Adaptive independent vector analysis for multi-subject complex-valued fMRI data
Introduction
Blind source separation (BSS) has been widely applied to multi-subject functional magnitude resonance imaging (fMRI) data analysis because of little requirement on prior information about the data. The resulting spatial maps (SMs) and time courses (TCs), being common components across all subjects or subject-specific, are vital for studying brain function. Among BSS techniques, tensor decomposition generates common SM and TC components across multiple subjects, but also demonstrates subject-specific intensity. Andersen and Rayens (2004) utilized canonical polyadic decomposition (CPD), which is a general tensor model that separates multi-trial fMRI data. Beckmann and Smith (2005) extended CPD to multi-subject fMRI data by combining it with ICA to deal with inter-subject SM variability. Since different subjects may generate different responses due to variation in response time or in their hemodynamic delay, inter-subject TC variability naturally also exists for multi-subject fMRI data. Kuang et al. (2015) proposed a solution by combining shift-invariant CPD (Mørup et al., 2008) and ICA to simultaneously consider inter-subject SM and TC variability. Compared to tensor decomposition, ICA-based analysis extracts subject-specific TCs and/or SMs for emphasizing inter-subject variability. Two such approaches are group ICA (Calhoun et al., 2001, Calhoun et al., 2008, Guo and Pagnonib, 2008, Erhardt et al., 2011, Calhoun and Adali, 2012b, Eloyan et al., 2013, Afshin-Pour et al., 2014) and independent vector analysis (IVA, a kind of joint ICA) (Lee et al., 2008a, Dea et al., 2011, Michael et al., 2014, Ma et al., 2014, Laney et al., 2015a, Laney et al., 2015b, Gopal et al., 2016a, Gopal et al., 2016b Adali et al., 2015). While group ICA provides individual TCs or SMs via ICA of temporally or spatially concatenated multi-subject fMRI datasets, IVA generates individual TCs and SMs via joint ICA of multi-subject fMRI datasets where similar SMs among different subjects were concatenated as source component vectors (SCVs). Multiple previous publications have shown that IVA outperforms group ICA for capturing inter-subject variability by exploiting both the dependence of similar SMs across multi-subject fMRI datasets and the independence of distinct SMs (Dea et al., 2011, Michael et al., 2014, Laney et al., 2015a, Laney et al., 2015b). Therefore, we focus on IVA methods in this study.
Real-valued IVA algorithms have been well developed and widely applied to magnitude-only multi-subject fMRI datasets. The first IVA algorithm for analyzing real-valued fMRI data was presented by Lee et al. (2007a) using a multivariate Laplace distribution (Lee et al., 2007a, Lee et al., 2008a) called IVA-L. With the development of IVA-G using multivariate Gaussian distribution (Anderson et al., 2012a), an IVA algorithm called IVA-GL was implemented by utilizing IVA-G to initialize the de-mixing matrix and IVA-L to perform the subsequent separation. IVA-GL emphasizes both second-order and higher-order statistics (Anderson et al., 2012a), and thus tends to be more efficient than IVA-L and IVA-G for fMRI analysis. IVA-GL was first tested using simulated fMRI data (Dea et al., 2011, Michael et al., 2014), and then applied to real-valued fMRI data for diverse applications: e.g., producing discriminative features for quantifying motor recovery after stroke (Laney et al., 2015a, Laney et al., 2015b); finding dynamic changes in spatial functional network connectivity in healthy individuals and schizophrenic patients (Ma et al., 2014, Calhoun and Adali, 2016); showing the spatial variation in fMRI brain networks (Gopal et al., 2016a, Gopal et al., 2016b); fusing multimodal data (Adali et al., 2015); and removing the gradient artifact in concurrently collected electroencephalogram (EEG) and fMRI data (Acharjee et al., 2015). In addition to IVA-GL, there are also other real-valued IVA algorithms that are promising for analyzing magnitude-only fMRI data, such as IVA with the Kotz family of distribution (Anderson et al., 2013), and IVA using an adaptive multivariate generalized Gaussian distribution (MGGD) (Boukouvalas et al., 2015).
However, there is a shortage of complex-valued IVA algorithms suitable for analyzing complex-valued fMRI data. Although magnitude-only fMRI data are extensively studied, fMRI data are initially acquired as complex-valued image pairs including magnitude and phase information (Adali and Calhoun, 2007, Calhoun and Adali, 2012a, Calhoun and Adali, 2012b). Phase fMRI data contain useful and unique information such as blood oxygenation levels during functional activation (Hoogenraad et al., 1998, Arja et al., 2010), the effects of macro- and micro-vessels (Menon, 2002, Tomasi and Caparelli, 2007), and the orientation of large blood vessels (Klassen and Menon, 2005). Analysis of complex-valued fMRI data provides additional insights beyond magnitude-only fMRI data (Rowe, 2005, Adali and Calhoun, 2007, Calhoun and Adali, 2012a, Calhoun and Adali, 2012b, Rodriguez et al., 2011, Rodriguez et al., 2012, Li et al., 2010, Li et al., 2011, Yu et al., 2015). The complex-valued method with pre-ICA de-noising (using observed phase images to identify and remove noisy voxels in original fMRI data) achieved higher sensitivity and specificity than the magnitude-only method (Rodriguez et al., 2011, Rodriguez et al., 2012, Li et al., 2011). By using post-ICA de-noising (using SM phase information to identify and remove noisy voxels in ICA estimates), the complex-valued method extracts more contiguous and reasonable activations than the magnitude-only method (Yu et al., 2015). This supports the potential of identifying useful brain information from complex-valued fMRI data beyond magnitude-only fMRI data. Although some complex-valued IVA algorithms exist, they are unsuitable for the analysis of complex-valued fMRI data for the following two reasons.
First, a good IVA application requires an appropriate multivariate probability density function to match SCVs distribution. Most existing algorithms were originally proposed and tuned to separate frequency-domain speech (Kim et al., 2006, Kim et al., 2007, Lee et al., 2007b, Lee et al., 2008b, Liang et al., 2014). Typical algorithms include the following: fast fixed-point IVA (FIVA) employing a spherically symmetric, exponential norm distribution (SEND), or spherically symmetric Laplace (SSL) distribution (Lee et al., 2007b); IVA assuming MGGD with a fixed shape parameter (Liang et al., 2014); and an adaptive IVA algorithm using a multivariate Gaussian mixture model for separating mixed speech and music signals (Lee et al., 2008b). These complex-valued IVA algorithms exhibited obvious degradation when applied to fMRI analysis due to different distributions between frequency-domain speech and fMRI data.
Secondly, there is large distribution variability for such a large number of fMRI SCVs (usually more than 40 SCVs). This study focuses on the analysis of full complex-valued fMRI data (without pre-ICA de-noising). In this case, many complicated noise components may be involved (e.g., head movement, respiration, cardiac pulsation) in addition to brain function-related components such as task-related components, transiently task-related components, and the default mode network (DMN). These brain function-related components generally exhibit a super-Gaussian distribution, while some noise components tend to have a sub-Gaussian distribution. Therefore, IVA using a fixed source distribution may not perform well due to mismatching distributions. The noncircular FIVA (non-FIVA) (Zhang et al., 2012) and complex-valued IVA-G (Anderson et al., 2012b) algorithms, which were proposed for noncircular sources by utilizing the information of a pseudo-covariance matrix, are candidates for extracting complex-valued fMRI sources with noncircular characteristics (Schreier et al., 2009, Schreier and Scharf, 2010, Lin et al., 2011).
This study proposes an adaptive IVA algorithm for group analysis of full complex-valued fMRI data. Preliminary results can be found in Kuang et al. (2016). We utilized MGGD, which contains multivariate super-Gaussian and sub-Gaussian distributions, and adaptively learned the MGGD shape parameter to match changing SCV distributions. Specifically, we derived a nonlinearity from an MGGD-based SCV distribution and estimated the shape parameter using maximum likelihood estimation (MLE). Furthermore, to decrease the noise effect, we adopted a subspace de-noising strategy (Na et al., 2013), and updated the MGGD-based nonlinearity in the dominant SCV subspace. Furthermore, we utilized a post-IVA phase de-noising strategy to remove noisy voxels from the SM estimates. Finally, we explicitly employed the noncircular characteristics of fMRI source signals (i.e., complex-valued SMs) by incorporating the fMRI data pseudo-covariance matrix into the IVA algorithm. Simulated and experimental fMRI data were then utilized to evaluate the proposed algorithm.
Section snippets
IVA model and cost function
Given K subjects of complex-valued fMRI data , where J is the number of time points, m = 1, …, M, and M is the total number of in-brain voxels obtained by flattening the volume image data. We first perform PCA and whitening preprocessing on for each subject to eliminate noise effects and to reduce dimensionality. Assume that is the matrix for reducing and whitening, we obtain a PCA-reduced and whitened mixture vector as , and
Experimental methods
We compared our proposed algorithm with 5 IVA algorithms: (1) IVA-GL (motivated by the good performance of real-valued IVA-GL, we generated complex-valued IVA-GL by using complex-valued IVA-G (Anderson et al., 2012b) for initialization of the de-mixing matrix, and IVA-L (Lee et al., 2007a, Lee et al., 2008a) for separation); (2) FIVA (Lee et al., 2007b); (3) non-FIVA (Zhang et al., 2012); (4) FIVA with subspace de-noising (FIVAs); and (5) non-FIVA with subspace de-noising (non-FIVAs). The
Effects of noise
Fig. 2 shows the noise influence (CNR = −10 dB to 10 dB) on the proposed algorithm, non-FIVAs, FIVAs, IVA-GL, non-FIVA, and FIVA. We computed , , , , and their standard deviations for the simulated fMRI data. The true number of components N = 12 was used. Our proposed method achieved the lowest , the highest , , , , and the smallest standard deviations for all cases, followed generally by non-FIVAs, FIVAs, non-FIVA, FIVA, and IVA-GL. We performed
Discussion
Our experimental results show that SCV shape parameters frequently changed with large amplitude for experimental fMRI data. These changes can be caused by the intrinsic differences between different SCVs, by noise, or by inter-subject spatial variability for a single SCV. As such, the proposed approach was more robust to the inter-subject variability and noise effect than non-FIVAs, FIVAs, non-FIVA, FIVA, and IVA-GL. Among these 5 IVA algorithm, non-FIVAs utilized β = 0.5 and achieved the best
Conclusions
This study proposed a new adaptive complex-valued IVA algorithm to analyze multi-subject complex-valued fMRI data. We proposed to use an MGGD-based nonlinear function with adaptively learned shape parameters to match varying SCV distributions, update the nonlinearity in the dominant SCV subspace to remove the effect of inter-subject spatial variability, and incorporate the pseudo-covariance matrix of fMRI data into the learning rule to emphasize the noncircularity of complex-valued fMRI
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 61379012, 61105008, 61331019, and 81471742, NSF grants 0840895 and 0715022, NIH grants R01EB005846 and 5P20GM103472, and the Fundamental Research Funds for the Central Universities (China, DUT14RC(3)037).
References (63)
- et al.
Evaluation of spatio-temporal decomposition techniques for group analysis of fMRI resting state datasets
Neuroimage
(2014) - et al.
Structure-seeking multilinear methods for the analysis of fMRI data
Neuroimage
(2004) - et al.
Complex-valued independent vector analysis: application to multivariate Gaussian model
Signal Process.
(2012) - et al.
Changes in fMRI magnitude data and phase data observed in block-design and event-related tasks
Neuroimage
(2010) - et al.
Tensorial extensions of independent component analysis for multisubject fMRI analysis
Neuroimage
(2005) - et al.
Key issues in decomposing fMRI during naturalistic and continuous music experience with independent component analysis
J. Neurosci. Methods
(2014) - et al.
Biophysical modeling of phase changes in BOLD fMRI
Neuroimage
(2009) - et al.
A unified framework for group independent component analysis for multi-subject fMRI data
Neuroimage
(2008) - et al.
The effect of physiological noise in phase functional magnetic resonance imaging: from blood oxygen level-dependent effects to direct detection of neuronal currents
Magn. Reson. Imaging
(2008) - et al.
Improving robustness and reliability of phase sensitive fMRI analysis using temporal off-resonance alignment of single-echo timeseries (TOAST)
Neuroimage
(2009)
Multi-subject fMRI analysis via combined independent component analysis and shift-invariant canonical polyadic decomposition
J. Neurosci. Methods
Capturing subject variability in fMRI data: a graph-theoretical analysis of GICA vs. IVA
J. Neurosci. Methods
Quantifying motor recovery after stroke using independent vector analysis and graph-theoretical analysis
NeuroImage: Clin.
Fast fixed-point independent vector analysis algorithms for convolutive blind source separation
Signal Process.
Independent vector analysis (IVA): multivariate approach for fMRI group study
Neuroimage
Independent vector analysis with a generalized multivariate Gaussian source prior for frequency domain blind source separation
Signal Process.
Semi-blind kurtosis maximization algorithm applied to complex-valued fMRI data
2011 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)
Shift-invariant multilinear decomposition of neuroimaging data
Neuroimage
Dynamic changes of spatial functional network connectivity in healthy individuals and schizophrenia patients using independent vector analysis
Neuroimage
De-noising, phase ambiguity correction and visualization techniques for complex-valued ICA of group fMRI data
Pattern Recogn.
Modeling both the magnitude and phase of complex-valued fMRI data
Neuroimage
ICA of full complex-valued fMRI data using phase information of spatial maps
J. Neurosci. Methods
Independent vector analysis for convolutive blind noncircular source separation
Signal Process.
Independent vector analysis for gradient artifact removal in concurrent EEG-fMRI data
IEEE Trans. Biomed. Eng.
Complex ICA of brain imaging data
IEEE Signal Process Mag.
Multimodal data fusion using source separation: two effective models based on ICA and IVA and their properties
Proc. IEEE
Joint blind source separation with multivariate Gaussian model: algorithms and performance analysis
IEEE Trans. Signal Process.
Independent vector analysis, the kotz distribution, and performance bounds
IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
A fast fixed-point algorithm for independent component analysis of complex valued signals
Int. J. Neural Syst.
An efficient multivariate generalized Gaussian distribution estimator: application to IVA
The 49th Annual Conference on Information Sciences and Systems (CISS)
Analysis of complex-valued functional magnetic resonance imaging data: are we just going through a phase?
Bull. Polish Acad. Sci. Tech. Sci.
Cited by (15)
Dynamic functional network connectivity based on spatial source phase maps of complex-valued fMRI data: Application to schizophrenia
2024, Journal of Neuroscience MethodsIndependent vector analysis: Model, applications, challenges
2023, Pattern RecognitionCoupled canonical polyadic decomposition of multi-group fMRI data with spatial reference and orthonormality constraints
2023, Biomedical Signal Processing and ControlSparse representation of complex-valued fMRI data based on spatiotemporal concatenation of real and imaginary parts
2021, Journal of Neuroscience MethodsCitation Excerpt :The proposed method detects 538.52 % (2586 vs. 405) more contiguous voxels than the magnitude-only approach, achieving successful detection of the ACC. These DMN results are also consistent with a previous ICA finding that complex-valued fMRI data provides additional brain information compared to magnitude-only fMRI data (Yu et al., 2015; Kuang et al., 2016, 2017a, b, 2018, 2020). Regarding comparison of the two SM denoising approaches for the proposed method, as shown in Fig. 5A, we found that phase denoising provided 48.60 % more Ѵin (2868 vs. 1930) and 26.79 % more Ѵout (2371 vs. 1870) than amplitude thresholding at Z > 1.5 in Fig. 5A(2), while removing a large number of high-amplitude (8.3 < Z ≤ 27.3) unwanted voxels located at the edges of the brain.
Characteristics and variability of functional brain networks
2020, Neuroscience Letters