ICA of full complex-valued fMRI data using phase information of spatial maps

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.

A new transfer entropy method for measuring directed connectivity from complex-valued fMRI data

Background

Inferring directional connectivity of brain regions from functional magnetic resonance imaging (fMRI) data has been shown to provide additional insights into predicting mental disorders such as schizophrenia. However, existing research has focused on the magnitude data from complex-valued fMRI data without considering the informative phase data, thus ignoring potentially important information.

Methods

We propose a new complex-valued transfer entropy (CTE) method to measure causal links among brain regions in complex-valued fMRI data. We use the transfer entropy to model a general non-linear magnitude-magnitude and phase-phase directed connectivity and utilize partial transfer entropy to measure the complementary phase and magnitude effects on magnitude-phase and phase-magnitude causality. We also define the significance of the causality based on a statistical test and the shuffling strategy of the two complex-valued signals.

Results

Simulated results verified higher accuracy of CTE than four causal analysis methods, including a simplified complex-valued approach and three real-valued approaches. Using experimental fMRI data from schizophrenia and controls, CTE yields results consistent with previous findings but with more significant group differences. The proposed method detects new directed connectivity related to the right frontal parietal regions and achieves 10.2-20.9% higher SVM classification accuracy when inferring directed connectivity using anatomical automatic labeling (AAL) regions as features.

Conclusion

The proposed CTE provides a new general method for fully detecting highly predictive directed connectivity from complex-valued fMRI data, with magnitude-only fMRI data as a specific case.

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.

Enhanced clinical task-based fMRI metrics through locally low-rank denoising of complex-valued data

OBJECTIVE

This study investigates a locally low-rank (LLR) denoising algorithm applied to source images from a clinical task-based functional MRI (fMRI) exam before post-processing for improving statistical confidence of task-based activation maps.

METHODS

Task-based motor and language fMRI was obtained in eleven healthy volunteers under an IRB approved protocol. LLR denoising was then applied to raw complex-valued image data before fMRI processing. Activation maps generated from conventional non-denoised (control) data were compared with maps derived from LLR-denoised image data. Four board-certified neuroradiologists completed consensus assessment of activation maps; region-specific and aggregate motor and language consensus thresholds were then compared with nonparametric statistical tests. Additional evaluation included retrospective truncation of exam data without and with LLR denoising; a ROI-based analysis tracked t-statistics and temporal SNR (tSNR) as scan durations decreased. A test-retest assessment was performed; retest data were matched with initial test data and compared for one subject.

RESULTS

fMRI activation maps generated from LLR-denoised data predominantly exhibited statistically significant (p = 4.88×10-4 to p = 0.042; one p = 0.062) increases in consensus t-statistic thresholds for motor and language activation maps. Following data truncation, LLR data showed task-specific increases in t-statistics and tSNR respectively exceeding 20 and 50% compared to control. LLR denoising enabled truncation of exam durations while preserving cluster volumes at fixed thresholds. Test-retest showed variable activation with LLR data thresholded higher in matching initial test data.

CONCLUSION

LLR denoising affords robust increases in t-statistics on fMRI activation maps compared to routine processing, and offers potential for reduced scan duration while preserving map quality.

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.

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.

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

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complex-valued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed l₀ norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shift-invariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations.

Spatial source phase: A new feature for identifying spatial differences based on complex-valued resting-state fMRI data

Spatial source phase, the phase information of spatial maps extracted from functional magnetic resonance imaging (fMRI) data by data-driven methods such as independent component analysis (ICA), has rarely been studied. While the observed phase has been shown to convey unique brain information, the role of spatial source phase in representing the intrinsic activity of the brain is yet not clear. This study explores the spatial source phase for identifying spatial differences between patients with schizophrenia (SZs) and healthy controls (HCs) using complex-valued resting-state fMRI data from 82 individuals. ICA is first applied to preprocess fMRI data, and post-ICA phase de-ambiguity and denoising are then performed. The ability of spatial source phase to characterize spatial differences is examined by the homogeneity of variance test (voxel-wise F-test) with false discovery rate correction. Resampling techniques are performed to ensure that the observations are significant and reliable. We focus on two components of interest widely used in analyzing SZs, including the default mode network (DMN) and auditory cortex. Results show that the spatial source phase exhibits more significant variance changes and higher sensitivity to the spatial differences between SZs and HCs in the anterior areas of DMN and the left auditory cortex, compared to the magnitude of spatial activations. Our findings show that the spatial source phase can potentially serve as a new brain imaging biomarker and provide a novel perspective on differences in SZs compared to HCs, consistent with but extending previous work showing increased variability in patient data.

Methods for cleaning the BOLD fMRI signal

Blood oxygen-level-dependent functional magnetic resonance imaging (BOLD fMRI) has rapidly become a popular technique for the investigation of brain function in healthy individuals, patients as well as in animal studies. However, the BOLD signal arises from a complex mixture of neuronal, metabolic and vascular processes, being therefore an indirect measure of neuronal activity, which is further severely corrupted by multiple non-neuronal fluctuations of instrumental, physiological or subject-specific origin. This review aims to provide a comprehensive summary of existing methods for cleaning the BOLD fMRI signal. The description is given from a methodological point of view, focusing on the operation of the different techniques in addition to pointing out the advantages and limitations in their application. Since motion-related and physiological noise fluctuations are two of the main noise components of the signal, techniques targeting their removal are primarily addressed, including both data-driven approaches and using external recordings. Data-driven approaches, which are less specific in the assumed model and can simultaneously reduce multiple noise fluctuations, are mainly based on data decomposition techniques such as principal and independent component analysis. Importantly, the usefulness of strategies that benefit from the information available in the phase component of the signal, or in multiple signal echoes is also highlighted. The use of global signal regression for denoising is also addressed. Finally, practical recommendations regarding the optimization of the preprocessing pipeline for the purpose of denoising and future venues of research are indicated. Through the review, we summarize the importance of signal denoising as an essential step in the analysis pipeline of task-based and resting state fMRI studies.

The role of diversity in complex ICA algorithms for fMRI analysis

BACKGROUND

The widespread use of data-driven methods, such as independent component analysis (ICA), for the analysis of functional magnetic resonance imaging data (fMRI) has enabled deeper understanding of neural function. However, most popular ICA algorithms for fMRI analysis make several simplifying assumptions, thus ignoring sources of statistical information, types of "diversity," and limiting their performance.

NEW METHOD

We propose the use of complex entropy rate bound minimization (CERBM) for the analysis of actual fMRI data in its native, complex, domain. Though CERBM achieves enhanced performance through the exploitation of the three types of diversity inherent to complex fMRI data: noncircularity, non-Gaussianity, and sample-to-sample dependence, CERBM produces results that are more variable than simpler methods. This motivates the development of a minimum spanning tree (MST)-based stability analysis that mitigates the variability of CERBM.

COMPARISON WITH EXISTING METHODS

In order to validate our method, we compare the performance of CERBM with the popular CInfomax as well as complex entropy bound minimization (CEBM).

RESULTS

We show that by leveraging CERBM and the MST-based stability analysis, we are able to consistently produce components that have a greater number of activated voxels in physically meaningful regions and can more accurately classify patients with schizophrenia than components generated using simpler models.

CONCLUSIONS

Our results demonstrate the advantages of using ICA algorithms that can exploit all inherent types of diversity for the analysis of fMRI data when coupled with appropriate stability analyses.