Two-stage decompositions for the analysis of functional connectivity for fMRI with application to Alzheimer’s disease risk

Abstract

Functional connectivity is the study of correlations in measured neurophysiological signals. Altered functional connectivity has been shown to be associated with a variety of cognitive and memory impairments and dysfunction, including Alzheimer’s disease. In this manuscript we use a two-stage application of the singular value decomposition to obtain data driven population-level measures of functional connectivity in functional magnetic resonance imaging (fMRI). The method is computationally simple and amenable to high dimensional fMRI data with large numbers of subjects. Simulation studies suggest the ability of the decomposition methods to recover population brain networks and their associated loadings. We further demonstrate the utility of these decompositions in a functional logistic regression model. The method is applied to a novel fMRI study of Alzheimer’s disease risk under a verbal paired associates task. We found a indication of alternative connectivity in clinically asymptomatic at-risk subjects when compared to controls, that was not significant in the light of multiple comparisons adjustment. The relevant brain network loads primarily on the temporal lobe and overlaps significantly with the olfactory areas and temporal poles.

1 Introduction

Functional connectivity is the study of correlations in measured neurophysiological signals. Disruptions in functional connectivity have been shown to be associated with many clinical sequelæ. However, methods for evaluating covariate-adjusted population level differences in functional connectivity associated with high throughput imaging modalities remains under current development. Matrix decompositions are common methods to summarize single-subject connectivity in functional magnetic resonance imaging (fMRI) for subsequent use in regression modeling. In this manuscript, we follow this approach and investigate a generalization of functional principal components for analyzing population fMRI-based connectivity data. We focus our analysis on distinguishing risk-status between subjects at high familial risk for Alzheimer’s disease and matched controls.

Functional connectivity is formally defined as “statistical dependencies among spatially remote neurophysiological events” (Friston et al., 2007). In practice, the study of functional connectivity is inherently tied to the methods used to evaluate the dependencies and the technology used for measurement (Horwitz, 2003). We focus entirely on functional connectivity as measured by BOLD (blood oxygen level dependent) fMRI using a two-stage singular value decomposition (SVD). The SVD is useful for summarizing the enormous number of correlations available into major directions of variation. We use the SVD to find major directions of both subject-specific and population-level variation in fMRI measurements and relate these directions to familial risk status in Alzheimer’s disease using a functional logistic regression model.

The SVD has been used frequently to study connectivity in fMRI. Friston (1994) states that “[the] SVD and equivalent devices are simple and powerful ways of decomposing a neuroimaging time-series into a series of orthogonal patterns that embody, in a stepdown fashion, the greatest amounts of functional connectivity”. Unlike seed voxel or ROI-based techniques, SVD based approaches do not require specifying a-priori anatomical regions or seeds. Moreover, as shown below, the SVD can be implemented quickly on modest computing infrastructures.

Below, we apply a nested application of the singular value decomposition to evaluate group differences in functional connectivity. This method is complimentary to existing factor-analytic group decompositions, such as independent components analysis and its tensor extensions (ICA Calhoun et al., 2001; Lukic et al., 2002; Svensén et al., 2002; Beckmann and Smith, 2005). Our approach produces orthogonal bases in time and space. These orthogonal bases permits us to connect these results to functional logistic regression. The method follows four steps: i., a subject-specific SVD, ii. a population level decomposition of aggregated subject-specific eigenvectors, iii. projecting the subject level data onto the population eigenvectors to obtain subject-specific loadings, iv. using the subject-specific loadings in a functional logistic regression model. This results in a direct approach for covariate adjustments when relating functional connectivity to group status. We apply these methods to a data set of subjects at high familial risk for Alzheimer’s disease and matched controls.

Our analysis of the example dataset builds on extensive existing research demonstrating anatomical, functional and effective connectivity differences between subjects with Alzheimer’s disease or cognitive impairment and non-diseased populations. Our study differs from others by considering subjects at high familial risk for Alzheimer’s disease that are clinically asymptomatic and matched controls (Bassett et al., 2006). In earlier studies (see Bowman et al., 2008; Caffo et al., 2009), we found group differences in these subjects when considering connectivity associated with regional task-related activation. In this manuscript, we consider more classical voxel-based connectivity using variations on the singular value decomposition. These methods do not rely on an anatomical parcellation of the brain. Moreover we connect the group SVD loadings to risk-status using a new form of covariate-adjusted functional regression.

2 Data

2.1 Study population

The data derive from an ongoing study of Alzheimer’s disease risk and biomarkers (Bassett et al., 2006; Yassa et al., 2008). The data compare subjects at high familial risk for Alzheimer’s disease and controls, usually low-risk spouses. Subjects were declared at risk if at least one parent had an autopsy-confirmed diagnosis of AD and at least one additional first degree relative received a clinical diagnosis of probable AD. Control subjects had no affected or diagnosed first degree relatives, screened negative on the Alzheimer Dementia Risk Questionaire (Breitner and Folstein, 1984) and the Dementia Questionaire (Silverman et al., 1986). Both control and at-risk subjects had no clinical AD symptoms. Specifically, all subjects were free of self reported memory complaints or treatments and scored in a normal range on the Telephone Interview for Cognitive Status (Brandt et al., 1988). At-risk subjects were an average of 11 years younger than the age of diagnosis for the affected parent. All subjects were over 50.

Two waves of data collection have been completed. Ninety five at-risk subjects and 90 controls were scanned in a first wave along with collection of important covariates, accompanying cognitive testing and blood for genetic typing. A second wave of data collection was performed approximately four years after baseline. A third wave of data collection is currently underway. In our example data set, we consider the second wave data.

2.2 Imaging protocol

The fMRI images were obtained via a 1.5 T Philips Intera-NT scanner (Philips Medical Systems, Best, The Netherlands) at the F.M. Kirby Functional Imaging Research Center (Kennedy Krieger Institute, Baltimore, MD). The system utilizes a Galaxy gradient (66 mT/m at 110 mT/m/s). A standard head coil was used in image acquisition. A sagittal localizer scan was collected for orientation. Two functional scans were acquired using echo-planar imaging (EPI) and a blood oxygenation level-dependent (BOLD) technique with repetition time (TR) = 1000 ms, echo time (TE) = 39 ms, flip angle = 90 degrees, field of view (FOV) = 230 mm in the x-y plane and matrix size = 64 × 64 reconstructed to 128 × 128. Eighteen coronal slices were acquired with a 4.5 mm thickness and an interslice gap of 0.5 mm, oriented perpendicular to the anteriorposterior commissure (ACPC) line. Slices were acquired along the z-axis, yielding a total coverage of 90 mm. Two sessions were performed, each with 370 time points. The data in this analysis considers only the first session. Total fMRI acquisition time was 12 minutes and 20 seconds.

The paradigm, programmed in E-prime 1.1 (Psychology Software Tools, Inc., Pittsburgh, PA, USA), was an auditory word-pair association task consisting of two six minute and ten second sessions. Each session consisted of six sets of three blocks. The types of blocks included encoding, recall, and rest. In the encoding block, subjects were presented with seven unrelated word pairs. In the recall block, subjects were presented with the first word of each pair and instructed to silently recall the second. In the baseline block, subjects were presented with an asterisk.

Data pre-processing was performed using Statistical Parametric Mapping (SPM99, Wellcome Department of Imaging Neuroscience, University College, London, UK) under MATLAB 7.0 (The Mathworks, Sherborn, MA, USA). Images were motion corrected by a six-parameter rigid-body realignment with the mean image across sessions. This was followed by re-slicing using a windowed-sinc interpolation. Non-linear normalization using 7 × 8 × 7 basis functions was used to warp each individual’s data into standard stereotaxic space. Template space was defined by SPM’s standard EPI template (Montreal Neurologic Institute, McGill University, Montreal, Canada). The template was manually cut to fit each individual scan in order to improve the quality of normalization on the partial-brain scans. Normalized scans were re-sliced to isotropic voxels (2 mm³), using trilinear interpolation and spatially smoothed with a full-width at half-maximum (FWHM) Gaussian kernel of 5 mm.

3 Methods

Let Y_i(v, t) represent the fMRI data for subject i = 1, … N, voxel v = 1, …, V and scan t = 1, …, T. Our goal is to obtain a parsimonious population value decomposition (PVD)

$$Y_i(v,t)=\sum _j\sum _k{\psi }_j(v){\xi }_k(t){\lambda }_{\mathit{ijk}},$$

where ψ_j(v) and ξ_k(t) represent orthonormal functional bases in space and time, respectively. Notice that ψ_j(v) and ξ_k(t) are population-level bases that do not vary by subject. In contrast, the loadings, λ_ijk, are subject-specific. We show how to use the λ_ijk in subsequent analyses as summaries of functional connectivity that achieve a great deal of data reduction.

Our approach utilizes two stages, subject-specific SVDs followed by population-level principal components analysis. This approach is particularly well suited to high-dimensional neuroimaging data and we further demonstrate how calculations can be performed on very modest computing resources. In the first stage, we obtain subject-specific singular value decompositions

$$Y_i(v,t)=\sum _k{\psi }_{ik}(v){\xi }_{ik}(t){\zeta }_{\mathit{ijk}},$$

where ζ_ijk are singular values and ψ_ik(v) and ξ_ik(t) are singular functions.

In the second stage, we retain a small number (say L) of ψ_ik(v) and consider the populations of spatial functions = {ψ_ij(v)}_{i=1,…,N,j=1,…,L} and time series ℬ = {ξ_ik(t)}_{i=1,…,N,k=1,…,L}. These collections of functions are then decomposed using a Karhunen/Loéve functional principal components decomposition (Karhunen, 1947; Loéve, 1945). Such decompositions write each subject’s functions as a linear combination of eigenfunctions. Specifically, we obtain orthonormal bases ψ_l(v) and ξ_l(t) so that ψ_ik(v) =Σ_l δ_iklψ_l(v) and ξ_ik(t) = Σ_l γ_iklξ_l(t). Here, ψ_l(v) are the eigenfunctions associated with , ξ_l(t) are the eigenfunctions associated with ℬ and δ_ijl and γ_ikl are associated eigenvalues. The eigenfunctions are then used to obtain the subject-specific loadings: λ_ijk =∫∫Y_i(v, t)ψ_j(v)ξ_k(t)dvdt, where λ_ijk is the subject-specific loading onto the left eigenfuction j and right eigenfunction k.

The benefits of this approach for summarizing connectivity information over related methods are numerous. Firstly, only standard matrix decompositions are needed to estimate the λ_ijk. This is in contrast with full tensor-based SVD methods (see Leibovici and Sabatier, 1998). Secondly, the process is performed iteratively in two stages. Thus, it mirrors standard two-stage random effect analyses of fMRI data and computing can be parallelized. Moreover, because of the two stage process, the method can be applied on very low memory systems. Thirdly, the parameters are uniquely interpretable. The ψ_j(v) are population-level eigenimages, summarizing areas of temporal synchronization across subjects. The ξ_k(t) are population-level eigenvariates, summarizing times of spatial synchronization. The λ_ijk represent the loading of subject i onto population eigenimages j and eigenvariates k. Hence, we hypothesize that these loadings will be a useful summary of connectivity, that may be useful as predictors. Moreover, we demonstrate how their use in regression models connects to functional regression. The loadings achieve a great deal of dimension reduction; in our example, we demonstrate interesting findings using only 25 of the loadings.

3.1 Implementation

Here we discuss implementation issues in dealing with high dimensional fMRI data. First, a brain mask is applied across subjects and only those voxels represented in all subjects are retained. This removes both background voxels as well as boundary voxels with incomplete data across subjects due to inexact registration. Let Y_i be the V × T data matrix for subject i. We assume that Y_i is centered in both time and space; i.e. ${\mathbf{Y}}_i=\{{\mathbf{I}}_V-{\mathbf{1}}_V^{\prime }{({\mathbf{1}}_V^{\prime }{\mathbf{1}}_V)}^{-1}{\mathbf{1}}_V^{\prime }\}{\stackrel{\sim }{\mathbf{Y}}}_i\{{\mathbf{I}}_T-{\mathbf{1}}_T^{\prime }{({\mathbf{1}}_T^{\prime }{\mathbf{1}}_T)}^{-1}{\mathbf{1}}_T^{\prime }\}$ where Ỹ_i (dimension V × T ) is the uncentered data matrix, I_R is an identity matrix of dimension R and 1_R is a vector of ones of length R.

We define the global connectivity matrix as the V × V matrix ${\mathbf{Y}}_i{\mathbf{Y}}_i^{\prime }/T$, which has (v₁, v₂) element ${}_{1\overline{T}}{\sum }_{t=1}^T{Y}_i(v_1,t)Y_i(v_2,t)$. This matrix completely summarizes temporally synchronous behavior in the fMRI data at each pair of voxels. Consider now, the T × T matrix $\frac{1}{V}{\mathbf{Y}}_i^{\prime }{\mathbf{Y}}_i$; this matrix is an autocovariance function calculated across voxels. That is, it has (t₁, t₂) element $\frac{1}{V}{\sum }_{v=1}^V{Y}_i(v,t_1)Y_i(v,t_2)$. This matrix summarizes spatially synchronous behavior in the fMRI data at each pair of scans.

Since finding spatial areas of temporal synchronization defines connectivity analysis, the global connectivity matrix is the core quantity of interest. However, having $\left(\begin{array}{c}V\\ 2\end{array}\right)$ unique elements, it necessarily must be summarized and is difficult to work with computationally. It is a useful numerical trick to calculate the SVD of Y_i by calculating the spectral decomposition of the T × T matrix ${\mathbf{Y}}_i^{\prime }{\mathbf{Y}}_i$ first.

Specifically, consider the eigenvalue decomposition of ${\mathbf{Y}}_i^{\prime }{\mathbf{Y}}_i={\mathbf{V}}_i{\mathbf{D}}_i^2{\mathbf{V}}_i^{\prime }$, where V_i is a T × T matrix so that ${\mathbf{V}}_i^{\prime }{\mathbf{V}}_i={\mathbf{I}}_T$ and ${\mathbf{D}}_i^2$ is a T × T diagonal matrix of eigenvalues. Here, the columns of V_i contain the subject-specific eigenvariates. Let U_i be the V × T matrix defined by ${\mathbf{U}}_i={\mathbf{Y}}_i{\mathbf{V}}_i{\mathbf{D}}_i^{-1}$. Notice that, performed in this order, U_i,D_i and V_i can be obtained quickly, without having to reserve memory or perform operations on the V × V global connectivity matrix. An added benefit is numerical stability, as calculation of ${\mathbf{Y}}_i^{\prime }{\mathbf{Y}}_i$ is linear in the number of voxels and only involves multiplication and addition, obtaining V_i and ${\mathbf{D}}_i^2$ involves only low-dimensional calculations and calculating U_i is also linear in the number of voxels and only involves multiplication and addition.

Defining U_i as such implies ${\mathbf{U}}_i^{\prime }{\mathbf{U}}_i={\mathbf{I}}_T,{\mathbf{Y}}_i={\mathbf{U}}_i{\mathbf{D}}_i{\mathbf{V}}_i^{\prime }$ and $\frac{1}{T}{\mathbf{Y}}_i{\mathbf{Y}}_i^{\prime }=\frac{1}{T}{\mathbf{U}}_i{\mathbf{D}}_i^2{\mathbf{U}}_i^{\prime }$. Hence, the latter point shows that the columns of U_i contain the eigenvectors of the global connectivity matrix.

The columns of U_i are referred to as eigenimages (Friston et al., 2007) and approximate the ψ_ik(v). Each U_i is a spatial map so that highly synchronous voxels will have large (in absolute value) corresponding values in some U_i. To elaborate, imagine a setting where the brain is comprised of four clusters so that, within each cluster, the fMRI time series are perfectly synchronous, while they are independent between clusters. Then, the SVD would result in four U_i vectors, each one having large (in absolute value) values for each cluster. Because of this, the U_i are often thought to estimate brain networks (Friston, 1994). Further, the global connectivity matrix satisfies $\frac{1}{T}{\mathbf{Y}}_i{\mathbf{Y}}_i^{\prime }=\frac{1}{T}{\sum }_j{D}_{ij}^2{\mathbf{U}}_{ij}{\mathbf{U}}_{ij}^{\prime }$ where D_ij is the j^th diagonal entry of D_i and U_ij is column (eigenimage) j from U_i. That is, the global connectivity matrix decomposes into a weighted sum of the outer products of the brain networks. The columns of V_i, referred to as eigenvariates, estimate ξ_ik(t). One can think of these as representing how the brain networks mix over time.

Consider retaining the first L columns of V_i and U_i for population level analyses, where we are assuming that the eigenvectors are ordered by decreasing eigenvalues. Note that necessarily L ≤ T; however, it will be typical for it to be much smaller. Let E be the (NL) × T matrix obtained by row-stacking the first L columns of V_i across subjects and H be the (NL) × V matrix obtained by row-stacking the first L columns of U_i across subjects. Let Σ̂_E be the T × T sample variance matrix defined by ${\mathbf{E}}^{\prime }\{{\mathbf{I}}_{NL}-{\mathbf{1}}_{NL}{({\mathbf{1}}_{NL}^{\prime }{\mathbf{1}}_{NL})}^{-1}{\mathbf{1}}_{NL}^{\prime }\}\mathbf{E}$. Let Σ_H be the V × V sample covariance matrix defined by ${\mathbf{\sum }}_H={\mathbf{H}}^{\prime }\{{\mathbf{I}}_{NL}-{\mathbf{1}}_{NL}{({\mathbf{1}}_{NL}^{\prime }{\mathbf{1}}_{NL})}^{-1}{\mathbf{1}}_{NL}^{\prime }\}\mathbf{H}$ be the corresponding matrix for H. We consider the eigenvalue decomposition of ${\widehat{\mathbf{\sum }}}_E=\mathbf{V}{\mathbf{D}}_E^2{\mathbf{V}}^{\prime }$ and ${\widehat{\mathbf{\sum }}}_H=\mathbf{U}{\mathbf{D}}_H^2{\mathbf{U}}^{\prime }$, where V is T × min(NL, T), D_E is min(NL, T) × min(NL, T ), U is V × NL and D_H is NL × NL. Now the columns of V represent population eigenvariates and the columns of U represent population eigenimages, estimating ψ_j(v) and ξ_j(v) respectively. In practice, calculating U requires a similar technique as outlined above to avoid creating the V × V matrix Σ_H.

We then project original subject specific data onto these bases. In specific, let Λ_i be the L × L matrix defined by Λ_i = [λ_ijk]_j,k = U′Y_iV. Here element (j, k) of Λ_i represents the subject-specific loading onto population eigenimage j and eigenvariate k. Hence, it represents the loading onto the specific brain network given by column j of U for the particular time series represented by column k of V. These our the proposed estimates for λ_ijk.

3.2 Regression models on the loadings

We consider an analysis with case-status as the outcome and the fMRI data and covariates as predictors. Let D_i ∈ {0, 1} represent the risk status for subject i with covariate values X_i. Consider the functional regression model:

$$\text{logit}\{P(D_{ij}=1)\}=\int \int Y_i(v,t)\beta (v,t)\mathit{dvdt}+{\mathbf{X}}_i\mathit{\gamma }.$$

Let φ_j(v) and ξ_k(t) be eigenfunctions. Then we have

$$\int \int Y_i(v,t)\beta (v,t)\mathit{dvdt}=\int \int \left\{\sum _j\sum _k{\phi }_j(v){\xi }_k(t){\lambda }_{\mathit{ijk}}\right\}\beta (v,t)\mathit{dvdt}=\sum _j\sum _k{\lambda }_{\mathit{ijk}}{\tau }_{jk},$$

where τ_jk =∫∫φ_j(v)ξ_k(t)β(v, t). Hence our model becomes

$$\text{logit}\{P(D_{ij}=1)\}=\sum _j\sum _k{\lambda }_{\mathit{ijk}}{\tau }_{jk}+{\mathbf{X}}_i^{\prime }\mathit{\gamma }.$$

Here τ_jk represents the change in the log-odds for risk status for brain network j and time series k.

To summarize, logistic regression models having the loadings as covariates result in a form of functional regression involving the entire subject-specific fMRI time series integrated over the bases derived by the population eigenimages and variates. In this way, covariate-adjusted regression relationships associated with connectivity can be explored easily. Moreover, to account for matching generalized linear mixed effect models (see McCulloch and Searle, 2004) with a fixed effect design matrix comprised of the λ_ijk and X_i and pair-specific random effects can be used.

4 Results

Table 1 displays demographic data for the AD At-risk data set. In this second wave of study there are 81 at-risk subjects and 68 controls. The groups are well matched on gender, age and education level. Unsurprisingly, there is a significant difference in the number of ε4 alleles of the Apolipoprotein E gene, as the number of such alleles has been associated with late onset Alzheimer’s disease (see Strittmatter and Roses, 1996, for a review).

Table 1.

Demographic data by risk status.

		At-risk	Control	P-value
Count		81	68
Gender	No. Male (%)	33 (41%)	36 (53%)	.19
Age	Mean (SD)	62 (6.68)	62 (7.5)	.90
APOE	No. Any 4 (%)	28* (35%)	12 (18%)	.04
Years of Educ.	No. < 12 (%)	5 ( 6%)	2 ( 3%)
	No. 12 (%)	18 (22%)	12 (18%)
	No. (12, 16) (%)	16 (20%)	11 (16%)
	No. 16 (%)	16 (20%)	12 (18%)
	No. > 16 (%)	26 (32%)	31 (46%)
				.51

^*One at-risk and one control subjects missing APOE status. Age P-values based on two group t-test while remaining were based on Chi-squared tests.

We then applied the two-stage decompositions outlined above. Figure 1 shows the first ten eigenvariates for four example subjects in a heatmap. Notice that the first eigenvariate is an overall mean shift. The same was true for the eigenimages. The next ten vary quite a bit across subjects containing both high- and low-variation signals. Figure 1 displays the subject-specific singular values for every subject. The values are rescaled to as $D_{ij}^2/{\sum }_{k=2}^{10}{D}_{ik}^2$ to highlight variance explained in the first 10 eigenvalues excluding the mean shift that was typified the first component.

Figure 1.

First 10 eigenvariates for four example subjects.

The subject-specific eigenimages had across-the-board signal changes being the first component. That is, like the eigenvariates, the first component represented uninteresting person-specific changes in the scanner signal. This heterogeneity likely estimates technological variation, such as scanner gain, that is not of interest when comparing group connectivity. Hence it is preferable to attempt to eliminate this variation to focus in on more interesting components. To address this, we subtracted the row means of E and H; i.e. performing the operation $\mathbf{E}\{{\mathbf{I}}_T-{\mathbf{1}}_T{({\mathbf{1}}_T^{\prime }{\mathbf{1}}_T)}^{-1}{\mathbf{1}}_T\}$ and $\mathbf{H}\{{\mathbf{I}}_V-{\mathbf{1}}_V{({\mathbf{1}}_V^{\prime }{\mathbf{1}}_V)}^{-1}{\mathbf{1}}_V\}$. This is equivalent to forcing each person-specific eigenimage and eigenvariate to have mean 0. Performing the analysis on these matrices removed this uninteresting component of variation.

Figure 3 displays the percentage of the population variation in the first level eigenimages and eigenvariates explained by the second level principal components decomposition. Both of these curves have a fairly slow rate of decay, suggesting a large degree of population-level heterogeneity in the subject-specific decompositions.

Figure 3.

Percent of the population variation in first level-eigenvariates (left) and eigenimages (right) explained by second-level eigenvalue decomposition.

Figure 4 displays renderings of the first ten population eigenimages. Thresholding was performed via visual inspection of the eigenimage histrogram to eliminate low-weighted areas. The areas displays had values orders of magnitude larger than the thresholded areas. For context, Figure 5 gives regions of interest (based on the anatomical parcellation given in Tzourio-Mazoyer et al., 2002) that have over 20% of their area overlapping with the eigenimage. Recall, eigenimages and variates are unique only up to scalings and therefore positive and negative values could be reversed with no loss of information.

Figure 4.

Three-D rendering of thresholded versions of the first six eigen images overlaid on a template. Red areas load positively while blue areas load negatively. The upper left is the first eigen image, the upper middle is the second, and so on.

Figure 5.

Regions with over 20% overlap with the specified eigenimage. Red areas load positively, blue negatively, purple have partial volumes loading positively and negatively. Abbreviations: Amyg. = Amygdala, Cer. = Cerebellum, Fr. = Frontal, Fus. = Fusiform gyrus, Inf. = Inferior, Ins. = Inusla, L. = Left, Mot. = Motor Area, Olf. = Olfactory, Op. = Opercular part, PHG = Para-Hippocampal Gyrus R. = Right, Rol. = Rolandic, Sup. = Superior, Supp. = Supplementary, Temp. = Temporal.

We summarize a subset of the population eigenimages. The first loads primarily on the superior portion of the temporal lobe. The second covers the majority of the imaging area. The third loads heavily on the temporal lobe and limbic substructures, such as the para-hippocampal gyrus. The fourth covers temporal and limbic areas and intersects with the small portion of the cerebellum in the imaging area. The eighth, which we will see is one of the more important eigenimages, loads specifically on temporal and limbic areas, especially covering olfactory areas. This is of interest as deficits in olfaction have been hypothesized to be connected with neurodegenerative disorders and AD in particular (see Mesholam et al., 1998, for a meta analysis and review).

Figure 6 displays the first ten population eigenvariates and their associated spectrum. The first eigenvariate represents a drift in the signal, which could represent biological or technological trends, such as learning effects or scanner drift. We reiterate that one must remember that the signs of such analysis are arbitrary and could represent either and increase or decrease in the signal over the session. The following two population eigenvariates represent slowly varying functions. The remaining have spectra that include spikes at the same frequency as the paradigm (see Figure 7), but also include higher frequency information. We further investigated if the eigenvariates separate between the two components of the task (encoding versus recall), which have the same spectra, but different phases. The fourth eigenvariate time is more correlated with the recall paradigm rather than encoding (−.02 versus .23). Eigenvariates 6 and 9 are more correlated with encoding than recall (correlations of −.14 versus −.04 and .19 versus .02, respectively). Eigenvariate 8, which is the most obviously associated with the paradigm, was more correlated with encoding (−.45 versus −.25), but retained significant correlation with the recall blocks. To elaborate, the peaks of this eigenvariate occur between the encoding and recall blocks, though are slightly closer to the recall blocks.

Figure 6.

First ten population eigenvariates for the at-risk AD data set. To the right of each plot is the associated spectrum in the −50 to 50 millihertz (mHz) range.

Figure 7.

Haemodynamically convolved encoding (top) and recall (bottom) design vectors with associated spectrum.

We next considered the use of the subject-specific loadings in functional logistic regression models. Before fitting fixed effect regression models, we first considered random effect models that accounted for spousal matching and familial aggregation. Fitted results suggested little or no correlation due to spousal matching or family. Therefore, we omit addressing this potential correlation in subsequent analyses.

Table 2 displays P-values for predicting risk-status treating each population loading in a separate model. Figure 8 shows 25^th, 50^th and 75^th percentiles by risk group for the ten most significant loadings. Of these, the most significant is the fourth population eigenimage and eighth eigenvariate. This appears to incorporate variation associated with the paradigm in the temporal poles. The next most significant loads heavily on the eighth population eigenimage and sixth eigenvariate. This eigenvariate includes slower variation contrasted between the superior temporal lobe and the olfactory areas of the temporal lobe. Given that the estimates are entirely empirical without a-priori hypotheses, multiplicity issues demand that these results must be interpreted with a grain of salt. To address this issue, we considered permutation testing. Here we permuted risk status 1,000 times, thus breaking any association between risk status and fMRI loadings. We refit the model for each permuted data set. Note, this does not require reobtaining the scores, which were constructed using only the images with no knowledge of risk status. For each refitted model, we retained the smallest P-value. These results suggests that there is a 50% chance of obtaining a minimum P-value as small as 0.01 and hence the possibility that the results may be due to chance associations can not be ruled out.

Table 2.

P-values comparing At-risk and control subject for each loading from functional linear models with a covariate term indicating the presence of any four APOE alleles.

	Eigenvariate
EigIm	1	2	3	4	5	6	7	8	9	10
1	0.822	0.779	0.264	0.791	0.850	0.235	0.210	0.200	0.379	0.987
2	0.759	0.734	0.792	0.326	0.329	0.699	0.265	0.735	0.076	0.692
3	0.774	0.528	0.361	0.579	0.507	0.189	0.240	0.981	0.512	0.183
4	0.710	0.603	0.900	0.549	0.696	0.166	0.953	0.010	0.153	0.186
5	0.819	0.754	0.767	0.774	0.381	0.417	0.162	0.525	0.512	0.849
6	0.735	0.721	0.716	0.483	0.941	0.303	0.091	0.931	0.715	0.398
7	0.582	0.686	0.706	0.818	0.996	0.314	0.910	0.713	0.560	0.474
8	0.305	0.930	0.165	0.968	0.743	0.050	0.354	0.681	0.262	0.299
9	0.684	0.742	0.675	0.097	0.718	0.052	0.822	0.053	0.348	0.674
10	0.945	0.678	0.529	0.145	0.845	0.574	0.996	0.078	0.158	0.828

Figure 8.

Normalized loadings’ 25^th, median and 75^th percentiles by group. Diamonds are at-risk, controls are squares. The (eigenimage, eigenvalue) pair are depicted to the left and gray bars are used to highlight groups.

5 Discussion

This paper shows the utility of two-stage decompositions for the analysis of population based fMRI data. Our approach first used the singular value decomposition to obtain subject-specific eigenimages (networks) and eigenvariates (time series). A small number of these are retained and aggregated. Separate second-level eigenvalue decompositions for the collections of eigenvariates and eigenimages, respectively, are used to form population-level brain networks and time series. We project the subject-level data onto these population eigenvectors to obtain a matrix of loadings onto each network/time series combination. We further showed how these loadings can be used in a generalized functional regression. We applied then in a matched logistic regression analysis of Alzheimer’s disease familial risk status.

The two-stage decomposition approach has several notable benefits as an exploratory method for discovering population brain networks and major directions of functional variation. Foremost is computational ease. Subject-specific decompositions are relatively easily obtained and, by retaining only a few of the networks and time series, the population values are similarly easily computed. We further explicitly demonstrated how calculations can be approached so that a high dimensional full connectivity matrix is never required to be loaded into memory. In addition, first level calculations can be easily made parallel by separately performing the SVD for each subject. Thus, this methodology will scale to next-generation studies involving hundreds or thousands of subjects. Another alternative would be the use of tensor extensions of the SVD and factor analysis (Leibovici and Sabatier, 1998; Kolda and Bader, 2009; De Lathauwer et al., 2000). While these methods offer more theoretically complete alternatives, they lack the simplicity and easy execution of two-stage decompositions.

By using the SVD as the basis for the decomposition, the most variable aspects of the population of fMRI data are used in the ensuing functional regression. This is useful, as more variable predictors will have lower standard errors. However, as the response was not used in finding these directions of variation, there is no guarantee that subtle, but important, directions of variation are ignored while large directions of variation that have no relationship with the response are retained.

We also demonstrate the inferential potential of these population networks and time series by projecting subject level data onto these bases. We then show how functional regression modeling can be used to assess significance of the loadings. Our work is influenced by work in functional regression for neuroimaging in Reiss and Ogden (2009).

A potential point of criticism is the difficulty in the choice of the number of components to be included from the subject-specific decompositions in the second level analysis. If important eigenvariates or eigenvectors are omitted from a majority of subjects on the first level, they will not be obtained in the second level analysis. Conceptually, one could use all T of the components. However, this would neither be parsimonious nor computationally feasible for large problems with hundreds of subjects (as with the Alzheimer’s disease data set). A related issue is a discussion on the potential need for regularization (such as in Witten et al., 2009) of the decomposition, at both the subject-specific and population level. However, like the penalization discussed above, regularization greatly increases the complexity of fitting, and we leave this as potential future work.

Our current approach uses visual inspection of the collection of subject-specific scree plots. Moreover, we were fortunate that interesting population-level directions of variation were apparent from choosing relatively few eigenvectors at the first stage. We fully stipulate that this could be improved upon, and such a strategy may miss interesting -yet subtle- variation embedded in the later eigenvectors. However, in our opinion, choosing the number of components is a subset of challenging model selection/Occam’s Razor problems that have stymied researchers for decades. Methods such as cross-validation or estimating prediction error may be of use. Alternative approaches use a large number of components and penalize at the regression level (Goldsmith et al., 2010). However, these solutions would be challenging to implement on problems of an interesting scale. Hence our recommendation is to acknowledge the degree of subjectivity in such an analysis and that choosing too few components will miss potential important features.

A second point of criticism is the lack of accounting for the multiple observations per subject contributing to the population level decomposition. We are less concerned with this aspect of the analysis, as this would affect inference based on the population eigenvectors more than estimation. However, in this manuscript, we focus on estimation and the use of the eigenvectors as predictors in functional regression models and do not make use of their measurement variation (though see Crainiceanu et al., 2009a). Another potential point of criticism is the lack of use of the variance ordering of subject-specific eigenvectors in the subsequent population analysis. That is, a subject’s first and fifth eigenvectors are treated equally in the population decomposition. We hypothesize that this criticism can be addressed by a weighting using the inverse of the associated eigenvalues. However, we relegate this approach to future research.

A final criticism is the lack of direct exploitation of the temporal or spatial structure of the data. In fact, the methods are invariant to the order of the rows or columns of the individual subject-specific data matrices, provided it is consistent across subjects. This has a potential negative effect in that accounting for spatio-temporal correlation may improve discovery of brain networks and hence resulting functional regression models. We currently address in an ad hoc, but standard and computationally feasible/fast method, using smoothing during the preprocessing steps. A more complete solution would incorporate this correlation more directly into a factor analysis model. However, this solution adds a great deal of complexity and computational burden.

This manuscript addresses decomposition methods to evaluate cross-sectional variation in brain networks. However, longitudinal functional imaging studies are becoming increasingly common. We have developed multilevel functional principal component methods for functions of one variable (time, for example) and are currently generalizing methods to consider hierarchical imaging data. However, the extension to extremely high dimensional imaging data remains a difficult task. Furthermore, connecting these decomposition methods with outcomes via functional regression is an area of active research (see Di et al., 2009; Crainiceanu et al., 2009b).

It is also worth noting limitations of using SVD to study brain networks. First, these decompositions guarantee orthogonal eigenimages and eigenvariates, which may or may not reflect actual biology. Our simulation study specifically assumed orthogonal networks and time series. Moreover, our simulation study imposed a large amount of variation when mixing over the time-series and images, also creating an ideal setting for the SVD. The method would struggle if signals were mixed largely in equal parts. In contrast, other methods, such as ICA, are more robust to these assumptions and hence are popular for analyzing brain connectivity (see Calhoun et al., 2003). However, unlike ICA, this two-stage SVD does not force a distinction between spatial and temporal decompositions. In addition, our two-stage method avoids the question of whether to stack rows or columns for group analysis (Calhoun et al., 2001; Lukic et al., 2002; Svensén et al., 2002; Guo and Pagnoni, 2008). More analogous tensor versions of ICA have been proposed (Beckmann and Smith, 2005); however, it is not clear whether computations will scale to large fMRI studies. Finally, though imposing orthogonality is rigid, it is very useful for creating a basis to decompose the fMRI signal for subsequent use in regression models.

The analysis of the AD risk data set yields interesting findings on altered connectivity in subjects with high familial risk for Alzheimer’s disease. The atypically large sample size for a functional imaging study and pre-clinical population including subjects at high familial risk are unique aspects of this study. This analysis corroborates differences in connectivity found using other methods on the same data (Caffo et al., 2009; Bowman et al., 2008). It is also of interest to note that, unlike the first wave (Bassett et al., 2006), group differences in paradigm-related activity were unremarkable in the second wave (see Caffo et al., 2009; Bowman et al., 2008). This change in paradigm-related activity may be due to a variety of reasons, including learning effects, differential attention to the task between the groups across visits and so on. A benefit of the study of functional connectivity is the lack of reliance on the paradigm, and hence potential robustness to these effects.

We demonstrate preliminary evidence for altered connectivity between asymptomatic at-risk subjects. Of primary interest is group segregation for the network encompassing the temporal poles and the olfactory areas of the temporal lobe. However, we caution over-interpretation of these results, as connectivity differences were not a primary a-priori hypothesis of the study and this effect did not survive multiplicity adjustment. For future work, we are investigating the robustness of the networks over time, both in the earlier phase and the third phase currently being collected. Stability of the networks and results over time would greatly bolster confidence in their validity. In addition, as better knowledge of actual eventual case status becomes available, potentially better results could be obtained. Further, potential weakness of our study is the narrow imaging area, which ignores possible long-range connectivity. However, we note that the imaging area focused on a band surrounding the medial temporal lobe, an area believed to be associated with AD (see the discussion in Bassett et al., 2006).

Our study compliments existing research on altered anatomical and functional connectivity between mild AD and mild cognitive impairment subjects and controls. Grady et al. (2001) studied 21 health elderly subjects and 11 mildly demented subjects using rCBF PET. They reported decreased correlations for the demented group between task-related areas in the prefontal cortex and hippocampus. Stam et al. (2007) considered small-world network hypotheses using EEG comparing 15 Alzheimer’s patients and 13 control subjects. They report decreased complexity of the network for the diseased group. Greicius et al. (2004) used ICA and fMRI to study default mode network differences between 13 mild AD cases and matched controls. They found decreased activation in the default mode network for the AD group in the posterior cingulate and hippocampus. Wang et al. (2006) studied connectivity between the hippocampus and other regions in 13 mild AD cases and matched controls and found disrupted and increased connectivity for the AD and control groups. Wang et al. (2007) considered inter-regional correlations between 14 AD subjects and matched controls in PET and found both increased and decreased inter-group connectivity differences.

In summary, the two-stage applications of the singular value decomposition along with functional logistic regression can shed considerable light on group fMRI studies. Estimates are easily calculated and computations scale to large studies. The functional logistic regression model allows for easy consideration of covariate effect.

Figure 2.

Rescaled percent variation explained by the second to tenth eigenvalues for all subjects.