS) expansion, we compared the before and during phases of a seizure attack in a patient diagnosed with left temporal epilepsy and successfully identified site T3 as the origin of the seizure attack.

MULTI-RESOLUTION STATISTICAL ANALYSIS ON GRAPH STRUCTURED DATA IN NEUROIMAGING

Statistical data analysis plays a major role in discovering structural and functional imaging phenotypes for mental disorders such as Alzheimer's disease (AD). The goal here is to identify, ideally early on, which regions in the brain show abnormal variations with a disorder. To make the method more sensitive, we rely on a multi-resolutional perspective of the given data. Since the underlying imaging data (such as cortical surfaces and connectomes) are naturally represented in the form of weighted graphs which lie in a non-Euclidean space, we introduce recent work from the harmonics literature to derive an effective multi-scale descriptor using wavelets on graphs that characterize the local context at each data point. Using this descriptor, we demonstrate experiments where we identify significant differences between AD and control populations using cortical surface data and tractography derived graphs/networks.

LARS NETWORK FILTRATION IN THE STUDY OF EEG BRAIN CONNECTIVITY

In a brain network, weak and non-significant edge weights between nodes signal spurious connections and are often discarded by thresholding the weights. Traditional practice of thresholding edge weights at an arbitrary value can be problematic. Network filtration provides an alternative by summarizing the changes in the network topology with respect to a broad range of thresholds. A well established network filtration approach depends on the graphical-LASSO (least absolute shrinkage and selection operator) model, where a sequence of binary networks are obtained based on non-zero sparse inverse covariance (IC) estimates of partial correlations at a range of sparsity parameters. The limitation of the graphical-LASSO network model is that it relies on the structural information rather than actual entries of the sparse IC matrices and therefore can only yield approximate dynamic topological changes in the network. In the current study, we propose a new network filtration approach based on least angle regression (LARS) that gives exact filtration values at which the filtration changes and apply it to study brain connectivity in response to emotional stimuli across different age groups via electroencephalographic (EEG) data.

Unified heat kernel regression for diffusion, kernel smoothing and wavelets on manifolds and its application to mandible growth modeling in CT images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel method is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, the method is applied to characterize the localized growth pattern of mandible surfaces obtained in CT images between ages 0 and 20 by regressing the length of displacement vectors with respect to a surface template.