LEICA: Laplacian eigenmaps for group ICA decomposition of fMRI data
Introduction
Functional magnetic resonance imaging (fMRI) studies have shed light on the overall functional organization of the brain. For example, resting state studies have revealed the existence of a number of intrinsic functional networks, including the “default mode” and “salience” networks Biswal et al., 1995, Cole et al., 2014, Greicius et al., 2003. These studies have also led to the discovery of specific patterns related to brain disorders Jafri et al., 2008, Greicius et al., 2004, Garrity et al., 2007, age Damoiseaux et al. (2008), and gender Weissman-Fogel et al. (2010).
The determination of large-scale patterns of brain activity is a challenging problem due to several factors, including the high-dimensionality of the data, various noise sources, and inter-subject variability. The most widely used approach to capture group-level functional connectivity is based on Independent Component Analysis (ICA) McKeown et al. (1998), which is a data-driven approach that assumes the existence of statistically independent latent features, called sources, which can generate the data through linear mixtures. However, ICA does not naturally provide a suitable method for drawing inferences about groups of subjects Erhardt et al. (2011), which has led to the development of several multi-subject approaches that are partly based on ICA. These approaches revolve around a few strategies. One method is to concatenate the data from multiple subjects along the temporal dimension Calhoun et al. (2001) followed by the use of Principal Component Analysis (PCA) Jolliffe (2002) to reduce the dimensionality of the data, and ending with the application of ICA to extract spatial features common at the group level. The tensorial extension to ICA Beckmann and Smith (2005) uses a three-dimensional tensor to estimate shared spatial patterns and time courses between subjects, and subject-specific loadings of the components to capture the multidimensional structure of the data. In Guo and Pagnoni (2008), a general framework is proposed for using the expectation maximization algorithm to estimate an unconstrained mixing matrix for the group-level independent components. Finally, canonical ICA Varoquaux et al. (2010) uses Canonical Correlation Analysis (CCA) to find components that maximize cross-correlation before applying the ICA step.
Two methods involving temporal concatenation of the data include GIFT Calhoun et al. (2001) (http://icatb.sourceforge.net/) and MELODIC Beckmann and Smith (2004) (http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/MELODIC/). GIFT approximates the PCA of the group temporally concatenated data by first reducing each subject's dataset to dimension m, and then running the PCA on the concatenation of the principal components across all subjects to further reduce temporal dimensionality. MELODIC, however, uses an incremental approach called MELODIC's Incremental Group PCA (MIGP) Smith et al. (2014) to implement the approximation. Although MELODIC and GIFT have been broadly used for group level fMRI analysis, they are not without limitations as will be explained next.
These ICA-based strategies involve the use of PCA to reduce the dimensionality of the data. PCA is basically a linear projection of the original data onto a low-dimensional subspace in such a way as to retain most of the variance of the data. However, PCA does not distinguish between variance due to noise versus variance due to inherent underlying signal variations. Moreover, there is no principled justification for using the variance as the basis for data reduction based solely on a linear transformation. The BOLD signal is a complex function of neural activity, oxygen metabolism, cerebral blood volume, cerebral blood flow, and other physiological parameters Wang et al. (2003). The dynamics underlying neural activity and hemodynamic physiology involve multiple nonlinearities Birn et al., 2001, Miller et al., 2001, Xie et al., 2008, a fact that has motivated the exploration of nonlinear approaches for fMRI analysis Thirion and Faugeras, 2004, Mannfolk et al., 2010. To the best of our knowledge, nonlinear models have not been proposed to extract common spatial patterns for multi-subject fMRI analysis.
Several nonlinear dimensionality reduction approaches have been successfully used to address a wide range of applications Van Der Maaten et al. (2009). Kernel PCA Schölkopf et al. (1998) maps data to a feature space using a nonlinear transformation (kernel) and performs linear PCA in the feature space. Isomap Tenenbaum et al. (2000) is based on computing a low dimensional representation that tries to preserve the pairwise geodesic distances, which are approximated by the pairwise shortest paths between corresponding points. Maximum variance unfolding (MVU) Weinberger and Saul (2006) computes a low-dimensional manifold such that local distances and angles are preserved. Locally linear embedding (LLE) Roweis and Saul (2000) is based on the geometric intuition of local linearity, which assumes that each point and its neighbors lie on an approximately linear patch of a low-dimensional manifold. The mapping is computed by trying to preserve linearity locally, such that each point has the same neighborhood structure as in the high-dimensional space. A technique of a different flavor is the t-distributed stochastic neighbor embedding (t-SNE) Maaten and Hinton (2008), which captures similarities between pairs of high-dimensional points through a probability distribution, followed by determining a mapping to a lower-dimensional space that achieves a similar distribution. Laplacian eigenmaps Belkin and Niyogi (2003) project data into a low dimensional Euclidean space such that local proximity relations are preserved as much as possible, mapping close points in the original space to close points in the low-dimensional space. Of these techniques, Laplacian eigenmaps appear to be the most suitable for fMRI data since its overall objective function can be expressed as minimizingwhere are the low-dimensional vectors corresponding to points in the original space, and is a proximity measure of points i and j in the original space. If we choose to be the zero-thresholded correlation between voxels i and j in the original space, this nonlinear projection maps voxels with strong correlations to nearby points in the low-dimensional Euclidean space. Correlations between voxels in fMRI data reveal functional relationships and hence contain the information that can be used to construct functional networks.
We note that Laplacian eigenmaps have been successfully used in analyzing fMRI data Marquand et al., 2017, Haak et al., 2016. Our approach differs from earlier studies in that they are ROI-based methods focusing on specific networks using prior brain-related knowledge, while our framework makes use of ICA to extract intrinsic networks for general fMRI datasets.
In this paper, we propose a group-level model that uses Laplacian eigenmaps as the main data reduction step, which preserves the correlation information in the original data as best as possible. The nonlinear map is robust relative to noise in the data and to inter-subject variability. After the Laplacian eigenmaps transformation, ICA is applied to the reduced data to estimate spatial features, followed by thresholding to extract group-level independent spatial maps. We call our algorithm LEICA (Laplacian Eigenmaps for group ICA decomposition).
The rest of the paper is organized as follows. In section 2, we present the details of our algorithm and inference method, and describe several tests used to validate our method and compare the results with state-of-the-art methods. In section 3, we show the corresponding test results on both resting state and working memory task fMRI datasets for 100 subjects from the Human Connectome Project. Finally, in section 4, we discuss our results.
Section snippets
Group ICA model
Spatial ICA assumes that the observed fMRI data are generated by a linear mixture of spatially independent components. The underlying model can be expressed as:where X is the observation matrix whose columns represent the time series corresponding to the voxels, A consists of the corresponding independent sources, and M is the mixing matrix. We note that (2) does not include a noise model, but fMRI data is inevitably confounded by noise. Moreover, as stated, the model is not suitable for
Group-level spatial maps
As described in section 2.6, an important parameter that is estimated from the data by the model is the number k of nearest neighbors used in constructing the similarity graph. Based on the resting state dataset, . Fig. 2 shows the 20 group-level thresholded spatial maps extracted by LEICA on the resting state dataset. The extracted independent spatial components were converted to Z scores and thresholded at .
Similarity score
We first visually compared the spatial maps from LEICA and MELODIC.
Discussion
In the present paper, we propose a nonlinear ICA-based model for extracting group-level spatial maps from multi-subject fMRI datasets. There are two key elements to our method. The first element is the construction of a k-nearest neighbor graph based on the group-average correlation matrix, which makes the final results less noisy and captures intrinsic functional connectivity. The second element is the use of a nonlinear dimensionality reduction stage based on Laplacian eigenmaps, which seeks
Conclusion
In this paper, we present a novel model – called LEICA – involving a non-linear dimensionality reduction procedure followed by ICA to identify group-level spatial features from multi-subject fMRI datasets, and applied it to resting state and working memory data. The spatial maps extracted by our model were shown to be related to meaningful functional networks, and are comparable or better than those generated by one of the current state-of-the-art models (MELODIC). Evaluation of the functional
Conflicts of interest
None.
Acknowledgment
The authors gratefully acknowledge the support provided by The University of Maryland/MPowering the State through the Center for Health-related Informatics and Bioimaging (CHIB) and by the NSF MRI Grant Number: CNS1429404. L.P. was supported by grants from the National Institute of Mental Health (MH071589 and MH112517). Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes
References (48)
- et al.
Tensorial extensions of independent component analysis for multisubject fmri analysis
Neuroimage
(2005) - et al.
Spatial heterogeneity of the nonlinear dynamics in the fmri bold response
Neuroimage
(2001) - et al.
Intrinsic dimension estimation: advances and open problems
Inf. Sci.
(2016) - et al.
Intrinsic and task-evoked network architectures of the human brain
Neuron
(2014) Afni: software for analysis and visualization of functional magnetic resonance neuroimages
Comput. Biomed. Res.
(1996)- et al.
Ica-based artefact removal and accelerated fmri acquisition for improved resting state network imaging
Neuroimage
(2014) - et al.
A unified framework for group independent component analysis for multi-subject fmri data
Neuroimage
(2008) - et al.
A method for functional network connectivity among spatially independent resting-state components in schizophrenia
Neuroimage
(2008) - et al.
Msm: a new flexible framework for multimodal surface matching
Neuroimage
(2014) - et al.
Automatic denoising of functional mri data: combining independent component analysis and hierarchical fusion of classifiers
Neuroimage
(2014)
Resting-state fmri in the human connectome project
Neuroimage
Group-pca for very large fmri datasets
Neuroimage
Advances in functional and structural mr image analysis and implementation as fsl
Neuroimage
The wu-minn human connectome project: an overview
Neuroimage
A group model for stable multi-subject ica on fmri datasets
Neuroimage
Temporal autocorrelation in univariate linear modeling of fmri data
Neuroimage
Spatiotemporal nonlinearity in resting-state fmri of the human brain
Neuroimage
Probabilistic independent component analysis for functional magnetic resonance imaging
IEEE Trans. Med. Imag.
Laplacian eigenmaps for dimensionality reduction and data representation
Neural Comput.
Functional connectivity in the motor cortex of resting human brain using echo-planar mri
Magn. Reson. Med.
A method for making group inferences from functional mri data using independent component analysis
Hum. Brain Mapp.
A whole brain fmri atlas generated via spatially constrained spectral clustering
Hum. Brain Mapp.
Reduced resting-state brain activity in the ”default network” in normal aging
Cerebr. Cortex
Functional connectivity of human striatum: a resting state fmri study
Cerebr. Cortex
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