Adaptive independent vector analysis for multi-subject complex-valued fMRI data

https://doi.org/10.1016/j.jneumeth.2017.01.017Get rights and content

Highlights

  • An adaptive IVA algorithm was proposed for multi-subject complex-valued fMRI data.

  • An MGGD-based nonlinear function was exploited to match varying SCV distributions.

  • The MGGD shape parameter was estimated using maximum likelihood estimation.

  • Subspace de-noising, post-IVA phase de-noising, and noncircularity were utilized.

  • Our method detected more contiguous activations than magnitude-only methods.

Abstract

Background

Complex-valued fMRI data can provide additional insights beyond magnitude-only data. However, independent vector analysis (IVA), which has exhibited great potential for group analysis of magnitude-only fMRI data, has rarely been applied to complex-valued fMRI data. The main challenges in this application include the extremely noisy nature and large variability of the source component vector (SCV) distribution.

New method

To address these challenges, we propose an adaptive fixed-point IVA algorithm for analyzing multiple-subject complex-valued fMRI data. We exploited a multivariate generalized Gaussian distribution (MGGD)- based nonlinear function to match varying SCV distributions in which the MGGD shape parameter was estimated using maximum likelihood estimation. To achieve our de-noising goal, we updated the MGGD-based nonlinearity in the dominant SCV subspace, and employed a post-IVA de-noising strategy based on phase information in the IVA estimates. We also incorporated the pseudo-covariance matrix of fMRI data into the algorithm to emphasize the noncircularity of complex-valued fMRI sources.

Results

Results from simulated and experimental fMRI data demonstrated the efficacy of our method.

Comparison with existing method(s)

Our approach exhibited significant improvements over typical complex-valued IVA algorithms, especially during higher noise levels and larger spatial and temporal changes. As expected, the proposed complex-valued IVA algorithm detected more contiguous and reasonable activations than the magnitude-only method for task-related (393%) and default mode (301%) spatial maps.

Conclusions

The proposed approach is suitable for decomposing multi-subject complex-valued fMRI data, and has great potential for capturing additional subject variability.

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 x˜(k)(m)J,k=1,,K, 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 x˜(k)(m) for each subject to eliminate noise effects and to reduce dimensionality. Assume that F(k)N×J(N<J) is the matrix for reducing and whitening, we obtain a PCA-reduced and whitened mixture vector as x(k)(m)=F(k)x˜(k)(m)N, and x(k)

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 ε¯, ρ¯sm_m, ρ¯sm_p, ρ¯tc_m, ρ¯tc_p and their standard deviations for the simulated fMRI data. The true number of components N = 12 was used. Our proposed method achieved the lowest ε¯, the highest ρ¯sm_m, ρ¯sm_p, ρ¯tc_m, ρ¯tc_p, 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).

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