Topological state-space estimation of functional human brain networks.

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

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.

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.

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.

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.

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.

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.

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.

Grading and interpretation of white matter hyperintensities using statistical maps.

BACKGROUND AND PURPOSE: We aimed to generate rigorous graphical and statistical reference data based on volumetric measurements for assessing the relative severity of white matter hyperintensities (WMHs) in patients with stroke. METHODS: We prospectively mapped WMHs from 2699 patients with first-ever ischemic stroke (mean age=66.8±13.0 years) enrolled consecutively from 11 nationwide stroke centers, from patient (fluid-attenuated-inversion-recovery) MRIs onto a standard brain template set. Using multivariable analyses, we assessed the impact of major (age/hypertension) and minor risk factors on WMH variability. RESULTS: We have produced a large reference data library showing the location and quantity of WMHs as topographical frequency-volume maps. This easy-to-use graphical reference data set allows the quantitative estimation of the severity of WMH as a percentile rank score. For all patients (median age=69 years), multivariable analysis showed that age, hypertension, atrial fibrillation, and left ventricular hypertrophy were independently associated with increasing WMH (0-9.4%, median=0.6%, of the measured brain volume). For younger (≤69) hypertensives (n=819), age and left ventricular hypertrophy were positively associated with WMH. For older (≥70) hypertensives (n=944), age and cholesterol had positive relationships with WMH, whereas diabetes mellitus, hyperlipidemia, and atrial fibrillation had negative relationships with WMH. For younger nonhypertensives (n=578), age and diabetes mellitus were positively related to WMH. For older nonhypertensives (n=328), only age was positively associated with WMH. CONCLUSIONS: We have generated a novel graphical WMH grading (Kim statistical WMH scoring) system, correlated to risk factors and adjusted for age/hypertension. Further studies are required to confirm whether the combined data set allows grading of WMH burden in individual patients and a tailored patient-specific interpretation in ischemic stroke-related clinical practice.

Tracing the evolution of multi-scale functional networks in a mouse model of depression using persistent brain network homology.

Many brain diseases or disorders, such as depression, are known to be associated with abnormal functional connectivity in neural networks in the brain. Some bivariate measures of electroencephalography (EEG) for coupling analysis have been used widely in attempts to explain abnormalities related with depression. However, brain network evolution based on persistent functional connections in EEG signals could not be easily unveiled. For a geometrical exploration of brain network evolution, here, we used persistent brain network homology analysis with EEG signals from a corticosterone (CORT)-induced mouse model of depression. EEG signals were obtained from eight cortical regions (frontal, somatosensory, parietal, and visual cortices in each hemisphere). The persistent homology revealed a significantly different functional connectivity between the control and CORT model, but no differences in common coupling measures, such as cross correlation and coherence, were apparent. The CORT model showed a more localized connectivity and decreased global connectivity than the control. In particular, the somatosensory and parietal cortices were loosely connected in the CORT model. Additionally, the CORT model displayed altered connections among the cortical regions, especially between the frontal and somatosensory cortices, versus the control. This study demonstrates that persistent homology is useful for brain network analysis, and our results indicate that the CORT-induced depression mouse model shows more localized and decreased global connectivity with altered connections, which may facilitate characterization of the abnormal brain network underlying depression.

Multi-resolutional shape features via non-Euclidean wavelets: applications to statistical analysis of cortical thickness.

Statistical analysis on arbitrary surface meshes such as the cortical surface is an important approach to understanding brain diseases such as Alzheimer's disease (AD). Surface analysis may be able to identify specific cortical patterns that relate to certain disease characteristics or exhibit differences between groups. Our goal in this paper is to make group analysis of signals on surfaces more sensitive. To do this, we derive multi-scale shape descriptors that characterize the signal around each mesh vertex, i.e., its local context, at varying levels of resolution. In order to define such a shape descriptor, we make use of recent results from harmonic analysis that extend traditional continuous wavelet theory from the Euclidean to a non-Euclidean setting (i.e., a graph, mesh or network). Using this descriptor, we conduct experiments on two different datasets, the Alzheimer's Disease NeuroImaging Initiative (ADNI) data and images acquired at the Wisconsin Alzheimer's Disease Research Center (W-ADRC), focusing on individuals labeled as having Alzheimer's disease (AD), mild cognitive impairment (MCI) and healthy controls. In particular, we contrast traditional univariate methods with our multi-resolution approach which show increased sensitivity and improved statistical power to detect a group-level effects. We also provide an open source implementation.

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

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

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

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

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

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

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

Structural variations in prefrontal cortex mediate the relationship between early childhood stress and spatial working memory.

A large corpus of research indicates that exposure to stress impairs cognitive abilities, specifically executive functioning dependent on the prefrontal cortex (PFC). We collected structural MRI scans (n = 61), well-validated assessments of executive functioning, and detailed interviews assessing stress exposure in humans to examine whether cumulative life stress affected brain morphometry and one type of executive functioning, spatial working memory, during adolescence-a critical time of brain development and reorganization. Analysis of variations in brain structure revealed that cumulative life stress and spatial working memory were related to smaller volumes in the PFC, specifically prefrontal gray and white matter between the anterior cingulate and the frontal poles. Mediation analyses revealed that individual differences in prefrontal volumes accounted for the association between cumulative life stress and spatial working memory. These results suggest that structural changes in the PFC may serve as a mediating mechanism through which greater cumulative life stress engenders decrements in cognitive functioning.

Bessel Fourier Orientation Reconstruction (BFOR): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices.

The ensemble average propagator (EAP) describes the 3D average diffusion process of water molecules, capturing both its radial and angular contents. The EAP can thus provide richer information about complex tissue microstructure properties than the orientation distribution function (ODF), an angular feature of the EAP. Recently, several analytical EAP reconstruction schemes for multiple q-shell acquisitions have been proposed, such as diffusion propagator imaging (DPI) and spherical polar Fourier imaging (SPFI). In this study, a new analytical EAP reconstruction method is proposed, called Bessel Fourier Orientation Reconstruction (BFOR), whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition, and is validated on both synthetic and real datasets. A significant portion of the paper is dedicated to comparing BFOR, SPFI, and DPI using hybrid, non-Cartesian sampling for multiple b-value acquisitions. Ways to mitigate the effects of Gibbs ringing on EAP reconstruction are also explored. In addition to analytical EAP reconstruction, the aforementioned modeling bases can be used to obtain rotationally invariant q-space indices of potential clinical value, an avenue which has not yet been thoroughly explored. Three such measures are computed: zero-displacement probability (Po), mean squared displacement (MSD), and generalized fractional anisotropy (GFA).

Early neglect is associated with alterations in white matter integrity and cognitive functioning.

Cognitive deficits have been reported in children who experienced early neglect, especially children raised in institutionalized settings. Previous research suggests that early neglect may differentially affect the directional organization of white matter in the prefrontal cortex (PFC). This may be one mechanism to explain cognitive deficits associated with neglect. To test this idea, properties of white matter and neurocognitive performance were assessed in children who suffered early neglect and those raised in typical environments (n = 63, Mage  = 11.75 years). As predicted, prefrontal white matter microstructure was affected, consistent with more diffuse organization, in children that suffered early neglect and this was related to neurocognitive deficits. Such findings underscore how early adversity may affect the PFC and explain cognitive deficits associated with neglect.

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.

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.