Evaluating the Strength of Structural Connectivity Underlying Brain Functional Networks

Abstract

In recent years, there has been strong interest in neuroscience studies to investigate brain organization through networks of brain regions that demonstrate strong functional connectivity (FC). Several well-known functional networks have been consistently identified in both task-related and resting-state functional magnetic resonance imaging (rs-fMRI) across different study populations. These networks are extracted from observed fMRI using data-driven analytic methods such as independent component analysis. A notable limitation of these FC methods is that they do not include or provide any information on the underlying structural connectivity (SC), which is believed to serve as the basis for interregional interactions in brain activity. We propose a new statistical measure of the strength of SC (sSC) underlying FC networks obtained from data-driven methods. The sSC is developed using information from diffusion tensor imaging (DTI) data. A key advantage of sSC is that it is a standardized coefficient that adjusts for the different number of voxels and baseline SC of various functional networks. Hence, sSC can be applied to compare the strength of structural connections across different FC networks. Furthermore, we propose a reliability index for data-driven FC networks to measure the reproducibility of the networks through resampling the observed data. By evaluating the association between the sSC and reliability index, we can investigate whether underlying SC informs the reliability of identified FC networks. To perform statistical inference such as hypothesis testing on the sSC, we develop a formal variance estimator of sSC based on a spatial semivariogram model with a novel distance metric. We demonstrate the performance of the sSC measure and its estimation and inference methods with simulation studies. For real data analysis, we apply our methods to a multimodal imaging study with rs-fMRI and DTI data from 20 healthy controls and 20 subjects with major depressive disorder. Results show that well-known resting-state networks all demonstrate higher SC within the network compared with the average structural connections across the brain. We also found that sSC is positively associated with the reliability index, indicating that FC networks that have stronger underlying SC are more reproducible across samples. These results provide evidence that structural connections do serve as structural basis for the FC networks and that the structural information from DTI data can be leveraged to inform the reliability of functional networks derived through data-driven methods.

Introduction

In recent years, network-oriented analysis has become a common approach in neuroimaging studies involving functional magnetic resonance imaging (fMRI). A long-standing neurophysiologic principle holds that neural processing utilizes functionally specialized areas in the brain, which interact as components of highly complex networks. An important objective of many neuroimaging analyses is to spatially organize brain activity into distributed systems by evaluating the relatedness or correlations between spatially remote neurophysiologic events (Friston et al., 1993), also called functional connectivity (FC). These distributed systems are known as brain functional networks (Smith et al., 2009; Wang et al., 2016), or FC networks, that represent a set of spatially disjoint regions in the brain that demonstrate coherent temporal dynamics in fMRI blood oxygen level-dependent (BOLD) signals. The identified functional networks may relate to resting-state brain activity or to neural activity associated with intrinsic neural processing and cognitive, emotional, visual, and motor functions.

Numerous statistical approaches have been applied to evaluate FC in network-oriented analysis. These methods can be generally grouped into two classes of approaches, namely correlation-based and partitioning methods. The correlation procedures quantify dependence between spatially distinct brain regions with correlations or partial correlations based on fMRI BOLD time series (Hampson et al., 2002; Kemmer et al., 2015; Wang et al., 2016). The second class of data-driven FC approaches uses partitioning algorithms to identify spatially distinct components or clusters in the brain, with each component containing voxels exhibiting similar neural processing characteristics over time. For example, cluster data analysis was one of the first partitioning methods that had applications in fMRI studies (Bowman and Patel, 2004; Cordes et al., 2002).

In recent years, independent component analysis (ICA) has become the most frequently used partitioning method to identify brain functional networks (Beckmann and Smith, 2004, 2005; Calhoun et al., 2001; Guo and Pagnoni, 2008; Lukemire et al., 2018; Shi and Guo, 2016; Wang and Guo, 2018). Based on the approaches such as ICA, some primary brain functional networks have been consistently identified across studies such as the default mode network, visual network, and sensorimotor network (Smith et al., 2009).

For the FC networks, it is presumed that axonal fiber tracts serve as the basis for interregional interactions in brain activity. However, the nature of this structure/function relationship is only beginning to be revealed. Evidence of the structural connectivity (SC) and FC relationship has emerged from computational work as well as clinical observations. For example, computational research has suggested that the underlying anatomical architecture of the cerebral cortex shapes resting-state FC on multiple timescales (Ghosh et al., 2008; Honey et al., 2007). Earlier clinical observations have revealed that interhemispheric FC is directly related to callosal integrity (Putnam et al., 2008; Quigley et al., 2003). Recent advances in diffusion MRI now provide a direct approach to empirically examine the structure/function relationship in human brains.

In recent work, joint analysis of fMRI and diffusion tensor imaging (DTI) data has provided evidence suggesting a general correspondence between FC (measured by correlation-based FC) and SC (measured by diffusion tractography) (Hagmann et al., 2008; Honey et al., 2009; Koch et al., 2002; Várkuti et al., 2011; Ward et al., 2014; Zhu et al., 2014). There has been work (Bowman et al., 2012; Xue et al., 2015) that jointly analyzes fMRI and DTI data to more accurately determine FC while incorporating direct SC. Goñi and associates (2014) utilize network metrics measured from anatomical network and predict FC network based on those metrics. Mueller and associates (2013) offer a new approach for comparing FC and SC by directly comparing their variability. Chamberland and associates (2017) jointly study SC and FC using a recently proposed test/retest approach that better quantifies intersubject variability by first correcting for intrasubject nuisance variability that can artificially influence FC and SC measures.

Several methods based on Gaussian graphical modeling have been developed to estimate sparse FC networks from fMRI data while using the SC to inform the selection of the shrinkage parameters for the FC precision matrix (Higgins et al., 2018; Hinne et al., 2014; Ng et al., 2012). In particular, the recently proposed structurally informed Gaussian graphical model (Higgins et al., 2018) provides a flexible approach for accommodating heterogeneity in the structure/function relationship and enables robust FC network estimation even in the presence of misspecified anatomical knowledge.

Given the significant development in joint analysis of FC and SC, some important questions still remain. First, it is important to examine whether and how the strength of SC (sSC) varies across functional networks, which will help us to understand the difference in anatomical structure underlying various FC networks. Second, it will be insightful to determine whether the sSC is associated with the reliability or robustness of FC networks. Researchers have observed that some functional networks are more reproducible or robust across studies and samples. It will be interesting to investigate whether the difference in underlying SC plays a role in this robustness. Third, it is beneficial to explore whether the sSC underlying each of the FC networks differs between subpopulations, such as subjects with psychiatric or neurologic diseases versus healthy controls.

To address these objectives, we need to develop a meaningful measure of within-network sSC that is comparable across FC networks and also across subjects. Currently, a commonly used approach to summarize the SC derived from a probabilistic tractography algorithm based on DTI data is to construct a structural network based on the SC and then calculate some network metrics such as average degree or clustering coefficient to quantify the SC property for functional networks (Achard and Bullmore, 2007; Li et al., 2009; Tao et al., 2015). These methods focus on summarizing network topology measures, which are different from our goal of quantifying the sSC within an FC network.

A more relevant approach to define the sSC is to use a raw measure of within-network SC, which is the sum of the count or proportion of fiber track connections from the probabilistic tractography for all pairs of nodes within an FC network. However, this raw within-network SC measure has limited comparability across networks and across subjects. First, it does not account for the difference in the size or number of voxels of the brain regions that constitute the FC networks. Second, the raw within-network SC measure does not account for the variations in the baseline SC across different FC networks.

In this study, baseline SC represents the average sSC between a brain location within an FC network and a random location in the brain, which can be either within or outside the FC network. FC networks that consist of regions adjacent to high-traffic white matter areas where there is a high density of fiber bundles often have higher baseline SC. Consequently, these FC networks tend to show higher raw SC measures compared with other FC networks that are not located adjacent to major fiber bundles, even if the other networks actually have stronger and more coherent within-network SC relative to their baseline SC. We have demonstrated the limitations of the raw within-network SC measure via simulation studies (Appendix A3). Moreover, the raw within-network SC measure may have limited comparability across subjects due to the differences in baseline SC between subjects.

We present a novel measure of the sSC underlying a functional network, which can help us understand the structural underpinnings of functional interactions in the brain. We use this measure to characterize the sSC underlying FC networks estimated by a data-driven FC method such as ICA. A key advantage of our sSC is that it is a standardized coefficient that adjusts for the different number of voxels and for baseline SC related to a functional network. This standardization makes sSC a more suitable measure to compare the strength of structural connections across different FC networks and to investigate differences in SC between subpopulations.

We develop an estimator for sSC based on DTI probabilistic tractography. We develop a variance estimator, based on a parametric semivariogram model with a novel distance metric, to quantify the variability of the estimated sSC. For inferences, we develop three statistical tests for addressing various questions in FC and SC analysis. The first test evaluates the significance of sSC of a given functional network. The second examines the difference in SC between functional networks. The third test evaluates whether there is a significant difference in sSC between subject groups. We conduct simulation studies to evaluate the performance of our proposed measure and apply our method to a resting-state fMRI (rs-fMRI) and DTI data set.

The proposed sSC measure is applicable to investigate whether underlying SC informs the reliability of FC networks derived from data-driven methods. One issue with data-driven FC analysis is that the estimated FC networks often vary across samples or methods. For example, the results from an ICA run may vary based on the choice of algorithm, starting values, subject variability, or data preprocessing steps (Calhoun et al., 2004). Some functional networks may not be reproducible in different analyses or in other data sets. Determining the reliability of the estimated ICs obtained from a given data set is important for appropriate interpretations of the extracted networks.

Previous work has proposed assessing reliability of the estimated networks by evaluating the reproducibility in repeated ICA runs, with different initial conditions, with data resampled from the original functional time series, or with simulated data having a known brain network structure (Duann et al., 2006; Himberg et al., 2004; Meinecke et al., 2002). These approaches measure the reliability of networks only through the algorithmic and statistical reproducibility of ICA applied to fMRI data. The goal in this article is to evaluate whether the structural connections underlying these FC networks are related to the reliability of these networks. To this end, we propose a reliability index that measures the reproducibility of the estimated functional networks based on bootstrapped samples. We then evaluate the association between the sSC and the reliability index across functional networks.

Several well-known functional networks have been consistently identified in both task and rs-fMRI studies across different populations (Buckner et al., 2008; Damoiseaux et al., 2006; Laird et al., 2011; Smith et al., 2009). Some networks are identified more robustly than others, although it is unclear why. Our proposed measure of sSC will allow us to investigate the underlying SC of different functional networks, and to better understand the differences in their anatomical underpinnings, reliability, and other aspects of their nature.

Materials and Methods

Data

Subjects

We apply our method to the rs-fMRI data from a total of 40 subjects, including 20 healthy controls and 20 patients with major depressive disorder (MDD), from the Predictors of Remission in Depression to Individual and Combined Treatments (PReDICT) study (Dunlop et al., 2012). We estimated FC networks from healthy subjects. They were an average age of 42.4 years (SD: 9.0) and were 50% male.

Using the rs-fMRI and DTI data collected from this group of subjects, we used the proposed sSC measure to evaluate the SC underlying the extracted FC networks and investigated the association between sSC and reliability measure. In addition, we compared sSC between the 20 healthy subjects versus the 20 MDD patients. The MDD patients were matched with healthy control subjects by age and gender, with an average age of 45.8 years (SD: 9.6) and 50% male. The mean Hamilton Depression Rating Scale (HAM-D) score for the MDD subjects was 19 (SD: 3.4), indicating “Severe Depression” (Hamilton, 1960). The average length of their current depression episode was 82 weeks.

Data acquisition and preprocessing

DTI, rs-fMRI, and T1-weighted data were collected in a single session with a 3T Siemens Tim Trio scanner. The DTI sequence consisted of 60 scans with different diffusion-weighted directions ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$b = 1000s / m{m^2}$$ \end{document} ) and four nondiffusion-weighted scans ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$b = 0$$ \end{document} ), acquired using a single-shot spin-echo echo planar imaging sequence. Additional DTI scanning parameters include the following: TR = 11,300 ms, TE = 104 ms, GRAPPA on, FOV = 256 mm, number of slices = 64, voxel size = 2 × 2 × 2 mm, and matrix size = 128 × 128.

For registration purposes, high-resolution T1-weighted images were collected using a 3D MPRAGE sequence with the following parameters: TR = 2600 ms, TI = 1100 ms, TE = 3 ms, number of slices = 176, voxel size = 111 mm, matrix size = 224256, and flip angle = \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${8^ \circ }$$ \end{document} . Functional images were collected over 150 time points, with a z-saga sequence to minimize artifacts in the medial prefrontal cortex and orbitofrontal cortex due to sinus cavities. rs-fMRI scans were acquired interleaved with the following parameters: TR = 2.92 sec, TE1 = 30 ms, TE2 = 66 ms, flip angle = \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${90^ \circ }$$ \end{document} , number of axial slices = 30, slice thickness = 4 mm, FOV = 220 mm, and total duration = 7.3 min.

Several standard preprocessing steps were applied to the rs-fMRI data, including despiking, slice timing correction, motion correction, registration to MNI 2 mm standard space, normalization to percent signal change, removal of linear trend, regressing out cerebrospinal fluid (CSF), white matter, and six movement parameters, bandpass filtering (0.009–0.08), and spatial smoothing with a 6 mm full width at half maximum Gaussian kernel. Preprocessing steps for the DTI data include brain extraction to remove nonbrain regions, phase reversal distortion correction, and aligning diffusion-weighted images to the average nondiffusion-weighted image by rigid body affine transformation to remove motion and eddy current-induced artifact. We then estimate the directional diffusion at each voxel based on a diffusion tensor model implemented via the diffusion toolbox in FSL (Behrens et al., 2003).

Functional network estimation

We estimated group-level functional networks from the rs-fMRI data using the Group ICA for fMRI Toolbox (GIFT) (Calhoun et al., 2001). We extracted 20 independent components from the data and identified a set of components that represent well-established resting-state functional networks (Smith et al., 2009).

SC estimation

SC is assessed using DTI data, a diffusion-weighted MRI technique that estimates the location of structural fibers in the brain by measuring the direction and magnitude of water diffusion, since water tends to diffuse along white matter fiber tracts (Johansen-Berg and Rushworth, 2009). We evaluated SC across the whole brain by implementing a widely used probabilistic tractography algorithm via the PROBTRACKX function in FSL's diffusion toolbox (Behrens et al., 2003, 2007). This procedure successively initiates streams from a seed voxel, which are intended to trace the paths of white matter fibers throughout the brain by sampling from the distributions of voxelwise principal diffusion directions.

To facilitate estimation of plausible white matter tracts, all seeding voxels used in the algorithm are located in white matter. There was an average of 900 white matter seeding voxels in each functional network; 5000 streams were initiated from each seed voxel and traced voxel-by-voxel as they passed through the diffusion tensor field, terminating according to a set of stopping rules. The following stopping rules were implemented to avoid estimation of implausible tracts. A white matter inclusion mask was imposed to select only streams that pass through white matter in the brain; a CSF exclusion mask was also imposed to exclude any streams that pass through CSF. Furthermore, a curvature threshold (0.2) was imposed to limit any pathways with sharp turns, and streams were terminated when they had traveled 2000 steps or looped back on themselves.

Methods

The sSC measure

We propose a new statistical measure for quantifying the sSC underlying a functional network. The sSC is defined based on the average sSC between brain locations that constitute the functional network, reflecting the strength of within-network SC. Suppose that based on a data-driven FC method such as group ICA, we extracted q functional networks of interests. In group ICA, spatial distribution of functional networks is reflected by the component maps estimated from ICA. We propose the following measure, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta_ \ell}$$ \end{document} , to quantify the sSC underlying the functional network represented by component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell = 1 , \ldots , q$$ \end{document} ) as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ \theta _ \ell } = \frac { { \mathop \sum \limits_ { { j , k \in{ \Omega _ \ell } } } [ { p_ { jk } } - ( { { \bar p } _j } + { { \bar p } _k } ) / 2 ] }} {{ \mathop \sum \limits_ { j , k \in { \Omega _ \ell } } [ 1 - ( { { \bar p } _j } + { { \bar p } _k } )/ 2 ] } }. \tag { 1 } \end{align*} \end{document}

where V is the total number of voxels within the brain. Similarly, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar p_k}$$ \end{document} is defined analogously to represent the average probability of structural connection between voxel k and the rest of the brain.

The raw SC probability, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${p_{jk}}$$ \end{document} , may be affected by network-specific factors such as the location of the network's voxels in the brain relative to white matter pathways and the average strength of structural connections across the brain, limiting the comparability of raw SC probability across functional networks and subjects. To help address this issue, we adjust the raw SC probability by the background SC strength, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( { \bar p_j} + { \bar p_k} ) / 2$$ \end{document} , which represents the average overall probability of SC between the voxels j, k in network \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} and a random location in the brain. The numerator of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta_ \ell}$$ \end{document} reflects the degree to which raw SC within the functional network exceeds the baseline SC between the network and the rest of the brain expected on average. This adjustment allows us to compare the underlying SC of networks located in high-traffic versus low-traffic SC areas and also improve the comparability of sSC across subjects who may be associated with different baseline SC.

We further standardize the sSC measure by dividing by the maximum possible value, in which there is complete SC between all voxel pairs within the functional network (i.e., \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${p_{jk}} = 1$$ \end{document} , for all \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j , k \in { \Omega_ \ell}$$ \end{document} ). The sSC is then within the standard scale of less than or equal to 1, which facilitates comparisons across FC networks and subjects. The definition of sSC is related to that of the Kappa coefficient (Cohen, 1968), in the sense that sSC represents the observed sSC, relative to the baseline SC expected by chance, divided by the maximum possible value. By definition, the sSC can potentially be negative. A negative sSC means the SC between the voxels within the functional network is weaker than the SC between voxels within the network and voxels outside the network. However, based on our experience, the sSC for established functional networks is usually above zero since voxels within these functional networks generally have stronger SC within the networks than with the rest of the brain on average.

In the sSC definition (1), j and k represent voxels in a functional network. Hence, the proposed sSC is a voxel-level measure of SC underlying an FC. Alternatively, we can define sSC in a similar way on the region level, where j and k represent regions within a functional network.

By adjusting for the network's baseline SC and standardizing the scale of the measure, the proposed sSC removes the effects from network-specific factors, such as network size (i.e., total number of voxels), the proportion of gray matter and white matter regions in the network, and the spatial location of a network. Therefore, sSC can potentially provide more accurate reflection of the relative within-network SC across different networks than the raw SC, which is demonstrated via simulation studies (Appendix A3).

We further illustrate the difference between the proposed sSC and the raw SC using real imaging data. We present comparisons between the two measures for three FC networks, including visual network, DMN, and executive control network for multiple subjects (Fig. 1). Across subjects, sSC consistently shows that it is highest in the visual network and lowest in the executive control network, which reflects the fact that the visual network has higher relative within-network SC than the executive control network. The raw SC, on the contrary, shows opposite results where it is highest for the executive control network and lowest for the visual network. The difference indicates that by adjusting the network-specific factors, the proposed sSC more accurately reflects the relative within-network SC across different FC networks.

FIG. 1.

Illustration of the difference between proposed sSC and raw SC. We show results of the two measures for three FC networks, including visual network, DMN, and EC network using data from five randomly selected subjects from a multimodality neuroimaging study. Across subjects, sSC consistently shows that the sSC is highest in the visual network and lowest in the EC network. The raw SC, on the contrary, shows the opposite results due to lack of adjustment for network-specific factors. DMN, default model network; EC, executive control; FC, functional connectivity; sSC, strength of structural connectivity.

Estimation and inference for the sSC measure

Estimate of the sSC measure

Plugging in the estimated probability of SC derived from probabilistic tractography yields the following estimator for the sSC measure of the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th functional network:

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} is the number of streams passing through voxels j and k, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$ { \bar N_j } = \frac { 1 } { { V - 1 } } \sum \nolimits_ { v = 1 , v \ne j } ^V { N_ { jv } } $$ \end{document} is the average streams connecting voxel j with the rest of the brain, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$ { \bar N_k } = \frac { 1 } { { V - 1 } } \sum \nolimits_ { v = 1 , v \ne k } ^V { N_ { kv } } $$ \end{document} is the average streams connecting voxel k with the rest of the brain, and N is the total number of streams initiated from each voxel in the probabilistic tractography procedure. The probability of SC for the voxel pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( j , k )$$ \end{document} , say \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${p_{jk}}$$ \end{document} , is thus estimated based on \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} (Behrens et al., 2007; Johansen-Berg and Rushworth, 2009). We note that when sSC is defined on a voxel level, rather than on a region level, the magnitude of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta_l}$$ \end{document} will tend to be relatively small.

Variance of the sSC estimate

To conduct statistical inference for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta_ \ell}$$ \end{document} , we propose a variance estimator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$var ( { \hat \theta_ \ell} )$$ \end{document} . We note that the sSC estimator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta_ \ell}$$ \end{document} is a function of the observed stream counts between pairwise voxels from a probabilistic tractography procedure. To facilitate deriving the variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta_ \ell}$$ \end{document} , we define \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} as a vector containing the stream counts \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} for all voxel pairs \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{ j , k \} , { \rm that \ is} ,$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \quad {\textbf{\textit{N}}^{ \bf{*}}} = \left[ { \begin{matrix} { \{ {N_{jk}} \} } \\ \end{matrix} } \right] = \left[ { \begin{matrix} {{N_{12}}} \\ {{N_{13}}} \\ \vdots \\ {{N_{V - 1 , V}}} \\ \end{matrix} } \right] \end{align*} \end{document}

We can re-express the proposed sSC estimate, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell}$$ \end{document} , as a function of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} as follows (Appendix for derivations): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ \hat \theta _ \ell } = { \frac { ( { \textbf { \textit { C } } _ \ell } - \textbf { \textit { A } } ) { \textbf { \textit { N } } ^ { \bf { * } } } } { b - \textbf { \textit { AN } } { ^ { \bf { * } } } } } , \tag { 3 } \end{align*} \end{document}

Here, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{C}}_ \ell }$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{C}}_j}$$ \end{document} are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$1 \times \left( { \begin{matrix} V \\ 2 \\ \end{matrix} } \right)$$ \end{document} row vectors of binary indicators. The z th \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\left( {z = 1 , \ldots , \left( { \begin{matrix} V \\ 2 \\ \end{matrix} } \right) } \right)$$ \end{document} element of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{C}}_ \ell}$$ \end{document} equals 1 if the z th voxel pair in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} belongs to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \Omega _ \ell }$$ \end{document} , and 0 otherwise; the z th \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\left( {z = 1 , \ldots , \left( { \begin{matrix} V \\ 2 \\ \end{matrix} } \right) } \right)$$ \end{document} element of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{C}}_j}$$ \end{document} is 1 if the z th voxel pair in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} involves voxel j, and 0 otherwise; and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${V_ \ell }$$ \end{document} is the number of voxels in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th network.

Equation (3) shows that the numerator and denominator of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta_ \ell}$$ \end{document} are each a linear function of the random vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} and some constants. Therefore, the variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta_ \ell}$$ \end{document} can be obtained from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma} }} = Var ( {{N}^{ \bf{*}}} )$$ \end{document} using the Delta method (Casella and Berger, 1990). Note that each element \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} is a binomial count of the number of streams passing the voxel pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\{ j , \;k \} $$ \end{document} out of a total of N initiations. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} follows a binomial distribution with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}} \sim { \rm{Bin}} ( N , {p_{jk}} )$$ \end{document} . Given that N is large, the binomial distribution is well approximated by normal distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}} \sim { \rm{N}} ( { \mu _{jk}} , \sigma _{jk}^2 )$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \mu _{jk}} = N{p_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\sigma_{jk}^2 = N{p_{jk}} ( 1 - {p_{jk}} )$$ \end{document} . The vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} approximately follows a multivariate normal distribution, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}} \sim MVN ( \bm{\mu} , {{\bm{\Sigma} }} )$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \bm{\mu} = \left[ { \begin{matrix} { \{ { \mu _{jk}} \} } \\ \end{matrix} } \right] = N \left[ { \begin{matrix} {{p_{12}}} \\ {{p_{13}}} \\ \vdots \\ {{p_{V - 1 , V}}} \\ \end{matrix} } \right] , \; { \rm and} \end{align*} \end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{{ \bm{\Sigma} }} = \left[ { \begin{matrix} { \{ Cov ( {N_{jk}} ,{N_{j \prime k \prime }} ) \} } \\ \end{matrix} } \right] \\ = \left[ { \begin{matrix} {var ( {N_{12}} ) } & {cov ( {N_{12}} ,{N_{13}} ) } & \cdots & {cov ( {N_{12}} , {N_{V - 1 , V}} ) } \\{cov ( {N_{13}} , {N_{12}} ) } & {var ( {N_{13}} ) } & \cdots &{cov ( {N_{13}} , {N_{V - 1 , V}} ) } \\ \vdots & \vdots & \ddots & \vdots \\ {cov ( {N_{V - 1 , V}} , {N_{12}} ) } & {cov( {N_{V - 1 , V}} , {N_{13}} ) } & \cdots & {var ( {N_{V - 1 , V}} ) } \\ \end{matrix} } \right] \end{align*} \end{document}

We have shown that the diagonal variance elements of the variance/covariance matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma}}}$$ \end{document} are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$var ( {N_{jk}} ) = \sigma _{jk}^2 = N{p_{jk}} ( 1 - {p_{jk}} )$$ \end{document} . The derivation of the off-diagonal covariance elements is more challenging. Given the high dimensionality of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma}}}$$ \end{document} , there are a large number of covariance terms. Furthermore, deriving the covariance between pairs of counts \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} is not straightforward because of the potential spatial dependence between them.

We use a novel approach based on a parametric semivariogram model (Bowman, 2007; Cressie, 1992; Gao et al., 2017) to estimate the covariance between pairs of stream counts from a probabilistic tractography procedure. Specifically, we model each covariance term, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$cov ( {N_{jk}} , {N_{j \prime k \prime }} )$$ \end{document} , as a function of the distance between voxel pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j , k$$ \end{document} and voxel pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j \prime , k \prime$$ \end{document} , denoted \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${d_{jk , j \prime k \prime }}$$ \end{document} , which decays as the distance between the observations increases.

One distinct feature in modeling spatial dependence between the tractography stream counts is that there is no straightforward way to define the spatial distance between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} . Unlike the standard case, these stream counts are not taken at single spatial locations but rather are outcomes observed between pairs of spatial locations, that is, voxels. In other words, we need a metric that quantifies the spatial distance between pairs of voxel pairs. We propose a novel distance metric between two voxel pairs \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( j , k )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( j \prime , k \prime )$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ d_ { jk , j \prime k \prime } } = \min \left[ { \frac { { { d^ * } ( j , j \prime ) + { d^ * } ( k , k \prime ) } } { 2 } , \frac { { { d^ * } ( j , k \prime ) + { d^ * } ( k , j \prime ) } } { 2 } } \right] , \tag { 4 } \end{align*} \end{document}

Biologically, the distance metric \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${d_{jk , j \prime k \prime }}$$ \end{document} represents the shortest average distance between the end-points of the streamlines in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} and helps inform the covariance or dependence of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} . Specifically, based on the semivariogram model, short distance in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${d_{jk , j \prime k \prime }}$$ \end{document} leads to high covariance between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} because they are more likely to represent anatomically similar white fiber tracts connecting the same pairs of brain regions. On the contrary, large distance in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${d_{jk , j \prime k \prime }}$$ \end{document} leads to low covariance between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{j \prime k \prime }}$$ \end{document} , because their streamlines are more likely to represent distinct white fiber tracts connecting different pairs of brain regions. We provide some examples in Appendix A1 to demonstrate biological interpretation of distance metric and covariance term.

The semivariogram offers a flexible approach to model the covariance as a function of spatial distance, since there are many different types of semivariogram models (e.g., exponential, Gaussian, spherical, and Matern class), each of which is characterized by shape parameters known as the range, sill, and the nugget effect. Thus, we use the semivariogram model to incorporate spatial dependence via our proposed distance metric (4) in the estimation of covariance between stream counts. The parametric model, and its associated parameters, is then estimated by fitting to the empirical semivariogram using the observed data (Cressie, 1992).

After obtaining the variance/covariance matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma}}}$$ \end{document} for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} , we then derive the approximate variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell}$$ \end{document} as follows, using the Delta method (Appendix A2 for details).

In addition to (5), we consider a nonparametric variance estimate based on the bootstrap. We will compare the performance of the parametric versus bootstrap variance estimators in simulation studies.

Hypothesis testing based on the sSC measure

Based on the estimated sSC \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell}$$ \end{document} and its variance estimator, we conduct statistical tests to compare the sSC across various functional networks and between groups of subjects. We present hypothesis tests for (1) testing the significance of sSC within a functional network, (2) comparing sSC between two functional networks, and (3) comparing the sSC underlying a given functional network between subject subgroups.

In the first case, we aim to test the significance of sSC within a given functional network. For a network represented by the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th component from ICA, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta _ \ell } = 0$$ \end{document} indicates that the observed sSC within the network is no higher than the average SC between the network and the rest of the brain. If an estimated component is representative of a true functional network, we expect the sSC within the network to be significantly above the average SC, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta _ \ell } > 0$$ \end{document} . Thus, we can test the significance of sSC within component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} by specifying the hypotheses as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_0}:{ \theta _ \ell } = 0$$ \end{document} versus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_1}:{ \theta _ \ell } > 0$$ \end{document} . We propose the following Wald-type test statistic: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ Z^* } = { \frac { { { \bar \hat \theta } _ \ell } } { \sqrt { \hat Var ( { { \hat \theta } _ \ell } ) / n } } } , \tag { 6 } \end{align*} \end{document}

with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar \hat \theta _ \ell } = \sum \nolimits_{i = 1}^n { \hat \theta _{i \ell }} / n$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _{i \ell}}$$ \end{document} represents the estimated sSC from the i th subject \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( i = 1 , \ldots , n )$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat Var ( { \hat \theta _ \ell } )$$ \end{document} is obtained from (5) by plugging in the sample estimates. If \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${Z^*} > {z_{1 - \alpha }}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${z_{1 - \alpha }}$$ \end{document} is the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$1 - \alpha$$ \end{document} percentile of the standard normal distribution, we would reject the null hypothesis H ₀ in favor of the alternative hypothesis H ₁ and conclude there is significant sSC within the network characterized by the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th IC.

The second test aims to compare the sSC between two networks characterized by components \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell \prime$$ \end{document} . The hypotheses in this case are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_0}:{ \theta _ \ell } = { \theta _{ \ell \prime }}$$ \end{document} versus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_1}:{ \theta _ \ell } \ne { \theta _{ \ell \prime }}$$ \end{document} . We reject the hypothesis when the test statistic \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\vert { \bar \hat \theta _ \ell } - { \bar \hat \theta _{ \ell \prime }} \vert$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar \hat \theta _ \ell}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar \hat \theta _{ \ell \prime }}$$ \end{document} are the average sSC across subjects for the two networks. We derive the p-value for this test statistic using a nonparametric permutation test. Specifically, we permute the network label within each subject to generate the null distribution for the test statistic \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\vert { \bar \hat \theta _ \ell } - { \bar \hat \theta _{ \ell \prime }} \vert$$ \end{document} and the p-value is then derived by comparing the observed test statistic from the data to the null distribution.

Third, we can test whether the sSC for a given network, for example, the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th network, differs between subject subgroups via hypotheses of the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_0}:{ \theta _{ \ell , 1}} = { \theta _{ \ell , 2}}$$ \end{document} versus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${H_1}:{ \theta _{ \ell , 1}} \ne { \theta _{ \ell , 2}}$$ \end{document} where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta _{ \ell , j}}$$ \end{document} is the sSC underlying the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th network for the j th subgroup, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$j = 1 , 2$$ \end{document} . We evaluate the group difference using the following test statistic: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ Z^* } = { \frac { { { \bar \hat \theta } _ { \ell , 1 } } - { { \bar \hat \theta } _ { \ell , 2 } } } { \sqrt { { \frac { \hat Var ( { { \hat \theta } _ { \ell , 1 } } ) } { { n_1 } } } + { \frac { \hat Var ( { { \hat \theta } _ { \ell , 2 } } ) } { { n_2 } } } } } } , \tag { 7 } \end{align*} \end{document}

where n ₁ and n ₂ are the number of subjects in each subgroup, and the definitions for the subgroup mean and variance terms are analogous to those in (6). If \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\vert {Z^*} \vert > {z_{1 - \alpha / 2}}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${z_{1 - \alpha / 2}}$$ \end{document} is the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$1 - \alpha / 2$$ \end{document} percentile of the standard normal distribution, we would reject the null hypothesis and conclude that there is a significant difference in the sSC between the two groups. Alternatively, we can test between-group hypotheses using a nonparametric permutation testing approach, in which we permute subjects' group labels to generate null distribution for between-group difference \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\vert { \bar \hat \theta _{ \ell , 1}} - { \bar \hat \theta _{ \ell , 2}} \vert$$ \end{document} .

Does sSC inform the reliability of functional networks?

It is often of interest to investigate the reliability of functional networks derived from data-driven methods such as ICA. The reproducibility of an estimated network varies across different runs of the algorithm or different samples of subjects. For example, due to the stochastic nature of the ICA algorithm and individual subject variability, the ICA-derived functional networks often vary across different fMRI studies or even across various ICA runs based on the same data. Researchers also noticed that some networks tend to be more reproducible compared with others (Zuo et al., 2010).

We investigate whether the strength of structural connections underlying functional networks is associated with their reliability. One hypothesis is that networks that have stronger direct structural connections may tend to be more reproducible given their underlying structure. Generally, a more reproducible functional network is interpreted as a network that is more consistently identified across different subject data sets sampled from the population and with different initial computational conditions. To aid our evaluation of potential links between sSC and reliability, we propose a novel reliability index, which basically assesses how reproducible each of the original components is in bootstrap samples from the original data. Please see the following Algorithm 1 for the definition of the proposed reliability index and the estimation procedure.

Algorithm 1 The algorithm for obtaining the reliability index for a functional network.

Step 1: generate B bootstrap samples each containing n subjects sampled with replacement from the original data set, and perform group ICA to extract q components in each bootstrap sample with different initial conditions.

Step 2: For the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th component extracted from the original data, we identify the corresponding component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \ell _b}$$ \end{document} in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${b^{th}}$$ \end{document} bootstrap sample by identifying the component from the bootstrap sample that shows the highest spatial correlation with the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th component from the original data.

Step 3: For the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell = 1 , \ldots , q$$ \end{document} ) component in the original data, we propose the following reliability index.

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${r_{ \ell { \ell _b}}}$$ \end{document} is the spatial correlation between original component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} and its corresponding bootstrap component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \ell _b}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${r_{ \ell \ell _b^*}}$$ \end{document} is the spatial correlation between component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} and the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell_b^*$$ \end{document} component in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${b^{th}}$$ \end{document} bootstrap sample.

In the proposed reliability index (8), the first term in the numerator of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${R_ \ell}$$ \end{document} represents the observed similarity between component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} and its corresponding component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \ell_b}$$ \end{document} in the bootstrap samples. This term reflects how reproducible the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th component is across these bootstrap samples. Since component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} can be correlated with any bootstrap component by chance, we evaluate by-chance correlation between the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th component and all the components from the bootstrap samples. The by-chance correlation is then removed from the observed similarity, and hence, the numerator of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${R_ \ell}$$ \end{document} represents the beyond-chance similarity between the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \ell_b}$$ \end{document} th original component and its counterparts in bootstrap samples.

We further standardize the measure by dividing by its maximum possible value, in which the original component \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} is perfectly reproduced in the bootstrap sample (i.e., \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${r_{ \ell { \ell _b}}} = 1$$ \end{document} ). The reliability index \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${R_ \ell}$$ \end{document} ranges from 0 to 1, where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${R_ \ell} = 0$$ \end{document} indicates the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} th IC is not reproducible in bootstrap samples after we correct for by-chance correlations across ICs and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${R_ \ell}$$ \end{document} close to 1 indicates that the component is highly reproducible.

Results

Simulation studies

We conduct simulation studies to evaluate the performance of the estimation and inference methods for the proposed sSC measure. We simulate fMRI data from two underlying source signals, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q = 2$$ \end{document} . We consider two sample sizes with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$n = 20 , 50$$ \end{document} subjects. For each source, the spatial maps for the source signals are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$10 \times 10$$ \end{document} images (i.e., \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V = 100$$ \end{document} ) with Gaussian background random variability of standard deviation of 0.5 with linearly added source signals (Fig. 2). For temporal dynamics, each source signal has a time series of length of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$T = 200$$ \end{document} that is generated based on time courses from real fMRI data and hence representing realistic fMRI temporal dynamics. We then generate the observed mixture by mixing the spatial source signals with the temporal mixing matrix based on the ICA model. Finally, Gaussian random variations are added to the mixed spatial sources to generate data for each subject.

FIG. 2.

True independent component maps for simulation studies. Source signal intensity for the true independent component maps for simulation studies. We simulated data from two underlying independent components (i.e., ICs). For each source, the spatial maps for the IC source signals are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$10 \times 10$$ \end{document} images with Gaussian background random variations with linearly added source signals.

We then simulate SC based on DTI data. To simulate the results of a probabilistic tractography procedure, we generate a \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\left( { \begin{matrix} V \\ 2 \\ \end{matrix} } \right)$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\times 1$$ \end{document} matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} for each subject, whose elements \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} represent the number of streams out of a total of N streams initiated that connect each voxel pair ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\,j , k$$ \end{document} ). To reduce the computation time for simulation, we consider \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$N = 20$$ \end{document} .

We use the model \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}} \sim { \rm{MVN}} ( \bm{\mu} ,{{ \bm{\Sigma} }} )$$ \end{document} to simulate these data, with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\bm{\mu}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma}}}$$ \end{document} defined as follows. For the mean vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\bm{\mu} = N\textbf{\textit{p}}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{p}}$$ \end{document} is the vector of voxel pair connection probabilities. Voxel pairs outside a component have a connection probability of 0.25, while voxel pairs inside IC 1 or 2 have connection probabilities of 0.5 and 0.75, respectively. The variance/covariance matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \bm{\Sigma}}}$$ \end{document} is defined based on the exponential semivariogram function, with parameters c ₀ (nugget), c_e (partial sill), and a_e (range). We generate the SC data in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} under both “low” (c ₀ = 1, c_e = 4, a_e = 1) and “high” (c ₀ = 2, c_e = 5, a_e = 1) noise levels. In this way, we simulate the SC data in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\textbf{\textit{N}}^{ \bf{*}}}$$ \end{document} and evaluate the sSC underlying each IC.

Our simulation study mainly aims to assess the accuracy of the proposed estimation and inference methods for sSC. Therefore, streamline counts are directly simulated from distributions instead of inferred from simulated DTI data. As a reviewer pointed out, there are now approaches, such as that used in ISMRM 2015, to generate more realistic SC data. For future work, we plan to investigate more sophisticated approaches for generating SC data that closely mimic white fiber tracts in the brain. We like to point out that a challenge for this more realistic simulation is that one will need to find ways to obtain the true values of some parameters such as the covariance of SC streamlines, which are unknown in this setting.

Once the subject-level fMRI and SC data have been simulated, we estimate the group-level IC maps using the GIFT method (Calhoun et al., 2001), and then estimate sSC for each of the estimated ICs. Table 1 summarizes the results under different simulation settings with varying sample sizes and noise levels, with 300 simulation runs for each setting. We estimate the sSC for each of the two ICs along with its variance using the proposed theoretical variance estimator (based on semivariogram model) as well as the nonparametric bootstrap variance estimator (based on 1000 bootstrap resamples). We also evaluate the coverage probabilities of two types of 95% confidence intervals (CI): the Wald-type CI based on the theoretical standard errors (SE), and the CI based on the bootstrap percentiles.

Table 1.

Simulation Results for Estimating the Strength of Structural Connectivity Based on 300 Simulated Data Sets

	Sample size	Noise level	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\theta$$ \end{document}	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} mean (SD)	Theoretical SE (SD)	Bootstrap SE (SD)	Cov. Prob. I ^a (Theoretical)	Cov. Prob. II ^a (Bootstrap)
IC 1	20	Low	0.3077	0.3081 (0.0091)	0.0083 (0.00035)	0.0093 (0.0016)	92.6	94.3
	20	High	0.3077	0.3074 (0.0104)	0.0093 (0.00043)	0.0105 (0.0018)	91.6	94
	50	Low	0.3077	0.3084 (0.0061)	0.0053 (0.00019)	0.0060 (0.00063)	90.7	93.7
	50	High	0.3077	0.3078 (0.0069)	0.0059 (0.00023)	0.0068 (0.00071)	91	94.7
IC 2	20	Low	0.64	0.6405 (0.0120)	0.0112 (0.00041)	0.0115 (0.0018)	93.3	93
	20	High	0.64	0.6389 (0.0134)	0.0126 (0.00040)	0.0127 (0.0020)	94.6	93.6
	50	Low	0.64	0.6409 (0.0080)	0.0071 (0.00019)	0.0074 (0.00073)	90.7	93
	50	High	0.64	0.6394 (0.0088)	0.0080 (0.00017)	0.0082 (0.00081)	93.7	93

Coverage probabilities I: Based on Wald-type CI using theoretical SE. Coverage probabilities II: Bootstrap percentile CI.

Results under the different simulation settings with varying sample sizes and noise levels, with 300 simulation runs for each setting. We estimate the sSC for each of the two ICs along with its variance using the proposed theoretical variance estimator based on semivariogram model (Theoretical SE) and the nonparametric bootstrap variance estimator based on 1000 bootstrap resamples (Bootstrap SE). We also evaluate the coverage probabilities of two types of intervals (CI): the Wald-type CI based on the theoretical SE (Cov. Prob. I), and the CI based on the bootstrap percentiles (Cov. Prob. II).

We evaluate the bias of our sSC estimator by comparing the mean \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} to the true sSC value, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\theta$$ \end{document} . There is very low bias in all simulation settings. We also assess the performance of the two variance estimators by comparing the mean estimated theoretical and bootstrap SE to the Monte Carlo standard deviation, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \rm{SD}} ( \hat \theta )$$ \end{document} . Both variance estimators are close to the Monte Carlo SD, with bootstrap SE demonstrating slightly better performance. Finally, we find that the coverage probabilities from both types of CIs are fairly close to the nominal level, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$95 \%$$ \end{document} .

Because calculation of the theoretical variance term requires estimating the large matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \hat{\bm{\Sigma}}}}$$ \end{document} , estimating the theoretical variance term \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$Var ( \hat \theta )$$ \end{document} can potentially pose a computational challenge when V is large. The bootstrap variance estimator, on the contrary, is more computationally efficient in this case since it avoids estimation of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${{ \hat{\bm{\Sigma} }}}$$ \end{document} , and shows good performance in our simulation studies. Thus, we recommend using the bootstrap method to conduct inference in real-data applications where V is large.

Based on a reviewer's suggestion, we have conducted additional simulation studies to demonstrate the advantages of the proposed sSC over the raw within-network SC measure. We show that the proposed sSC has better comparability across FC networks with a different number of voxels or different baseline SC. Details are presented in Appendix A3.

Application to real data set

sSC analysis

We apply our sSC method to an fMRI and DTI data set from the PReDICT study. First, we evaluate and compare the sSC underlying various functional networks. We run a group ICA using fMRI data from 20 normal controls, since studies of MDD have shown resting-state FC differences in MDD patients (Greicius et al., 2007; Northoff et al., 2011; Veer, 2010). We extract \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$q = 15$$ \end{document} group-level IC maps, nine of which appear to represent well-known resting-state networks (RSNs) (Laird et al., 2009; Smith et al., 2009) (Fig. 3). For each of the nine ICs, we create a thresholded white matter mask, consisting of about 900 voxels, for investigating the SC underlying the IC.

FIG. 3.

Brain functional networks estimated based on group ICA of subjects' fMRI data in the PReDICT study. We extract 14 ICs from the data and 9 of them correspond to well-known resting-state networks. fMRI, functional magnetic resonance imaging; ICA, independent component analysis. PReDICT, Predictors of Remission in Depression to Individual and Combined Treatments.

To evaluate the SC distribution of each IC, we conduct a probabilistic tractography procedure using voxels in the thresholded IC mask as seed locations. We initiate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$N = 5000$$ \end{document} streams from each seed voxel in the IC and trace the streams across the brain. The tractography procedure provides us \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar N_j}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \bar N_k}$$ \end{document} for each voxel pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( \,j , k )$$ \end{document} in IC \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell ( \ell = 1 , \ldots , 9 )$$ \end{document} , from which we estimate the sSC underlying IC \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\ell$$ \end{document} , that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell }$$ \end{document} . We conduct inference for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \theta_ \ell}$$ \end{document} using the bootstrap.

Table 2 shows sSC results for nine estimated ICs from healthy controls. The \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} values are all fairly small, since our analysis is conducted on the voxel level, yet all ICs have strength of underlying SC that is highly significantly greater than 0 ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p < 0.0001$$ \end{document} ), indicating that the sSC within the functional networks is higher than the background sSC in the brain.

Table 2.

Results of Hypothesis Testing for the sSC for ICA-Derived Brain Functional Networks Among Control Subjects in PReDICT Study

IC		Mean( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell}$$ \end{document} )	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$SE_{{ \rm{boot}}}$$ \end{document} ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \hat \theta _ \ell}$$ \end{document} )	Bootstrap CI	Bootstrap p-value
2	Motor	0.0071	0.0013	(0.0066–0.0077)	<0.0001
3	FP	0.0081	0.0010	(0.0077–0.0086)	<0.0001
4	EC	0.0048	0.0004	(0.0046–0.0049)	<0.0001
5	FP	0.0077	0.0008	(0.0074–0.0081)	<0.0001
8	Visual	0.0098	0.0014	(0.0092–0.0103)	<0.0001
10	EC	0.0052	0.0008	(0.0048–0.0055)	<0.0001
11	Visual	0.0085	0.0009	(0.0082–0.0089)	<0.0001
12	Motor	0.0058	0.0005	(0.0056–0.0060)	<0.0001
13	DMN	0.0078	0.0016	(0.0071–0.0085)	<0.0001

All ICs have strength of underlying SC that is highly significantly greater than 0 (p < 0: 0001), indicating that the sSC within the functional networks is higher than the background sSC in the brain. Among the networks, the visual network demonstrates the highest sSC, while the executive control network displays the lowest sSC (see bolded values). SEboot( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} ): bootstrap standard error of the sSC estimate. Bootstrap CI: bootstrap confidence interval for the sSC. Bootstrap p-value: p-value for the bootstrap test on whether the sSC is equal to zero.

ICA, independent component analysis; PReDICT, Predictors of Remission in Depression to Individual and Combined Treatments; sSC, strength of structural connectivity.

Among the networks, the visual network demonstrates the highest sSC, while the executive control network displays the lowest sSC. Figure 4 displays the SC distribution for these two networks. It shows that the visual network has a much higher within-network SC compared with its background SC across the brain, while the executive control network has low within-network SC relative to the rest of the brain. Permutation testing reveals that the sSC underlying the visual network is significantly higher than the sSC underlying the executive control network ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$p < 0.0001$$\end{document} ).

FIG. 4.

The structural connections of two brain functional networks (visual network and executive control network) estimated from the PReDICT study. Here, the red area represents the IC maps of each brain network estimated via the group ICA, and the blue region represents the structural connections of the IC region with the rest of the brain. (A) IC 8 (a visual network) has the highest sSC, which is confirmed by the high degree of SC within the IC. (B) IC 4 (an executive control network) has the lowest sSC, which is consistent with the low degree of SC within the IC relative to the rest of the brain.

Next, we examine the difference in sSC for each IC between the control and MDD subject groups (Table 3). Results show each group's mean sSC for each IC, along with the bootstrap-based 95% CI and p-values (uncorrected) for the group difference in sSC. p-Values are derived based on both the bootstrap SE and permutation testing. The tests provide moderate evidence suggesting the sSC is higher among control subjects than the MDD in the frontal parietal left network. However, no significant between-group differences were revealed for the other extracted networks.

Table 3.

Results of Hypothesis Testing for Comparing sSC Between MDD and Controls in the PReDICT Study

IC		Controls		MDD		Bootstrap CI (mdd–con)	Bootstrap p-value	Perm. test p-value
IC		Mean( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} )	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$SE_{{ \rm{boot}}}$$ \end{document} ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} )	Mean( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} )	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$SE_{{ \rm{boot}}}$$ \end{document} ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} )	Bootstrap CI (mdd–con)	Bootstrap p-value	Perm. test p-value
2	Motor	0.0071	0.0013	0.0070	0.0012	(−0.0009 to 0.0005)	0.702	0.698
3	FP	0.0081	0.0010	0.0079	0.0012	(−0.0009 to 0.0005)	0.525	0.562
4	EC	0.0048	0.0004	0.0047	0.0004	(−0.0003 to 0.0002)	0.820	0.825
5	FP	0.0077	0.0008	0.0072	0.0010	(−0.0011 to −0.0001)	0.018	0.054
8	Visual	0.0098	0.0014	0.0095	0.0016	(−0.0012 to 0.0007)	0.639	0.608
10	EC	0.0052	0.0008	0.0053	0.0008	(−0.0003 to 0.0006)	0.451	0.475
11	Visual	0.0085	0.0009	0.0080	0.0008	(−0.0010 to 0.0000)	0.084	0.077
12	Motor	0.0058	0.0005	0.0060	0.0007	(−0.0002 to 0.0006)	0.277	0.299
13	DMN	0.0078	0.0016	0.0073	0.0012	(−0.0015 to 0.0003)	0.251	0.252

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm SE}_{{ \rm{boot}}}$$ \end{document} ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat \theta$$ \end{document} ): bootstrap standard error of the sSC estimate. Bootstrap CI (mdd-con): bootstrap confidence interval for the difference of sSC between the MDD and control group. Bootstrap p-value and Perm. test p-value: p-value for testing the difference of sSC between the MDD and control group based on the bootstrap test and permutation test, respectively.

MDD, major depressive disorder.

Finally, we investigate whether the sSC informs the reliability of the functional networks estimated by data-driven methods such as ICA. We plot the sSC measure versus the proposed reliability index for each of the nine ICA-derived functional networks in the PReDICT data, and find that these measures are positively associated (Fig. 5). That is, ICs with stronger underlying SC are more likely to be consistently estimated by the ICA algorithm. This suggests that we can leverage SC information from DTI data to inform the FC networks estimated from fMRI data.

FIG. 5.

The association between sSC and network reliability. We plot the sSC measure versus the proposed reliability index for each of the nine ICA-derived functional networks in the PReDICT data, and find that the reliability of functional networks is positively associated with the underlying sSC.

Discussion

Our analysis framework integrates information from the fMRI and DTI data modalities to calculate a novel statistical measure of the sSC underlying a functional network. Our simulation studies and data application demonstrate the utility of the sSC measure, and reveal a positive association between sSC and the reliability of functional networks estimated by ICA in the PReDICT study.

The analysis of sSC in this article is conducted on the voxel level, while region-level analysis is feasible too. The proposed sSC measure could be defined on the region level, where j and k in the sSC definition (1) represent a pair of regions rather than voxels within an FC network. The estimation of the region-level sSC is similar to that of the voxel-level sSC. The main difference is that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${N_{jk}}$$ \end{document} would then represent the number of streams connecting two regions j and k in a probabilistic tractography procedure. We note that when the sSC measure is defined on the voxel level, the magnitude of sSC is inherently small relative to its upper bound of 1. The magnitude of the region-level sSC is expected to be larger because the probability of SC between two regions is higher due to multiple voxels contained in each region.

In this article, we propose to model the covariance between two counts of streamlines connecting two sets of voxel pairs using a semivariogram model, where the covariance decays with the increase of distance between the two sets of voxel pairs. We propose a novel metric to quantify the distance between the end-points of the two streamlines. The metric is defined based on Euclidean distance because it is the most commonly used function in literature to approximate the anatomical distance between brain locations (Alexander-Bloch et al., 2012; Bellec et al., 2006; Dia et al., 2015; Salvador et al., 2005). As a reviewer rightly pointed out, the Euclidean distance has its limitations and does not always fully capture the anatomical distance between voxel pairs. For example, it does not taken into account the corpus callosum connection between right and left brain. The proposed distance metric for two sets of voxel pairs can be improved if a more appropriate distance function that truly reflects anatomical distance in the brain becomes available.

We propose a reliability index for data-driven FC networks to measure the reproducibility of networks through resampling the observed data. Our results show that the sSC is positively associated with the reliability of the FC network. As a reviewer pointed out, another way to assess the reliability of FC is to use test/retest data sets. It is noteworthy that there are some differences between the reliability assessed using the bootstrap approach and the reliability assessed using test/retest data sets. Specifically, the bootstrap-based reliability reflects reproducibility of FC networks across different subject data sets sampled from the population, that is, it reflects the across-sample reliability. In comparison, the test/retest reliability reflects reproducibility of FC networks within the same subjects across replicated fMRI scans, that is, it reflects within-subject reliability.

In this article, we are mainly interested in studying the across-sample FC reproducibility. Hence, the reliability accessed via the bootstrap approach in this article is more appropriate for our purpose. In future work, it will also be of interest to evaluate within-subject FC reproducibility using the test/retest data sets such as that in the HCP study and to investigate how it is associated with the sSC.

There has been extensive research devoted to FC over the past several years, with substantial interest focusing on resting-state FC. A set of RSNs have been consistently identified in these investigations (Smith et al., 2009; Damoiseaux et al., 2006; Laird et al., 2011), most prominently the default mode network (DMN; Buckner et al., 2008). While our work focused on resting-state connectivity, our proposed methods extend to other studies involving task-related fMRI.

Footnotes

Acknowledgments

Research reported in this publication was supported by the National Institute of Mental Health of the National Institutes of Health under award number R01MH105561 and R01MH079448 and by the National Center for Advancing Translational Sciences of the National Institutes of Health under award number UL1TR002378. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Author Disclosure Statement

No competing financial interests exist.

Appendix

References

Achard

, Bullmore

. 2007. Efficiency and cost of economical brain functional networks. PLoS Comput Biol, 3:e17.

Alexander-Bloch

, Vertes

, Stidd

, Lalonde

, Clasen

, Rapoport

, et al. 2012. The anatomical distance of functional connections predicts brain network topology in health and schizophrenia. Cereb Cortex, 23:127–138.

Beckmann

, Smith

. 2004. Probabilistic independent component analysis for functional magnetic resonance imaging. IEEE Trans Med Imaging, 23:137–152.

Beckmann

, Smith

. 2005. Tensorial extensions of independent component analysis for multisubject FMRI analysis. Neuroimage, 25:294–311.

Behrens

TEJ

, Berg

, Jbabdi

, Rushworth

MFS

, Woolrich

. 2007. Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?. Neuroimage, 34:144–155.

Behrens

TEJ

, Woolrich

, Jenkinson

, Johansen-Berg

, Nunes

, Clare

, et al. 2003. Characterization and propagation of uncertainty in diffusion-weighted MR imaging. Magn Reson Med, 50:1077–1088.

Bellec

, Perlbarg

, Jbabdi

, Pélégrini-Issac

, Anton

J-L

, Doyon

, Benali

. 2006. Identification of large-scale networks in the brain using fmri. Neuroimage, 29:1231–1243.

Bowman

. 2007. Spatiotemporal models for region of interest analyses of functional neuroimaging data. J Am Stat Assoc, 102:442–453.

Bowman

, Patel

. 2004. Identifying spatial relationships in neural processing using a multiple classification approach. NeuroImage, 23:260–268.

10.

Bowman

, Zhang

, Derado

, Chen

. 2012. Determining functional connectivity using fmri data with diffusion-based anatomical weighting. NeuroImage, 62:1769–1779.

11.

Buckner

, Andrews-Hanna

, Schacter

. 2008. The brain's default network: anatomy, function, and relevance to disease. Ann N Y Acad Sci, 1124:1–38.

12.

Calhoun

, Pearlson

, Adali

. 2004. Independent component analysis applied to fMRI data: a generative model for validating results. J VLSI Signal Process Syst Signal Image Video Technol, 37:281–291.

13.

Calhoun

, Adali

, Pearlson

, Pekar

. 2001. A method for making group inferences from functional MRI data using independent component analysis. Hum Brain Mapp, 14:140–151.

14.

Casella

, Berger

. 1990. Statistical Inference. Number. 241–245. Belmont, CA: Duxbury Press.

15.

Chamberland

, Girard

, Bernier

, Fortin

, Descoteaux

, Whittingstall

. 2017. On the origin of individual functional connectivity variability: The role of white matter architecture. Brain Connectivity, 7:491–503.

16.

Cohen

. 1968. Weighted kappa: nominal scale agreement with provision for scaled disagreement or partial credit. Psychol Bull, 70:213–220.

17.

Cordes

, Haughton

, Carew

, Arfanakis

, Maravilla

. 2002. Hierarchical clustering to measure connectivity in fMRI resting-state data. Magn Reson Imaging, 20:305–317.

18.

Cressie

. 1992. Statistics for spatial data. Terra Nova, 4:613–617.

19.

Dai

, Yan

, Li

, Wang

, Cao

, et al. 2015. Identifying and mapping connectivity patterns of brain network hubs in alzheimer's disease. Cereb Cortex, 25:3723–3742.

20.

Damoiseaux

, Rombouts

, Barkhof

, Scheltens

, Stam

, Smith

, Beckmann

. 2006. Consistent resting-state networks across healthy subjects. Proc Natl Acad Sci USA, 103:13848–13853.

21.

Duann

J-R

, Jung

T-P

, Makeig

, Sejnowski

. 2006. Repeated decompositions reveal the stability of infomax decomposition of fmri data. Conf Proc IEEE Eng Med Biol Soc, 5: 5324–5327.

22.

Dunlop

, Binder

, Cubells

, Goodman

, Kelley

, Kinkead

, et al. 2012. Predictors of remission in depression to individual and combined treatments (PReDICT): study protocol for a randomized controlled trial. Trials, 13:106.

23.

Friston

, Frith

, Liddle

, Frackowiak

RSJ

. 1993. Functional connectivity: the principal-component analysis of large (PET) data sets. J Cereb Blood Flow Metab, 13:5.

24.

Gao

, Shahbaba

, Ombao

. 2017. Modeling binary time series using gaussian processes with application to predicting sleep states. J Classif, 35:549–579.

25.

Ghosh

, Rho

, McIntosh

, Kötter

, Jirsa

. 2008. Noise during rest enables the exploration of the brain's dynamic repertoire. PLoS Comput Biol, 4:e1000196.

26.

Goñi

, van den Heuvel

, Avena-Koenigsberger

, de Mendizabal

, Betzel

, Griffa

, et al. 2014. Resting-brain functional connectivity predicted by analytic measures of network communication. Proc Natl Acad Sci USA, 111:833–838.

27.

Greicius

, Flores

, Menon

, Glover

, Solvason

, Kenna

, et al. 2007. Resting-state functional connectivity in major depression: abnormally increased contributions from subgenual cingulate cortex and thalamus. Biol Psychiatry, 62:429–437.

28.

Guo

, Pagnoni

. 2008. A unified framework for group independent component analysis for multi-subject fMRI data. NeuroImage, 42:1078–1093.

29.

Hagmann

, Cammoun

, Gigandet

, Meuli

, Honey

, Wedeen

, Sporns

. 2008. Mapping the structural core of human cerebral cortex. PLoS Biol, 6:e159.

30.

Hamilton

. 1960. A rating scale for depression. J Neurol Neurosurg Psychiatry, 23:56–62.

31.

Hampson

, Peterson

, Skudlarski

, Gatenby

, Gore

. 2002. Detection of functional connectivity using temporal correlations in MR images. Hum Brain Mapp, 15:247–262.

32.

Higgins

, Kundu

, Guo

. 2018. Integrative bayesian analysis of brain functional networks incorporating anatomical knowledge. NeuroImage, 181:263–278.

33.

Himberg

, Hyvärinen

, Esposito

. 2004. Validating the independent components of neuroimaging time series via clustering and visualization. Neuroimage, 22:1214–1222.

34.

Hinne

, Ambrogioni

, Janssen

, Heskes

, van Gerven

. 2014. Structurally-informed bayesian functional connectivity analysis. Neuroimage, 86:294–305.

35.

Honey

, Kötter

, Breakspear

, Sporns

. 2007. Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc Natl Acad Sci USA, 104:10240–10245.

36.

Honey

, Sporns

, Cammoun

, Gigandet

, Thiran

J-P

, Meuli

, Hagmann

. 2009. Predicting human resting-state functional connectivity from structural connectivity. Proc Natl Acad Sci USA, 106:2035–2040.

37.

Johansen-Berg

, Rushworth

MFS

. 2009. Using diffusion imaging to study human connectional anatomy. Ann Rev Neurosci, 32:75–94.

38.

Kemmer

, Guo

, Wang

, Pagnoni

. 2015. Network-based characterization of brain functional connectivity in Zen practitioners. Front Psychol, 6:1–14.

39.

Koch

, Norris

, Hund-Georgiadis

. 2002. An investigation of functional and anatomical connectivity using magnetic resonance imaging. Neuroimage, 16:241–250.

40.

Laird

, Eickhoff

, Li

, Robin

, Glahn

, Fox

. 2009. Investigating the functional heterogeneity of the default mode network using coordinate-based meta-analytic modeling. J Neurosci, 29:14496–14505.

41.

Laird

, Fox

PTMT

, Eickhoff

, Turner

, Ray

, Mckay

, et al. 2011. Behavioral interpretations of intrinsic connectivity networks. J Cogn Neurosci, 23:4022–4037.

42.

, Liu

, Li

, Qin

, Li

, Yu

, Jiang

. 2009. Brain anatomical network and intelligence. PLoS Comput Biol, 5:e1000395.

43.

Lukemire

, Wang

, Verma

, Guo

. 2018. Hint: A toolbox for hierarchical modeling of neuroimaging data. arXiv preprint arXiv:1803.07587.

44.

Meinecke

, Ziehe

, Kawanabe

, Muller

K-R

. 2002. A resampling approach to estimate the stability of one-dimensional or multidimensional independent components. IEEE Trans Biomed Eng, 49:1514–1525.

45.

Mueller

, Wang

, Fox

, Yeo

, Sepulcre

, Sabuncu

, et al. 2013. Individual variability in functional connectivity architecture of the human brain. Neuron, 77:586–595.

46.

, Varoquaux

, Poline

J-B

, Thirion

. 2012. A novel sparse graphical approach for multimodal brain connectivity inference. Med Image Comput Comput Assist Interv, 15(Pt 1):707–714.

47.

Northoff

, Wiebking

, Feinberg

, Panksepp

. 2011. The'resting-state hypothesis' of major depressive disorder-a translational subcortical-cortical framework for a system disorder. Neurosci Biobehav Rev, 35:1929–1945.

48.

Putnam

, Wig

, Grafton

, Kelley

, Gazzaniga

. 2008. Structural organization of the corpus callosum predicts the extent and impact of cortical activity in the nondominant hemisphere. J Neurosci, 28:2912–2918.

49.

Quigley

, Cordes

, Turski

, Moritz

, Haughton

, Seth

, Meyerand

. 2003. Role of the corpus callosum in functional connectivity. Am J Neuroradiol, 24:208–212.

50.

Salvador

, Suckling

, Coleman

, Pickard

, Menon

, Bullmore

. 2005. Neurophysiological architecture of functional magnetic resonance images of human brain. Cereb Cortex, 15:1332–1342.

51.

Shi

, Guo

. 2016. Investigating differences in brain functional networks using hierarchical covariate-adjusted independent component analysis. Ann Appl Stat, 10:1930–1957.

52.

Smith

, Fox

, Miller

, Glahn

, Fox

, Mackay

, et al. 2009. Correspondence of the brain's functional architecture during activation and rest. Proc Natl Acad Sci U S A, 106:13040–13045.

53.

Tao

, Liu

, Zhang

, Li

, Qin

, Yu

, Jiang

. 2015. The structural connectivity pattern of the default mode network and its association with memory and anxiety. Front Neuroanat, 9:152.

54.

Várkuti

, Cavusoglu

, Kullik

, Schiffler

, Veit

, Yilmaz

, et al. 2011. Quantifying the link between anatomical connectivity, gray matter volume and regional cerebral blood flow: an integrative mri study. PLoS One, 6:e14801.

55.

Veer

. 2010. Whole brain resting-state analysis reveals decreased functional connectivity in major depression. Front Syst Neurosci, 4:1–10.

56.

Wang

, Guo

. 2018. A hierarchical independent component analysis model for longitudinal neuroimaging studies. arXiv preprint arXiv:1808.01557.

57.

Wang

, Kang

, Kemmer

, Guo

. 2016. An efficient and reliable statistical method for estimating functional connectivity in large scale brain networks using partial correlation. Front Neurosci, 10:123.

58.

Ward

, Schultz

, Huijbers

, Van Dijk

, Hedden

, Sperling

. 2014. The parahippocampal gyrus links the default-mode cortical network with the medial temporal lobe memory system. Hum Brain Mapp, 35:1061–1073.

59.

Xue

, Bowman

, Pileggi

, Mayer

. 2015. A multimodal approach for determining brain networks by jointly modeling functional and structural connectivity. Front Comput Neurosci, 9:22.

60.

Zhu

, Zhang

, Jiang

, Hu

, Chen

, Yang

, et al. 2014. Fusing dti and fmri data: a survey of methods and applications. NeuroImage, 102:184–191.

61.

Zuo

X-N

, Kelly

, Adelstein

, Klein

, Castellanos

, Milham

. 2010. Reliable intrinsic connectivity networks: test–retest evaluation using ica and dual regression approach. Neuroimage, 49:2163–2177.