Persistent Homology in Sparse Regression and Its Application to Brain Morphometry

A Taxonomy of Neuroscientific Strategies Based on Interaction Orders

In recent decades, neuroscience has advanced with increasingly sophisticated strategies for recording and analysing brain activity, enabling detailed investigations into the roles of functional units, such as individual neurons, brain regions and their interactions. Recently, new strategies for the investigation of cognitive functions regard the study of higher order interactions-that is, the interactions involving more than two brain regions or neurons. Although methods focusing on individual units and their interactions at various levels offer valuable and often complementary insights, each approach comes with its own set of limitations. In this context, a conceptual map to categorize and locate diverse strategies could be crucial to orient researchers and guide future research directions. To this end, we define the spectrum of orders of interaction, namely, a framework that categorizes the interactions among neurons or brain regions based on the number of elements involved in these interactions. We use a simulation of a toy model and a few case studies to demonstrate the utility and the challenges of the exploration of the spectrum. We conclude by proposing future research directions aimed at enhancing our understanding of brain function and cognition through a more nuanced methodological framework.

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.

Adversarially Trained Persistent Homology Based Graph Convolutional Network for Disease Identification Using Brain Connectivity

Brain disease propagation is associated with characteristic alterations in the structural and functional connectivity networks of the brain. To identify disease-specific network representations, graph convolutional networks (GCNs) have been used because of their powerful graph embedding ability to characterize the non-Euclidean structure of brain networks. However, existing GCNs generally focus on learning the discriminative region of interest (ROI) features, often ignoring important topological information that enables the integration of connectome patterns of brain activity. In addition, most methods fail to consider the vulnerability of GCNs to perturbations in network properties of the brain, which considerably degrades the reliability of diagnosis results. In this study, we propose an adversarially trained persistent homology-based graph convolutional network (ATPGCN) to capture disease-specific brain connectome patterns and classify brain diseases. First, the brain functional/structural connectivity is constructed using different neuroimaging modalities. Then, we develop a novel strategy that concatenates the persistent homology features from a brain algebraic topology analysis with readout features of the global pooling layer of a GCN model to collaboratively learn the individual-level representation. Finally, we simulate the adversarial perturbations by targeting the risk ROIs from clinical prior, and incorporate them into a training loop to evaluate the robustness of the model. The experimental results on three independent datasets demonstrate that ATPGCN outperforms existing classification methods in disease identification and is robust to minor perturbations in network architecture. Our code is available at https://github.com/CYB08/ATPGCN.

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 analysis of functional connectivity in Parkinson's disease

Parkinson's disease (PD) is a clinically heterogeneous disorder, which mainly affects patients' motor and non-motor function. Functional connectivity was preliminary explored and studied through resting state functional magnetic resonance imaging (rsfMRI). Through the topological analysis of 54 PD scans and 31 age-matched normal controls (NC) in the Neurocon dataset, leveraging on rsfMRI data, the brain functional connection and the Vietoris-Rips (VR) complex were constructed. The barcodes of the complex were calculated to reflect the changes of functional connectivity neural circuits (FCNC) in brain network. The 0-dimensional Betti number β0 means the number of connected branches in VR complex. The average number of connected branches in PD group was greater than that in NC group when the threshold δ ≤ 0.7. Two-sample Mann-Whitney U test and false discovery rate (FDR) correction were used for statistical analysis to investigate the FCNC changes between PD and NC groups. In PD group, under threshold of 0.7, the number of FCNC involved was significantly differences and these brain regions include the Cuneus_R, Lingual_R, Fusiform_R and Heschl_R. There are also significant differences in brain regions in the Frontal_Inf_Orb_R and Pallidum_R, when the threshold increased to 0.8 and 0.9 (p < 0.05). In addition, when the length of FCNC was medium, there was a significant statistical difference between the PD group and the NC group in the Neurocon dataset and the Parkinson's Progression Markers Initiative (PPMI) dataset. Topological analysis based on rsfMRI data may provide comprehensive information about the changes of FCNC and may provide an alternative for clinical differential diagnosis.

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 analysis of brain dynamics in autism based on graph and persistent homology

Autism spectrum disorder (ASD) is a heterogeneous disorder with a rapidly growing prevalence. In recent years, the dynamic functional connectivity (DFC) technique has been used to reveal the transient connectivity behavior of ASDs' brains by clustering connectivity matrices in different states. However, the states of DFC have not been yet studied from a topological point of view. In this paper, this study was performed using global metrics of the graph and persistent homology (PH) and resting-state functional magnetic resonance imaging (fMRI) data. The PH has been recently developed in topological data analysis and deals with persistent structures of data. The structural connectivity (SC) and static FC (SFC) were also studied to know which one of the SC, SFC, and DFC could provide more discriminative topological features when comparing ASDs with typical controls (TCs). Significant discriminative features were only found in states of DFC. Moreover, the best classification performance was offered by persistent homology-based metrics and in two out of four states. In these two states, some networks of ASDs compared to TCs were more segregated and isolated (showing the disruption of network integration in ASDs). The results of this study demonstrated that topological analysis of DFC states could offer discriminative features which were not discriminative in SFC and SC. Also, PH metrics can provide a promising perspective for studying ASD and finding candidate biomarkers.

Method for persistent topological features extraction of schizophrenia patients' electroencephalography signal based on persistent homology

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris-Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors.

Potential grouping of nodes induced by higher-order structures in complex networks

Complex networks display an organization of elements into nontrivial structures at versatile inherent scales, imposing challenges on a more complete understanding of their behavior. The interest of the research presented here is in the characterization of potential mesoscale structures as building blocks of generalized communities in complex networks, with an integrated property that goes beyond the pairwise collections of nodes. For this purpose, a simplicial complex is obtained from a mathematical graph, and indirectly from time series, producing the so-called clique complex from the complex network. As the higher-order organizational structures are naturally embedded in the hierarchical strata of a simplicial complex, the relationships between aggregation of nodes are stored in the higher-order combinatorial Laplacian. Based on the postulate that aggregation of nodes represents integrated configuration of information, the observability parameter is defined for the characterization of potential configurations, computed from the entries of the combinatorial Laplacian matrix. The framework introduced here is used to characterize nontrivial inherent organizational patterns embedded in two real-world complex networks and three complex networks obtained from heart rate time series recordings of three different subject's meditative states.

From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research.

Representation Learning for Fine-Grained Change Detection

Fine-grained change detection in sensor data is very challenging for artificial intelligence though it is critically important in practice. It is the process of identifying differences in the state of an object or phenomenon where the differences are class-specific and are difficult to generalise. As a result, many recent technologies that leverage big data and deep learning struggle with this task. This review focuses on the state-of-the-art methods, applications, and challenges of representation learning for fine-grained change detection. Our research focuses on methods of harnessing the latent metric space of representation learning techniques as an interim output for hybrid human-machine intelligence. We review methods for transforming and projecting embedding space such that significant changes can be communicated more effectively and a more comprehensive interpretation of underlying relationships in sensor data is facilitated. We conduct this research in our work towards developing a method for aligning the axes of latent embedding space with meaningful real-world metrics so that the reasoning behind the detection of change in relation to past observations may be revealed and adjusted. This is an important topic in many fields concerned with producing more meaningful and explainable outputs from deep learning and also for providing means for knowledge injection and model calibration in order to maintain user confidence.

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.

Effect of APOE ε4 on multimodal brain connectomic traits: a persistent homology study

BACKGROUND

Although genetic risk factors and network-level neuroimaging abnormalities have shown effects on cognitive performance and brain atrophy in Alzheimer's disease (AD), little is understood about how apolipoprotein E (APOE) ε4 allele, the best-known genetic risk for AD, affect brain connectivity before the onset of symptomatic AD. This study aims to investigate APOE ε4 effects on brain connectivity from the perspective of multimodal connectome.

RESULTS

Here, we propose a novel multimodal brain network modeling framework and a network quantification method based on persistent homology for identifying APOE ε4-related network differences. Specifically, we employ sparse representation to integrate multimodal brain network information derived from both the resting state functional magnetic resonance imaging (rs-fMRI) data and the diffusion-weighted magnetic resonance imaging (dw-MRI) data. Moreover, persistent homology is proposed to avoid the ad hoc selection of a specific regularization parameter and to capture valuable brain connectivity patterns from the topological perspective. The experimental results demonstrate that our method outperforms the competing methods, and reasonably yields connectomic patterns specific to APOE ε4 carriers and non-carriers.

CONCLUSIONS

We have proposed a multimodal framework that integrates structural and functional connectivity information for constructing a fused brain network with greater discriminative power. Using persistent homology to extract topological features from the fused brain network, our method can effectively identify APOE ε4-related brain connectomic biomarkers.

[Brain functional network reconstruction based on compressed sensing and fast iterative shrinkage-thresholding algorithm]

The construction of brain functional network based on resting-state functional magnetic resonance imaging (fMRI) is an effective method to reveal the mechanism of human brain operation, but the common brain functional network generally contains a lot of noise, which leads to wrong analysis results. In this paper, the least absolute shrinkage and selection operator (LASSO) model in compressed sensing is used to reconstruct the brain functional network. This model uses the sparsity of L1-norm penalty term to avoid over fitting problem. Then, it is solved by the fast iterative shrinkage-thresholding algorithm (FISTA), which updates the variables through a shrinkage threshold operation in each iteration to converge to the global optimal solution. The experimental results show that compared with other methods, this method can improve the accuracy of noise reduction and reconstruction of brain functional network to more than 98%, effectively suppress the noise, and help to better explore the function of human brain in noisy environment.

White Matter Brain Network Research in Alzheimer's Disease Using Persistent Features

Despite the severe social burden caused by Alzheimer’s disease (AD), no drug than can change the disease progression has been identified yet. The structural brain network research provides an opportunity to understand physiological deterioration caused by AD and its precursor, mild cognitive impairment (MCI). Recently, persistent homology has been used to study brain network dynamics and characterize the global network organization. However, it is unclear how these parameters reflect changes in structural brain networks of patients with AD or MCI. In this study, our previously proposed persistent features and various traditional graph-theoretical measures are used to quantify the topological property of white matter (WM) network in 150 subjects with diffusion tensor imaging (DTI). We found significant differences in these measures among AD, MCI, and normal controls (NC) under different brain parcellation schemes. The decreased network integration and increased network segregation are presented in AD and MCI. Moreover, the persistent homology-based measures demonstrated stronger statistical capability and robustness than traditional graph-theoretic measures, suggesting that they represent a more sensitive approach to detect altered brain structures and to better understand AD symptomology at the network level. These findings contribute to an increased understanding of structural connectome in AD and provide a novel approach to potentially track the progression of AD.

Topological View of Flows Inside the BOLD Spontaneous Activity of the Human Brain

Spatio-temporal brain activities with variable delay detectable in resting-state functional magnetic resonance imaging (rs-fMRI) give rise to highly reproducible structures, termed cortical lag threads, that propagate from one brain region to another. Using a computational topology of data approach, we found that persistent, recurring blood oxygen level dependent (BOLD) signals in triangulated rs-fMRI videoframes display previously undetected topological findings, i.e., vortex structures that cover brain activated regions. Measure of persistence of vortex shapes in BOLD signal propagation is carried out in terms of Betti numbers that rise and fall over time during spontaneous activity of the brain. Importantly, a topology of data given in terms of geometric shapes of BOLD signal propagation offers a practical approach in coping with and sidestepping massive noise in neurodata, such as unwanted dark (low intensity) regions in the neighborhood of non-zero BOLD signals. Our findings have been codified and visualized in plots able to track the non-trivial BOLD signals that appear intermittently in a sequence of rs-fMRI videoframes. The end result of this tracking of changing lag structures is a so-called persistent barcode, which is a pictograph that offers a convenient visual means of exhibiting, comparing, and classifying brain activation patterns.

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.

Metabolic Brain Network Analysis of FDG-PET in Alzheimer's Disease Using Kernel-Based Persistent Features

Recent research of persistent homology in algebraic topology has shown that the altered network organization of human brain provides a promising indicator of many neuropsychiatric disorders and neurodegenerative diseases. However, the current slope-based approach may not accurately characterize changes of persistent features over graph filtration because such curves are not strictly linear. Moreover, our previous integrated persistent feature (IPF) works well on an rs-fMRI cohort while it has not yet been studied on metabolic brain networks. To address these issues, we propose a novel univariate network measurement, kernel-based IPF (KBI), based on the prior IPF, to quantify the difference between IPF curves. In our experiments, we apply the KBI index to study fluorodeoxyglucose positron emission tomography (FDG-PET) imaging data from 140 subjects with Alzheimer's disease (AD), 280 subjects with mild cognitive impairment (MCI), and 280 healthy normal controls (NC). The results show the disruption of network integration in the progress of AD. Compared to previous persistent homology-based measures, as well as other standard graph-based measures that characterize small-world organization and modular structure, our proposed network index KBI possesses more significant group difference and better classification performance, suggesting that it may be used as an effective preclinical AD imaging biomarker.

A concise and persistent feature to study brain resting-state network dynamics: Findings from the Alzheimer's Disease Neuroimaging Initiative

Alzheimer's disease (AD) is the most common type of dementia in the elderly with no effective treatment currently. Recent studies of noninvasive neuroimaging, resting-state functional magnetic resonance imaging (rs-fMRI) with graph theoretical analysis have shown that patients with AD and mild cognitive impairment (MCI) exhibit disrupted topological organization in large-scale brain networks. In previous work, it is a common practice to threshold such networks. However, it is not only difficult to make a principled choice of threshold values, but also worse is the discard of potential important information. To address this issue, we propose a threshold-free feature by integrating a prior persistent homology-based topological feature (the zeroth Betti number) and a newly defined connected component aggregation cost feature to model brain networks over all possible scales. We show that the induced topological feature (Integrated Persistent Feature) follows a monotonically decreasing convergence function and further propose to use its slope as a concise and persistent brain network topological measure. We apply this measure to study rs-fMRI data from the Alzheimer's Disease Neuroimaging Initiative and compare our approach with five other widely used graph measures across five parcellation schemes ranging from 90 to 1,024 region-of-interests. The experimental results demonstrate that the proposed network measure shows more statistical power and stronger robustness in group difference studies in that the absolute values of the proposed measure of AD are lower than MCI and much lower than normal controls, providing empirical evidence for decreased functional integration in AD dementia and MCI.

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.

On the Reliability of Individual Brain Activity Networks

There is intense interest in fMRI research on whole-brain functional connectivity, and however, two fundamental issues are still unresolved: the impact of spatiotemporal data resolution (spatial parcellation and temporal sampling) and the impact of the network construction method on the reliability of functional brain networks. In particular, the impact of spatiotemporal data resolution on the resulting connectivity findings has not been sufficiently investigated. In fact, a number of studies have already observed that functional networks often give different conclusions across different parcellation scales. If the interpretations from functional networks are inconsistent across spatiotemporal scales, then the whole validity of the functional network paradigm is called into question. This paper investigates the consistency of resting state network structure when using different temporal sampling or spatial parcellation, or different methods for constructing the networks. To pursue this, we develop a novel network comparison framework based on persistent homology from a topological data analysis. We use the new network comparison tools to characterize the spatial and temporal scales under which consistent functional networks can be constructed. The methods are illustrated on Human Connectome Project data, showing that the DISCOH₂ network construction method outperforms other approaches at most data spatiotemporal resolutions.

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.

Dynamic network model with continuous valued nodes for longitudinal brain morphometry

Longitudinal brain morphometry probes time-related brain morphometric patterns. We propose a method called dynamic network modeling with continuous valued nodes to generate a dynamic brain network from continuous valued longitudinal morphometric data. The mathematical framework of this method is based on state-space modeling. We use a bootstrap-enhanced least absolute shrinkage operator to solve the network-structure generation problem. In contrast to discrete dynamic Bayesian network modeling, the proposed method enables network generation directly from continuous valued high-dimensional short sequence data, being free from any discretization process. We applied the proposed method to a study of normal brain development.

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.

Structure-Function Network Mapping and Its Assessment via Persistent Homology

Understanding the relationship between brain structure and function is a fundamental problem in network neuroscience. This work deals with the general method of structure-function mapping at the whole-brain level. We formulate the problem as a topological mapping of structure-function connectivity via matrix function, and find a stable solution by exploiting a regularization procedure to cope with large matrices. We introduce a novel measure of network similarity based on persistent homology for assessing the quality of the network mapping, which enables a detailed comparison of network topological changes across all possible thresholds, rather than just at a single, arbitrary threshold that may not be optimal. We demonstrate that our approach can uncover the direct and indirect structural paths for predicting functional connectivity, and our network similarity measure outperforms other currently available methods. We systematically validate our approach with (1) a comparison of regularized vs. non-regularized procedures, (2) a null model of the degree-preserving random rewired structural matrix, (3) different network types (binary vs. weighted matrices), and (4) different brain parcellation schemes (low vs. high resolutions). Finally, we evaluate the scalability of our method with relatively large matrices (2514x2514) of structural and functional connectivity obtained from 12 healthy human subjects measured non-invasively while at rest. Our results reveal a nonlinear structure-function relationship, suggesting that the resting-state functional connectivity depends on direct structural connections, as well as relatively parsimonious indirect connections via polysynaptic pathways.