Stochastic geometric network models for groups of functional and structural connectomes.

Structural and functional connectomes are emerging as important instruments in the study of normal brain function and in the development of new biomarkers for a variety of brain disorders. In contrast to single-network studies that presently dominate the (non-connectome) network literature, connectome analyses typically examine groups of empirical networks and then compare these against standard (stochastic) network models. The current practice in connectome studies is to employ stochastic network models derived from social science and engineering contexts as the basis for the comparison. However, these are not necessarily best suited for the analysis of connectomes, which often contain groups of very closely related networks, such as occurs with a set of controls or a set of patients with a specific disorder. This paper studies important extensions of standard stochastic models that make them better adapted for analysis of connectomes, and develops new statistical fitting methodologies that account for inter-subject variations. The extensions explicitly incorporate geometric information about a network based on distances and inter/intra hemispherical asymmetries (to supplement ordinary degree-distribution information), and utilize a stochastic choice of network density levels (for fixed threshold networks) to better capture the variance in average connectivity among subjects. The new statistical tools introduced here allow one to compare groups of networks by matching both their average characteristics and the variations among them. A notable finding is that connectomes have high "smallworldness" beyond that arising from geometric and degree considerations alone.

Resting-state networks and the functional connectome of the human brain in agenesis of the corpus callosum.

The corpus callosum is the largest white matter fiber bundle connecting the two cerebral hemispheres. In this work, we investigate the effect of callosal dysgenesis on functional magnetic resonance imaging (fMRI) resting-state networks and the functional connectome. Since alternate commissural routes between the cerebral hemispheres exist, we hypothesize that bilateral cortical networks can still be maintained in partial or even complete agenesis of the corpus callosum (AgCC). However, since these commissural routes are frequently indirect, requiring polysynaptic pathways, we hypothesize that quantitative measurements of interhemispheric functional connectivity in bilateral networks will be reduced in AgCC compared with matched controls, especially in the most highly interconnected cortical regions that are the hubs of the connectome. Seventeen resting-state networks were extracted from fMRI of 11 subjects with partial or complete AgCC and 11 matched controls. The results show that the qualitative organization of resting-state networks is very similar between controls and AgCC. However, interhemispheric functional connectivity of precuneus, posterior cingulate cortex, and insular-opercular regions was significantly reduced in AgCC. The preserved network organization was confirmed with a connectomic analysis of the resting-state fMRI data, showing five functional modules that are largely consistent across the control and AgCC groups. Hence, the reduction or even complete absence of callosal connectivity does not affect the qualitative organization of bilateral resting-state networks or the modular organization of the functional connectome, although quantitatively reduced functional connectivity can be demonstrated by measurements within bilateral cortical hubs, supporting the hypothesis that indirect polysynaptic pathways are utilized to preserve interhemispheric temporal synchrony.

The structural connectome of the human brain in agenesis of the corpus callosum.

Adopting a network perspective, the structural connectome reveals the large-scale white matter connectivity of the human brain, yielding insights into cerebral organization otherwise inaccessible to researchers and clinicians. Connectomics has great potential for elucidating abnormal connectivity in congenital brain malformations, especially axonal pathfinding disorders. Agenesis of the corpus callosum (AgCC) is one of the most common brain malformations and can also be considered a prototypical genetic disorder of axonal guidance in humans. In this exploratory study, the structural connectome of AgCC is mapped and compared to that of the normal human brain. Multiple levels of granularity of the AgCC connectome are investigated, including summary network metrics, modularity analysis, and network consistency measures, with comparison to the normal structural connectome after simulated removal of all callosal connections ("virtual callostomy"). These investigations reveal four major findings. First, global connectivity is abnormally reduced in AgCC, but local connectivity is increased. Second, the network topology of AgCC is more variable than that of the normal human connectome, contradicting the predictions of the virtual callostomy model. Third, modularity analysis reveals that many of the tracts that comprise the structural core of the cerebral cortex have relatively weak connectivity in AgCC, especially the cingulate bundles bilaterally. Finally, virtual lesions of the Probst bundles in the AgCC connectome demonstrate that there is consistency across subjects in many of the connections generated by these ectopic white matter tracts, and that they are a mixture of cortical and subcortical fibers. These results go beyond prior diffusion tractography studies to provide a systems-level perspective on anomalous connectivity in AgCC. Furthermore, this work offers a proof of principle for the utility of the connectome framework in neurodevelopmental disorders.

Independent component analysis of DTI reveals multivariate microstructural correlations of white matter in the human brain.

It has recently been demonstrated that specific patterns of correlation exist in diffusion tensor imaging (DTI) parameters across white matter tracts in the normal human brain. These microstructural correlations are thought to reflect phylogenetic and functional similarities between different axonal fiber pathways. However, this earlier work was limited in three major respects: (1) the analysis was restricted to only a dozen selected tracts; (2) the DTI measurements were averaged across whole tracts, whereas metrics such as fractional anisotropy (FA) are known to vary considerably within single tracts; and (3) a univariate measure of correlation was used. In this investigation, we perform an automated multivariate whole-brain voxel-based study of white matter FA correlations using independent component analysis (ICA) of tract-based spatial statistics computed from 3T DTI in 53 healthy adult volunteers. The resulting spatial maps of the independent components show voxels for which the FA values within each map co-vary across individuals. The strongest FA correlations were found in anatomically recognizable tracts and tract segments, either singly or in homologous pairs. Hence, ICA of DTI provides an automated unsupervised decomposition of the normal human brain into multiple separable microstructurally correlated white matter regions, many of which correspond to anatomically familiar classes of white matter pathways. Further research is needed to determine whether whole-brain ICA of DTI represents a novel alternative to tractography for feature extraction in studying the normal microstructure of human white matter as well as the abnormal white matter microstructure found in neurological and psychiatric disorders.

Order Selection of the Linear Mixing Model for Complex-valued FMRI Data.

Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, which motivates the use of complex-valued data analysis methods. Due to the high dimension and high noise level of fMRI data, order selection and dimension reduction are important procedures for multivariate analysis methods such as independent component analysis (ICA). In this work, we develop a complex-valued order selection method to estimate the dimension of signal subspace using information-theoretic criteria. To correct the effect of sample dependence to information-theoretic criteria, we develop a general entropy rate measure for complex Gaussian random process to calibrate the independent and identically distributed (i.i.d.) sampling scheme in the complex domain. We show the effectiveness of the approach for order selection on both simulated and actual fMRI data. A comparison between the results of order selection and ICA on real-valued and complex-valued fMRI data demonstrates that a fully complex analysis extracts more information about brain activation.

Group Study of Simulated Driving fMRI Data by Multiset Canonical Correlation Analysis.

In this work, we apply a novel statistical method, multiset canonical correlation analysis (M-CCA), to study a group of functional magnetic resonance imaging (fMRI) datasets acquired during simulated driving task. The M-CCA method jointly decomposes fMRI datasets from different subjects/sessions into brain activation maps and their associated time courses, such that the correlation in each group of estimated activation maps across datasets is maximized. Therefore, the functional activations across all datasets are extracted in the order of consistency across different dataset. On the other hand, M-CCA preserves the uniqueness of the functional maps estimated from each dataset by avoiding concatenation of different datasets in the analysis. Hence, the cross-dataset variation of the functional activations can be used to test the hypothesis of functional-behavioral association. In this work, we study 120 simulated driving fMRI datasets and identify parietal-occipital regions and frontal lobe as the most consistently engaged areas across all the subjects and sessions during simulated driving. The functional-behavioral association study indicates that all the estimated brain activations are significantly correlated with the steering operation during the driving task. M-CCA thus provides a new approach to investigate the complex relationship between the brain functions and multiple behavioral variables, especially in naturalistic tasks as demonstrated by the simulated driving study.

On entropy rate for the complex domain.

We derive the entropy rate formula for a complex Gaussian random process by using a widely linear model. The resulting expression is general and applicable to both circular and noncircular Gaussian processes, since any second-order stationary process can be modeled as the output of a widely linear system driven by a circular white noise. Furthermore, we demonstrate application of the derived formula to an order selection problem. We extend a scheme for independent and identically distributed (i.i.d.) sampling to the complex domain to improve the estimation performance of information-theoretic criteria when samples are correlated. We show the effectiveness of the approach for order selection for simulated and actual functional magnetic resonance imaging (fMRI) data that are inherently complex valued.

Microstructural correlations of white matter tracts in the human brain.

The purpose of this study is to investigate whether specific patterns of correlation exist in diffusion tensor imaging (DTI) parameters across different white matter tracts in the normal human brain, and whether the relative strengths of these putative microstructural correlations might reflect phylogenetic and functional similarities between tracts. We performed quantitative DTI fiber tracking on 44 healthy adult volunteers to obtain tract-based measures of mean diffusivity (MD), fractional anisotropy (FA), axial diffusivity (AD), and radial diffusivity (RD) from four homologous pairs of neocortical association pathways (arcuate fasciculi, inferior fronto-occipital fasciculi, inferior longitudinal fasciculi, and uncinate fasciculi bilaterally), a homologous pair of limbic association pathways (left and right dorsal cingulum bundles), and a homologous pair of cortical-subcortical projection pathways (left and right corticospinal tracts). From the resulting inter-tract correlation matrices, we show that there are statistically significant correlations of DTI parameters between tracts, and that there are statistically significant variations among these inter-tract correlations. Furthermore, we observe that many, but by no means all, of the strongest correlations are between homologous tracts in the left and right hemispheres. Even among homologous pairs of tracts, there are wide variations in the degree of coupling. Finally, we generate a data-driven hierarchical clustering of the fiber pathways based on pairwise FA correlations to demonstrate that the neocortical association pathways tend to group separately from the limbic pathways at trend-level statistical significance, and that the projection pathways of the left and right corticospinal tracts comprise the most distant outgroup with high confidence (p<0.01). Hence, specific patterns of microstructural correlation exist between tracts and may reflect phylogenetic and functional similarities between tracts. The study of these microstructural relationships between white matter pathways might aid research on the genetic basis and on the behavioral effects of axonal connectivity, as well as provide a revealing new perspective with which to investigate neurological and psychiatric disorders.

Multi-set canonical correlation analysis for the fusion of concurrent single trial ERP and functional MRI.

Functional magnetic resonance imaging (fMRI) data and electroencephalography (EEG) data provide complementary spatio-temporal information about brain function. Methods to couple the relative strengths of these modalities usually involve two stages: first forming a feature set from each dataset based on one criterion followed by exploration of connections among the features using a second criterion. We propose a data fusion method for simultaneously acquired fMRI and EEG data that combines these steps using a single criterion for finding the cross-modality associations and performing source separation. Using multi-set canonical correlation analysis (M-CCA), we obtain a decomposition of the two modalities, into spatial maps for fMRI data and a corresponding temporal evolution for EEG data, based on trial-to-trial covariation across the two modalities. Additionally, the analysis is performed on data from a group of subjects in order to make group inferences about the covariation across modalities. Being multivariate, the proposed method facilitates the study of brain connectivity along with localization of brain function. M-CCA can be easily extended to incorporate different data types and additional modalities. We demonstrate the promise of the proposed method in finding covarying trial-to-trial amplitude modulations (AMs) in an auditory task involving implicit pattern learning. The results show approximately linear decreasing trends in AMs for both modalities and the corresponding spatial activations occur mainly in motor, frontal, temporal, inferior parietal, and orbito-frontal areas that are linked both to sensory function as well as learning and expectation--all of which match activations related to the presented paradigm.

Joint Blind Source Separation by Multi-set Canonical Correlation Analysis.

In this work, we introduce a simple and effective scheme to achieve joint blind source separation (BSS) of multiple datasets using multi-set canonical correlation analysis (M-CCA) [1]. We first propose a generative model of joint BSS based on the correlation of latent sources within and between datasets. We specify source separability conditions, and show that, when the conditions are satisfied, the group of corresponding sources from each dataset can be jointly extracted by M-CCA through maximization of correlation among the extracted sources. We compare source separation performance of the M-CCA scheme with other joint BSS methods and demonstrate the superior performance of the M-CCA scheme in achieving joint BSS for a large number of datasets, group of corresponding sources with heterogeneous correlation values, and complex-valued sources with circular and non-circular distributions. We apply M-CCA to analysis of functional magnetic resonance imaging (fMRI) data from multiple subjects and show its utility in estimating meaningful brain activations from a visuomotor task.

Canonical Correlation Analysis for Feature-Based Fusion of Biomedical Imaging Modalities and Its Application to Detection of Associative Networks in Schizophrenia.

Typically data acquired through imaging techniques such as functional magnetic resonance imaging (fMRI), structural MRI (sMRI), and electroencephalography (EEG) are analyzed separately. However, fusing information from such complementary modalities promises to provide additional insight into connectivity across brain networks and changes due to disease. We propose a data fusion scheme at the feature level using canonical correlation analysis (CCA) to determine inter-subject covariations across modalities. As we show both with simulation results and application to real data, multimodal CCA (mCCA) proves to be a flexible and powerful method for discovering associations among various data types. We demonstrate the versatility of the method with application to two datasets, an fMRI and EEG, and an fMRI and sMRI dataset, both collected from patients diagnosed with schizophrenia and healthy controls. CCA results for fMRI and EEG data collected for an auditory oddball task reveal associations of the temporal and motor areas with the N2 and P3 peaks. For the application to fMRI and sMRI data collected for an auditory sensorimotor task, CCA results show an interesting joint relationship between fMRI and gray matter, with patients with schizophrenia showing more functional activity in motor areas and less activity in temporal areas associated with less gray matter as compared to healthy controls. Additionally, we compare our scheme with an independent component analysis based fusion method, joint-ICA that has proven useful for such a study and note that the two methods provide complementary perspectives on data fusion.

Estimating the number of independent components for functional magnetic resonance imaging data.

Multivariate analysis methods such as independent component analysis (ICA) have been applied to the analysis of functional magnetic resonance imaging (fMRI) data to study brain function. Because of the high dimensionality and high noise level of the fMRI data, order selection, i.e., estimation of the number of informative components, is critical to reduce over/underfitting in such methods. Dependence among fMRI data samples in the spatial and temporal domain limits the usefulness of the practical formulations of information-theoretic criteria (ITC) for order selection, since they are based on likelihood of independent and identically distributed (i.i.d.) data samples. To address this issue, we propose a subsampling scheme to obtain a set of effectively i.i.d. samples from the dependent data samples and apply the ITC formulas to the effectively i.i.d. sample set for order selection. We apply the proposed method on the simulated data and show that it significantly improves the accuracy of order selection from dependent data. We also perform order selection on fMRI data from a visuomotor task and show that the proposed method alleviates the over-estimation on the number of brain sources due to the intrinsic smoothness and the smooth preprocessing of fMRI data. We use the software package ICASSO (Himberg et al. [ 2004]: Neuroimage 22:1214-1222) to analyze the independent component (IC) estimates at different orders and show that, when ICA is performed at overestimated orders, the stability of the IC estimates decreases and the estimation of task related brain activations show degradation.

A feature-selective independent component analysis method for functional MRI.

In this work, we propose a simple and effective scheme to incorporate prior knowledge about the sources of interest (SOIs) in independent component analysis (ICA) and apply the method to estimate brain activations from functional magnetic resonance imaging (fMRI) data. We name the proposed method as feature-selective ICA since it incorporates the features in the sample space of the independent components during ICA estimation. The feature-selective scheme is achieved through a filtering operation in the source sample space followed by a projection onto the demixing vector space by a least squares projection in an iterative ICA process. We perform ICA estimation of artificial activations superimposed into a resting state fMRI dataset to show that the feature-selective scheme improves the detection of injected activation from the independent component estimated by ICA. We also compare the task-related sources estimated from true fMRI data by a feature-selective ICA algorithm versus an ICA algorithm and show evidence that the feature-selective scheme helps improve the estimation of the sources in both spatial activation patterns and the time courses.