Shift-Invariant Canonical Polyadic Decomposition of Complex-Valued Multi-Subject fMRI Data With a Phase Sparsity Constraint

Estimation of complete mutual information exploiting nonlinear magnitude-phase dependence: Application to spatial FNC for complex-valued fMRI data

BACKGROUND

Real-valued mutual information (MI) has been used in spatial functional network connectivity (FNC) to measure high-order and nonlinear dependence between spatial maps extracted from magnitude-only functional magnetic resonance imaging (fMRI). However, real-valued MI cannot fully capture the group differences in spatial FNC from complex-valued fMRI data with magnitude and phase dependence.

METHODS

We propose a complete complex-valued MI method according to the chain rule of MI. We fully exploit the dependence among magnitudes and phases of two complex-valued signals using second and fourth-order joint entropies, and propose to use a Gaussian copula transformation with a lower bound property to avoid inaccurate estimation of joint probability density function when computing the joint entropies.

RESULTS

The proposed method achieves more accurate MI estimates than the two histogram-based (normal and symbolic approaches) and kernel density estimation methods for simulated signals, and enhances group differences in spatial functional network connectivity for experimental complex-valued fMRI data.

COMPARISON WITH EXISTING METHODS

Compared with the simplified complex-valued MI and real-valued MI, the proposed method yields higher MI estimation accuracy, leading to 17.4 % and 145.5 % wider MI ranges, and more significant connectivity differences between healthy controls and schizophrenia patients. A unique connection between executive control network (EC) and right frontal parietal areas, and three additional connections mainly related to EC are detected than the simplified complex-valued MI.

CONCLUSIONS

With capability in quantifying MI fully and accurately, the proposed complex-valued MI is promising in providing qualified FNC biomarkers for identifying mental disorders such as schizophrenia.

Dynamic functional network connectivity based on spatial source phase maps of complex-valued fMRI data: Application to schizophrenia

BACKGROUND

Dynamic spatial functional network connectivity (dsFNC) has shown advantages in detecting functional alterations impacted by mental disorders using magnitude-only fMRI data. However, complete fMRI data are complex-valued with unique and useful phase information.

METHODS

We propose dsFNC of spatial source phase (SSP) maps, derived from complex-valued fMRI data (named SSP-dsFNC), to capture the dynamics elicited by the phase. We compute mutual information for connectivity quantification, employ statistical analysis and Markov chains to assess dynamics, ultimately classifying schizophrenia patients (SZs) and healthy controls (HCs) based on connectivity variance and Markov chain state transitions across windows.

RESULTS

SSP-dsFNC yielded greater dynamics and more significant HC-SZ differences, due to the use of complete brain information from complex-valued fMRI data.

COMPARISON WITH EXISTING METHODS

Compared with magnitude-dsFNC, SSP-dsFNC detected additional and meaningful connections across windows (e.g., for right frontal parietal) and achieved 14.6% higher accuracy for classifying HCs and SZs.

CONCLUSIONS

This work provides new evidence about how SSP-dsFNC could be impacted by schizophrenia, and this information could be used to identify potential imaging biomarkers for psychotic diagnosis.

Denoising brain networks using a fixed mathematical phase change in independent component analysis of magnitude-only fMRI data

Brain networks extracted by independent component analysis (ICA) from magnitude-only fMRI data are usually denoised using various amplitude-based thresholds. By contrast, spatial source phase (SSP) or the phase information of ICA brain networks extracted from complex-valued fMRI data, has provided a simple yet effective way to perform the denoising using a fixed phase change. In this work, we extend the approach to magnitude-only fMRI data to avoid testing various amplitude thresholds for denoising magnitude maps extracted by ICA, as most studies do not save the complex-valued data. The main idea is to generate a mathematical SSP map for a magnitude map using a mapping framework, and the mapping framework is built using complex-valued fMRI data with a known SSP map. Here we leverage the fact that the phase map derived from phase fMRI data has similar phase information to the SSP map. After verifying the use of the magnitude data of complex-valued fMRI, this framework is generalized to work with magnitude-only data, allowing use of our approach even without the availability of the corresponding phase fMRI datasets. We test the proposed method using both simulated and experimental fMRI data including complex-valued data from University of New Mexico and magnitude-only data from Human Connectome Project. The results provide evidence that the mathematical SSP denoising with a fixed phase change is effective for denoising spatial maps from magnitude-only fMRI data in terms of retaining more BOLD-related activity and fewer unwanted voxels, compared with amplitude-based thresholding. The proposed method provides a unified and efficient SSP approach to denoise ICA brain networks in fMRI data.

Multi-Subject Analysis for Brain Developmental Patterns Discovery via Tensor Decomposition of MEG Data

Identification of informative signatures from electrophysiological signals is important for understanding brain developmental patterns, where techniques such as magnetoencephalography (MEG) are particularly useful. However, less attention has been given to fully utilizing the multidimensional nature of MEG data for extracting components that describe these patterns. Tensor factorizations of MEG yield components that encapsulate the data's multidimensional nature, providing parsimonious models identifying latent brain patterns for meaningful summarization of neural processes. To address the need for meaningful MEG signatures for studies of pediatric cohorts, we propose a tensor-based approach for extracting developmental signatures of multi-subject MEG data. We employ the canonical polyadic (CP) decomposition for estimating latent spatiotemporal components of the data, and use these components for group level statistical inference. Using CP decomposition along with hierarchical clustering, we were able to extract typical early and late latency event-related field (ERF) components that were discriminative of high and low performance groups ([Formula: see text]) and significantly correlated with major cognitive domains such as attention, episodic memory, executive function, and language comprehension. We demonstrate that tensor-based group level statistical inference of MEG can produce signatures descriptive of the multidimensional MEG data. Furthermore, these features can be used to study group differences in brain patterns and cognitive function of healthy children. We provide an effective tool that may be useful for assessing child developmental status and brain function directly from electrophysiological measurements and facilitate the prospective assessment of cognitive processes.

Constrained CPD of Complex-Valued Multi-Subject fMRI Data via Alternating Rank-R and Rank-1 Least Squares

Complex-valued shift-invariant canonical polyadic decomposition (CPD) under a spatial phase sparsity constraint (pcsCPD) shows excellent separation performance when applied to band-pass filtered complex-valued multi-subject fMRI data. However, some useful information may also be eliminated when using a band-pass filter to suppress unwanted noise. As such, we propose an alternating rank- R and rank-1 least squares optimization to relax the CPD model. Based upon this optimization method, we present a novel constrained CPD algorithm with temporal shift-invariance and spatial sparsity and orthonormality constraints. More specifically, four steps are conducted until convergence for each iteration of the proposed algorithm: 1) use rank- R least-squares fit under spatial phase sparsity constraint to update shared spatial maps after phase de-ambiguity; 2) use orthonormality constraint to minimize the cross-talk between shared spatial maps; 3) update the aggregating mixing matrix using rank- R least-squares fit; 4) utilize shift-invariant rank-1 least-squares on a series of rank-1 matrices reconstructed by each column of the aggregating mixing matrix to update shared time courses, and subject-specific time delays and intensities. The experimental results of simulated and actual complex-valued fMRI data show that the proposed algorithm improves the estimates for task-related sensorimotor and auditory networks, compared to pcsCPD and tensorial spatial ICA. The proposed alternating rank- R and rank-1 least squares optimization is also flexible to improve CPD-related algorithm using alternating least squares.

SSPNet: An interpretable 3D-CNN for classification of schizophrenia using phase maps of resting-state complex-valued fMRI data

Convolutional neural networks (CNNs) have shown promising results in classifying individuals with mental disorders such as schizophrenia using resting-state fMRI data. However, complex-valued fMRI data is rarely used since additional phase data introduces high-level noise though it is potentially useful information for the context of classification. As such, we propose to use spatial source phase (SSP) maps derived from complex-valued fMRI data as the CNN input. The SSP maps are not only less noisy, but also more sensitive to spatial activation changes caused by mental disorders than magnitude maps. We build a 3D-CNN framework with two convolutional layers (named SSPNet) to fully explore the 3D structure and voxel-level relationships from the SSP maps. Two interpretability modules, consisting of saliency map generation and gradient-weighted class activation mapping (Grad-CAM), are incorporated into the well-trained SSPNet to provide additional information helpful for understanding the output. Experimental results from classifying schizophrenia patients (SZs) and healthy controls (HCs) show that the proposed SSPNet significantly improved accuracy and AUC compared to CNN using magnitude maps extracted from either magnitude-only (by 23.4 and 23.6% for DMN) or complex-valued fMRI data (by 10.6 and 5.8% for DMN). SSPNet captured more prominent HC-SZ differences in saliency maps, and Grad-CAM localized all contributing brain regions with opposite strengths for HCs and SZs within SSP maps. These results indicate the potential of SSPNet as a sensitive tool that may be useful for the development of brain-based biomarkers of mental disorders.

Discovering hidden brain network responses to naturalistic stimuli via tensor component analysis of multi-subject fMRI data

The study of brain network interactions during naturalistic stimuli facilitates a deeper understanding of human brain function. To estimate large-scale brain networks evoked with naturalistic stimuli, a tensor component analysis (TCA) based framework was used to characterize shared spatio-temporal patterns across subjects in a purely data-driven manner. In this framework, a third-order tensor is constructed from the timeseries extracted from all brain regions from a given parcellation, for all participants, with modes of the tensor corresponding to spatial distribution, time series and participants. TCA then reveals spatially and temporally shared components, i.e., evoked networks with the naturalistic stimuli, their time courses of activity and subject loadings of each component. To enhance the reproducibility of the estimation with the adaptive TCA algorithm, a novel spectral clustering method, tensor spectral clustering, was proposed and applied to evaluate the stability of the TCA algorithm. We demonstrated the effectiveness of the proposed framework via simulations and real fMRI data collected during a motor task with a traditional fMRI study design. We also applied the proposed framework to fMRI data collected during passive movie watching to illustrate how reproducible brain networks are evoked by naturalistic movie viewing.

Low-Rank Tucker-2 Model for Multi-Subject fMRI Data Decomposition With Spatial Sparsity Constraint

Tucker decomposition can provide an intuitive summary to understand brain function by decomposing multi-subject fMRI data into a core tensor and multiple factor matrices, and was mostly used to extract functional connectivity patterns across time/subjects using orthogonality constraints. However, these algorithms are unsuitable for extracting common spatial and temporal patterns across subjects due to distinct characteristics such as high-level noise. Motivated by a successful application of Tucker decomposition to image denoising and the intrinsic sparsity of spatial activations in fMRI, we propose a low-rank Tucker-2 model with spatial sparsity constraint to analyze multi-subject fMRI data. More precisely, we propose to impose a sparsity constraint on spatial maps by using an l_p norm ( ), in addition to adding low-rank constraints on factor matrices via the Frobenius norm. We solve the constrained Tucker-2 model using alternating direction method of multipliers, and propose to update both sparsity and low-rank constrained spatial maps using half quadratic splitting. Moreover, we extract new spatial and temporal features in addition to subject-specific intensities from the core tensor, and use these features to classify multiple subjects. The results from both simulated and experimental fMRI data verify the improvement of the proposed method, compared with four related algorithms including robust Kronecker component analysis, Tucker decomposition with orthogonality constraints, canonical polyadic decomposition, and block term decomposition in extracting common spatial and temporal components across subjects. The spatial and temporal features extracted from the core tensor show promise for characterizing subjects within the same group of patients or healthy controls as well.

Three-way parallel group independent component analysis: Fusion of spatial and spatiotemporal magnetic resonance imaging data

Advances in imaging acquisition techniques allow multiple imaging modalities to be collected from the same subject. Each individual modality offers limited yet unique views of the functional, structural, or dynamic temporal features of the brain. Multimodal fusion provides effective ways to leverage these complementary perspectives from multiple modalities. However, the majority of current multimodal fusion approaches involving functional magnetic resonance imaging (fMRI) are limited to 3D feature summaries that do not incorporate its rich temporal information. Thus, we propose a novel three-way parallel group independent component analysis (pGICA) fusion method that incorporates the first-level 4D fMRI data (temporal information included) by parallelizing group ICA into parallel ICA via a unified optimization framework. A new variability matrix was defined to capture subject-wise functional variability and then link it to the mixing matrices of the other two modalities. Simulation results show that the three-way pGICA provides highly accurate cross-modality linkage estimation under both weakly and strongly correlated conditions, as well as comparable source estimation under different noise levels. Results using real brain imaging data identified one linked functional-structural-diffusion component associated to differences between schizophrenia and controls. This was replicated in an independent cohort, and the identified components were also correlated with major cognitive domains. Functional network connectivity revealed visual-subcortical and default mode-cerebellum pairs that discriminate between schizophrenia and controls. Overall, both simulation and real data results support the use of three-way pGICA to identify multimodal spatiotemporal links and to pursue the study of brain disorders under a single unifying multimodal framework.

Sparse representation of complex-valued fMRI data based on spatiotemporal concatenation of real and imaginary parts

BACKGROUND

Spatial sparsity has been found to be in line with the intrinsic characteristic of brain activation. However, identifying a sparse representation of complex-valued fMRI data is challenging due to high noise within the phase data.

NEW METHODS

We propose to reduce the noise by combining real and imaginary parts of complex-valued fMRI data along spatial and temporal dimensions to form a real-valued spatiotemporal concatenation model. This model not only enables flexible usage of existing real-valued sparse representation algorithms but also allows for the reconstruction of complex-valued spatial and temporal components from their real and imaginary estimates. We propose to select components from both real and imaginary estimates to reconstruct the complex-valued component, using phase denoising to recover weak brain activity from high-amplitude noise.

RESULTS

The K-SVD algorithm was used to obtain a sparse representation within the spatiotemporal concatenation model. The results from simulated and experimental complex-valued fMRI datasets validated the efficacy of our method.

COMPARISON WITH EXISTING METHODS

Compared to a magnitude-only approach, the proposed method detected additional voxels manifest within several specific regions expected to be involved but likely missing from the magnitude-only data, e.g., in the anterior cingulate cortex region. Simulation results showed that the additional voxels were accurate and unique information from the phase data. Compared to a complex-valued dictionary learning algorithm, our method exhibited lower noise for both magnitude and phase maps.

CONCLUSIONS

The proposed method is robust to noise and effective for identifying a sparse representation of the natively complex-valued fMRI data.