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Journal of Neuroscience Methods 191 (2010) 126–137

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Journal of Neuroscience Methods

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Revealing voxel correlation cliques by functional holography analysis of fMRI

Yael Jacoba,b,c, Amir Rapsona, Michal Kafri c, Itay Baruchia, Talma Hendlerb,c, Eshel Ben Jacoba,d,∗

a School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israelb Sackler Faculty of Medicine, Tel Aviv University, 69978 Tel Aviv, Israelc Functional Brain Imaging Unit, Wohl Institute for Advanced Imaging, Tel Aviv Sourasky Medical Center, 64239 Tel Aviv, Israeld Center for Theoretical and Biological Physics, University of California San Diego, La Jolla, CA 92093, United States

a r t i c l e i n f o

Article history:Received 4 February 2010Received in revised form 3 June 2010Accepted 7 June 2010

Keywords:Multivariate BOLD analysisFunctional holographyPCAMotor activationDominant handednessShannon entropy

a b s t r a c t

Data acquired by functional brain imaging are of a multivariate and complex nature. Selecting relevanttopographically specific information for system-level analysis is a highly non-trivial task. This chal-lenge has traditionally been addressed by hypothesis-driven approaches. Recently, data-driven methodsmaking no a priori assumptions about the signal were developed. Here, we present a hybrid approach,selecting data-driven voxels in paradigm-driven measurements in order to identify functional connec-tivity motifs in the voxel correlations. Our tool is the functional holography (FH) method, originallydeveloped for analyzing electrophysiological recordings and based on analyzing the voxel–voxel cor-relation matrices. The algorithm selects the relevant voxels using a dendrogram clustering methodcombined with a unique standard deviation (STD) filter, identifying the voxels with high STD correla-tions. Functional connectivity motifs are revealed through a dimension-reduction procedure by principalcomponent analysis (PCA) allowing for a reduced three-dimensional holographic presentation space.Information loss due to PCA is retrieved by connecting voxels in the reduced space with lines that arecolor-coded according to the correlations. Our results show that the FH analysis performed for a sin-gle trial reveals interesting motifs, even in a simple motor task: unilateral hand movements yieldedtwo clusters, one in the contralateral M1 region showing neuronal activation and one in the ipsilat-eral homologues region showing deactivation. Thus, according to a single trial level analysis, of 12-timepoints alone, we can determine which hand the subject moved. Moreover, using cluster quantificationbased on eigenvalue entropy calculation, we obtained good separation between right- and left-handedsubjects.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Functional magnetic resonance imaging (fMRI) enables moni-toring of the entire human brain of subjects performing given tasks,with relatively high spatial resolution. Each scanning session detailsactivity of thousands of ∼1 mm3 volumes, known as voxels. Selec-tion of the significant information-carrying voxels, extraction ofrelevant system-level information, particularly concerning func-tional connectivity between different brain areas, and classificationof behavioral activity based on brain activity alone – all representmajor computational challenges.

Over the years, much computational effort in fMRI data anal-ysis has been devoted to the development of advanced methodsfor localizing brain activity. Both hypothesis-driven, data-driven,and hybrid approaches have been developed. In hypothesis-driven

∗ Corresponding author at: School of Physics and Astronomy, Tel Aviv University,69978 Tel Aviv, Israel. Tel.: +972 3 6407845; fax: +972 3 6425787.

E-mail address: eshelbj@gmail.com (E. Ben Jacob).

approaches, prior paradigm-based assumptions about the fMRI sig-nal are made. In the commonly used voxel-based analysis, thecorrelations between the recorded signal of each voxel and theparadigm conditions are computed. One of the most frequentlyused methods for analyzing measurements in hypothesis-drivenparadigms is the general linear model (GLM) (Friston et al., 1995).

As most standard methods, GLM requires very long epochs ofbrain activity and averaging of the data, typically over entire blocksof trials. While averaged data may provide statistical power, impor-tant specific nuances and underlying structures of the system mightbe overlooked.

Extracting meaningful information from high-dimension mul-tivariate neuroimaging data is a highly non-trivial task, usuallyrequiring one to reduce the dimensionality of the data. Hypothesis-driven approaches, which usually adopt univariate analysismethods, reduce a priori the dimensionality of the data. While thisfacilitates the extraction of information from the data, it may alsolead to loss of important information.

When strong assumptions concerning the time course of theobserved activity cannot be formulated a priori, researchers must

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Y. Jacob et al. / Journal of Neuroscience Methods 191 (2010) 126–137 127

resort to data-driven approaches. Two commonly used data-drivenmethods are principal component analysis (PCA) (Jolliffe, 1986) andindependent component analysis (ICA) (Bell and Sejnowski, 1995)algorithms. Both analyses can be used as dimension-reductionalgorithms, and be used to search for clusters (subgroups) of voxelsin the covariance space of the signals showing a common response(Andersen et al., 1999).

When the research goal is to reveal functional connectiv-ity between different brain locations, approaches using bothhypothesis- and data-driven methods can be applied. In onecommon strategy, the signal at a single voxel (or a represen-tative single averaged signal over several voxels) in a specifichypothesis-driven location of interest is chosen as the seed region.Then, correlation coefficients between the time course of theseed voxel and that of all the voxels in the brain are calcu-lated, from which a functional connectivity network between thedifferent locations can be inferred (Friston, 1994; Rogers et al.,2007). This technique was mostly used to delineate the couplingbetween regions under different conditions (Bleich-Cohen et al.,2009) and more recently to characterize connectivity during thebrain’s “resting state” (Rogers et al., 2007) and to assess cross-subjects voxel correlations (Bartels and Zeki, 2005; Hasson et al.,2004).

Functional connectivity does not say much about causal rela-tionships in the network (Friston et al., 1997). Methods to dealwith this limitation have been developed, e.g. covariance structuralequation modeling (McIntosh and Gonzales-Lima, 1994), dynamiccausal modeling (DCM) method (Friston et al., 2003) and GrangerCausality (Friston and Buchel, 2000); however, these are beyondthe scope of the present paper.

The goal of this present paper is to present a hybrid methodfor the extraction of functional connectivity motifs from functionalimaging datasets. The method can operate on a single trial level,avoiding the loss of information resulting from a priori assump-tions, dimension reduction, and statistical averaging.

We illustrate the computational power and functional validityof our method by analyzing the fMRI BOLD signal obtained for asimple motor task in which participants intermittently clenchedtheir right or left hand according to auditory instructions.

Our method is based on functional holography (FH) a methodoriginally developed to analyze electrophysiological recordings(Baruchi et al., 2006). The basic idea underlying FH is that a com-plex system behaves like a hologram. In a hologram we can takeout a little piece and still see the “big picture”, only in lower reso-lution.

The FH is an effective clustering method, capable of cap-turing system-level networks, from the voxel–voxel correlationmatrices. The algorithm identifies voxel clusters using a den-drogram clustering method combined with a standard deviation(STD) filter. The networks are then visualized in a reduced holo-graphic space generated by a dimension-reduction algorithm. Toretrieve information lost in the dimension-reduction process, weconnect the voxels in the reduced space with lines color-codedaccording to the original correlations. These extracted networksprovide a holographic presentation of the functional relationsbetween the hemodynamic responses at different brain loca-tions.

Consistent with a previous fMRI study of finger tapping tasks(Bandettini et al., 1993), our analysis revealed two independentclusters located in the motor area M1 contralateral and ipsilat-eral to the moving hand. One cluster exhibited activation, i.e. anincrease in the BOLD signal, while the other exhibited a mirror-reversed behavior, with a decrease in the BOLD signal. Inspection ofrepetitive (two consequent movements of the same hand) vs. alter-nating hand movements suggested the existence of an inhibitionmechanism of the ipsilateral hemisphere.

2. Materials and methods

2.1. Experimental design

2.1.1. Imaging acquisitionFifteen healthy volunteers (10 men and 5 women), aged 24–53

(mean 32.7 ± 8.6), signed an informed consent form, required toparticipate in this study, which was approved by the Helsinkiethical committee. Nine subjects were right-handed and six left-handed, as assessed by the Edinburgh inventory (Oldfield, 1971).Structural data were acquired with 3D axial spoiled gradienteco (SPGR) T1-wighted sequence with the following parameters:TR = 5.57 ms, TE = 1.38 ms, Flip angle = 12◦, 146 axial slices withthickness of 1 mm, no gaps, 256 × 256 matrix with a FOV of 25 cm2.

Functional MRI was performed on a 3.0 T GE scanner at theSourasky MC, Tel-Aviv. T2*-weighted images were acquired usinggradient-echo echo-planar imaging (GE-EPI) pulse sequence withthe following parameters: TR/TE = 3000/30 ms, Flip angle = 90◦, 32axial slices with thickness of 4 mm, no gaps, 96/96 matrix (recon-structed to 128/128) with a FOV of 24 cm2.

2.1.2. Motor activation taskWe examined the effectiveness of the proposed method by ana-

lyzing a simple hand movement motor task. The subjects wereinstructed to clench and open either their left or right hand, accord-ing to an auditory cue. The paradigm consisted of 11 block of114 volumes each consisted of a resting period with cross fixation(6–15 s), an auditory instruction period regarding hand movement(right or left; 3 s), and a period of hand movement execution (15 s).The blocks were presented in a constant order across subjectswith regard to which hand to move; i.e. RRLRLRLLRL (R – right,L – left).

Two types of sequences can be examined from our paradigm:alternating hand movements (referred to as Case I) and repetitivehand movements (referred to as Case II). To simplify the presenta-tion we use the following notation to indicate the order of the handmovements – LR, RL RR and LL. For right after left, left after right,right after right and left after left, respectively.

2.1.3. Data preprocessingEach image was preprocessed in SPM2 (http://www.fil.ion.

ucl.ac.uk/spm) for head motion correction. In order to minimizeprocessing of data neither normalization nor spatial smoothing wasimplemented. Following a standard GLM data analysis of right vs.left hand contrast, we chose a representative motor activity slice,including primary motor area (designated as M1) and supplemen-tary motor area (SMA). In Fig. 1 we demonstrate the chosen motorregion of interests (ROIs) marked in red on a single slice of a singlesubject. These regions were defined on a single or multiple axialslices. Approximately 600–1100 voxels were chosen for each sub-ject and then applied to the FH analysis (see Appendix A for moredetails).

Each block was separately defined to include the pre-task rest,the movement, and the post-task rest periods (which cover thehemodynamic delay). Each block related time course contains 12-time points. The analysis was done separately for each of the 10blocks (tasks) performed by each subject with no averaging.

2.2. FH analysis

2.2.1. Voxel correlationsThe first stage in the functional holography analysis is computa-

tion of the voxel correlation matrices – the matrices of correlationsbetween the dynamical responses of all pairs of voxels. We used thePearson formula (Pearson, 1901) to calculate the correlation C(i, j)

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Fig. 1. (A) The experiment’s paradigm time course (taken from Brain Voyager application). The horizontal axis represents the time in TR units and the vertical axis the BOLDsignal value. The magenta and pink bars represent the right and the left movement blocks respectively. The marked section is an example of the time course chosen for oneblock. In this case the block is of a right hand movement. (B) Selection of the ROIs locations. Following a GLM data analysis, an EPI image of the 5th axial slice of this singlesubject (BEZO) was chosen as a representative motor activity slice. The selected rough ROIs – the SMA and the homologues M1, colored red, were drawn extensively aroundregions of known activity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

between voxels (i) and (j):

C(i, j) =

T∑k=1

(X(i, k) − �(i))(X(j, k) − �(j))

�(i)�(j)(1)

where X(i) and X(j) are the recorded time (BOLD) signals of vox-els (i) and (j), with corresponding means �(i), �(j) and standarddeviations �(i), �(j).

The values of the correlations vary between −1.0 (strong neg-ative correlations) and 1.0 (strong positive correlations). Due toreasons that will be discussed below, we converted the correlationvalues from the interval [−1,1] to the interval [0,1]. For N voxels, thepair-wise correlations define a symmetric N × N correlation matrix.In Fig. 2A we show the correlation matrix representing for the ROIsvoxels of a single subject. We note that the correlation matrix canbe associated with an N − 1 dimensional space of correlations, orthe “correlation space” (Baruchi et al., 2004, 2006). We note thatthe correlation space does not represent a real space in the sensethat the eigenvectors do not create an orthogonal mathematicalspace.

In order to reveal subgroups in the correlation matrix, we makeuse of the commonly used dendrogram clustering algorithm (Dubesand Jain, 1980). This algorithm reorders the correlation matrixsuch that highly correlated voxels are closely located. This is per-formed using the correlation distance D(i,j) between voxels (i) and(j), which is the Euclidean distance between the rows i and j in thecorrelation matrix (the vectors of correlations of each one of thevoxels with all other ones)

D(i, j) =∥∥�C(i) − �C(j)

∥∥ =

√√√√ N∑k=1

(C(i, k) − C(j, k))2, (2)

where �C(i) is the correlation vector between voxel (i) and all othervoxels. Next, the algorithm reorders the correlation matrix by sort-ing it according to the hierarchical tree of correlation distances(Fig. 2B and C). We note that others (Cordes et al., 2002; Stanberry etal., 2003) previously analyzed frequency-based correlations eval-uated for fMRI resting state data by means of a dendrogramclustering algorithm. Analyses presented here are performed in thetime domain. Furthermore, contrary to the method used by Cordeset al., which uses correlation distance, i.e. one minus the samplecorrelation between voxels, as the distance measure for the clus-tering between pairs of voxels, the correlation distances here arethe Euclidean distances. In such a way we produce a real metricthat satisfies the triangle inequality. Finally, voxels are selected bySTD filtering of the correlations (Baruchi et al., 2006).

The clusters observed in the reordered correlation matrix corre-spond to subgroups of highly correlated voxels. We selected fromthe ROI voxels that form statistically significant clusters. The auto-matic selection process is composed of two stages: 1. Correlationdistance – using the dendrogram hierarchical tree (Fig. 2C), weselected voxels with a correlation distance below a set threshold,which was chosen to be the STD of the correlation distances. 2. STDfiltering (Fig. 2D) – we searched for subgroups of highly correlatedvoxels, which have low (or negative) correlations with other vox-els outside the subgroup. The correlations of the voxels within thesubgroups are expected to have high STD. Therefore we computedthe distribution of the correlations’ STD and selected voxels whichhad higher STD than the mean of the distribution.

The second stage of the selection process was repeated iter-atively until only the significant clusters remained (Fig. 2E). Thestopping criterion of this selection process was based on the calcu-lation of the cophenetic correlation coefficient (CPCC).

The CPCC criterion (Sokal and Rohlf, 1962) is determined fromthe correlation distance; the CPCC between the set of voxel loca-tions before reordering the matrix (Y) and after reordering (Z) isgiven by:

CPCC =

∑i<j

(Y(i, j) − y)(Z(i, j) − z)√∑i<j

(Y(i, j) − y)2∑i<j

(Z(i, j) − z)2

(3)

Y(i,j) and Z(i,j) are the distances between voxels (i) and (j), and (y)and (z) are the corresponding averaged locations.

Maximal statistical significance of the clustering corresponds toCPCC = 1. Therefore, we used CPCC = 0.95 as the significance thresh-old in the selection process. By setting this threshold we were ableto select 170 voxels from the 689 voxels in the case shown in Fig. 2.

To gain additional confidence about the validation of the iden-tified clusters, we calculated inter- and intra-cluster ratios, asdetailed in Appendix B. Clusters with a negative ratio (due tonegative off-diagonal correlations across the two cliques) were con-sidered significant. Fig. 2F shows the brain locations of the voxelsselected by the CPCC criteria and the inter intra-cluster ratio. As canbe seen, the selected voxels belong to significant clusters and cor-respond to the M1 regions of the two hemispheres and the SMA in amidline region. In order to validate the clusters anatomical locationwe conducted co-registration of the results with the subject’s SPGRimage using the SPM8 (http://www.fil.ion.ucl.ac.uk/spm). Next, weprojected the new image on the anatomical image (e.g. see Fig. 3Fin Section 3.1). We note that the transformation the clusters hadundergone in the co-registration processes makes them appearslightly different in the SPGR space.

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Y. Jacob et al. / Journal of Neuroscience Methods 191 (2010) 126–137 129

Fig. 2. Selection of relevant voxels. (A) Voxel correlation matrix. This matrix represents the correlation coefficients between all voxel pairs. Here we have 689 voxels whichare represented in each axis. The color code is as follows: red – positive correlation; green – no correlation; blue – negative correlation (the color scale is rescaled to 0–1range). Notice that the diagonal coefficients representing the correlation of a voxel with itself have a perfect positive correlation and are indicated by the red color. (B) Thesame correlations matrix shown in panel (A) after reordering by the dendrogram clustering algorithm. (C) The dendrogram tree. The vertical axis is the correlation distancebetween the voxels (the Euclidian distance between the vectors of correlations of each voxel with all the others, or the row in the correlation matrix that corresponds tothe voxel). Longer/shorter distances correspond to lower/higher correlations. The existence of two significant clusters is obvious. The voxels that belong to these clusters areidentified as those between which the correlation distance is shorter than the threshold distance (0.08) defined by the red dashed line. (D) The histogram of the standarddeviation of the correlation of each voxel with all others – the STD of each row in the correlation matrix. The red dashed line represents the STD filtering threshold (0.25).We crop the voxels below the threshold to select the voxels with STD above the threshold. The outcome of this process (including the CPCC described in the text) is that 176out of 689 voxels are selected. More specifically, the iterative cropping process in this example was done first by applying a correlation distance threshold of 0.08 and then,to refine the separation of the clusters, by applying an STD threshold of 0.25. (E) The correlation matrix and (F) the brain locations of the 176 selected voxels. The significanceof the clusters is (−0.88) using the method described in Appendix B – the ratio between the inter- and intra-cluster correlations. The obtained value indicates strong positive(close to 1) intra-correlations and strong negative (close to −1) correlations between the clusters. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of the article.)

Finally, the statistical significance of the obtained clusters isdetermined according to their spatial distribution. Looking at eachhemisphere’s selected voxels separately, a statistical proportiontest was conducted for each with the null hypothesis that all pro-portions are equal. Cluster with p-value smaller than 0.05 in one ofthe hemispheres is considered significant.

For additional validation of the results, we performed theanalysis on one thousand datasets, each consisting of 12 shuffledtime points. All of the shuffled data analyses resulted in spatiallynon-significant clusters. The mean cluster size (number of voxelsin each cluster normalized to the total amount of voxels analyzed)for the shuffled data is very small (1.33%) suggesting these arerandom clusters, whereas the mean cluster size for the originaldata is (11.5%).

2.2.2. Collective normalizationThe next step of the analysis is to perform the affinity transfor-

mation and compute the affinity matrix, a collective normalizationof the correlations (cross-correlation). The motivation for this is

to capture mutual or relative effects between several voxels, vianormalization of the correlations between every pair of voxelsaccording to their correlations with all the other pairs in the matrix.The affinity transformation represents a collective property of allchannels, and can help to capture hidden collective motifs relatedto functional connectivity in the network behaviors (Baruchi et al.,2004, 2006). There are many different metrics for this kind of nor-malization, however we chose to calculate the affinity matrix usingthe meta-correlations matrix MC(i,j), which is the Pearson’s corre-lation between the rows of the reordered correlation matrix of anytwo components (i) and (j) as described in Eq. (4). The affinity col-lective normalized matrix is the product of the correlation matrixand the meta-correlation matrix as defined in Eq. (5).

MC(i, j) =

N∑k /= i,j

(C(i, k) − �c(i))(C(j, k) − �c(j))√〈C(i)2〉〈C(j)2〉

, (4)

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Fig. 3. Closer inspection of the clusters (voxel cliques). (A) The new dendrogram tree for the two clusters identified in Fig. 2. (B) The corresponding voxel correlations matrix.The two clusters that were found are framed with their color code: green for the right hemisphere cluster (voxels 1–64); magenta for the left hemisphere cluster (voxels65–176). In this case, of a right hand movement, the magenta cluster represents the contralateral left hemisphere, while the green cluster represents the ipsilateral righthemisphere. The bars on the plots denote the STD. (C) The holographic representation of the two clusters in the 3D space, the axes of which are the three leading principalPCA vectors of the correlation matrix. Each voxel is located in this space according to its eigenvalues corresponding to the leading principal vectors. (D) The brain axial sliceimage. The display of the two clusters detailed in B with the same color code. In order to ensure the clusters comprise the homologues M1 anatomical regions, we conductedco-registration on the results, and then overlaid the new image on the SPGR image (E). Note that the two clusters have a distinct separation in the brain. A fraction of 0.87of the voxels of the magenta cluster’s voxels is located at the left hemisphere, and a fraction of 0.77 green cluster’s voxels are located at the right hemisphere. The averagedBOLD signals over time of all the voxels in the magenta and green clusters with the STD are shown in (F) and (G), respectively. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of the article.)

A(i, j) =√

C(i, j)MC(i, j). (5)

This MC matrix is calculated on the reshuffled rows of the matrixin a way that all the elements between the voxels (i) and (j) them-selves are not included in the calculation.

As discussed above, we rescale the range of the correlations tothe [0,1] scale, in order to perform the affinity transformation.

After realizing that the correlation and the affinity matrices con-tain crucial information about the functionality of the network, wedecided to use a dimensional reduction algorithm to simplify thedimensionality of the system.

2.2.3. Dimension-reduction and construction of the holographicnetworks

Searching for hidden functional relations between the motortasks and the selected voxels associated with the hemodynamicresponse of the motor cortex, we performed dimension reductionof the correlation matrices. Here we used the PCA dimension-reduction algorithm to extract the maximal relevant informationembedded in the voxel correlation matrices, but other algorithmsmay also be used. Using the PCA algorithm, the relevant informa-tion is presented in a 3-dimensional PCA space (Baruchi et al., 2004,2006) whose axes are the three leading PCA principal vectors. Each

voxel is placed in this space according to its three eigenvalues forthe three leading principal vectors. We found that reduction tothree dimensions (projection on the three leading principal com-ponents) extracts most of the relevant information (above 85%), aswas found for subdural EEG recorded brain activity (Baruchi et al.,2006) and for gene expression data (Madi et al., 2008).

To retrieve information that might be lost during the dimen-sion reduction, we link each pair of voxels by lines color-codedaccording to the correlations between them (Baruchi et al., 2004,2006; Madi et al., 2008). The result is a holographic representation(Fig. 4 in Section 3) of a holographic networks (or manifolds) oflinked nodes (voxels) in the PCA space. We chose to present strongcorrelations (above 0.8) and strong anti-correlations (below 0.2).

2.2.4. Calculation of cluster entropyEntropy has been used in statistics and information the-

ory to develop measures of the information content of signals(Shannon, 1948). Entropy can be used to quantify the devia-tion of the cluster’s eigenvalue distribution from a uniform one(Alter et al., 2000). This idea was used in the context of bio-logical systems (Varshavsky et al., 2006, 2007) and economicsystems (Shapira et al., 2009). We calculated the eigenvalue entropy

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Y. Jacob et al. / Journal of Neuroscience Methods 191 (2010) 126–137 131

Fig. 4. Correspondence between the holographic (in the PCA space) and the anatomical presentations of the clusters. (A) and (B) show the holographic presentations of thevoxels or the holographic networks while the voxels with correlations above 0.8 are linked in (A) and voxels with correlations below 0.2 (are linked in (B)). We note thatsince we used 0–1 scale, correlations below 0.2 correspond to strong negative correlations (below −0.4). Displays of the corresponding holographic networks for differentcorrelation coefficients on the brain slice images are shown in (C) and (D). (C) Corresponds to (A), but voxels with correlations with a specific range of 0.98–0.99 are linked.(D) Corresponds to (B), but connecting only voxels with very strong anti-correlation in the range 0–0.05 (or −0.9 to −1.0 using the scale −1.0 to 1.0).

defined by,

S = − 1log(N)

N∑i=1

˝(i) log[˝(i)], (6)

where ˝ is given by,

˝(i) = �(i)2∑Ni=1�(i)2

. (7)

N the number of voxels and �(i) denotes the matrix eigenvalues.S ranges from 0 to 1. Note that 1/log(N) is a normalization

factor ensuring that S reaches its maximum (S = 1) for a uniformeigenvalue distribution (i.e. random correlations matrix). Latentinformation, LI = 1 − S, is higher for lower entropies.

3. Results

3.1. Functional relations between ipsi- and contralateralhemodynamic responses in M1

As expected, the task selectively activated motor areas. The FHanalysis conducted for each subject and each trial yielded twoindependent clusters. One cluster revealed signal consistent withhemodynamic response function (HRF) activation (Huettel et al.,2004) and was located in the M1 region contralateral to the move-ment. The additional cluster revealed an opposite signal behavior;a decrease in the BOLD signal, and was located in the homologueM1 of the ipsilateral hemisphere. Two clusters could be identifiedin 82% of the trials across subjects, and 90% of the clusters werefound to be significant (p < 0.05) in the spatial statistical proportiontest (see Section 2.2.1).

The two clusters in every block analysis had strong negativecorrelations between them (∼−0.30 ± 0.16) with a negative inter-and intra-cluster ratio (∼−0.62).

To understand the hemodynamic relations between the twoobtained clusters, we computed the signal average over all the vox-

els for each cluster, as shown in Fig. 3F and G. The averaged signalof the contralateral and ipsilateral M1 clusters showed respectivelyactivation and deactivation with respect to resting baseline. Thedynamics of the ipsilateral and contralateral M1 time courses werealmost the mirror image of each other, explaining the strong neg-ative correlations between the two clusters (Fig. 3B).

As seen in Fig. 3D and E some scattered voxels belongingto the activation cluster appeared in the ipsilateral hemisphere,while some voxels belonging to the deactivation cluster appearedin the contralateral hemisphere. However, the scattering ofthese voxels renders their neuro-physiological interpretationproblematic.

The holographic representation of the two identified clusters inthe affinity PCA space is presented in Fig. 3C. Additional functionalmotifs are illustrated in Fig. 4; positive and negative voxel correla-tions are shown both as a holographic network in the affinity PCAspace (Fig. 4A and B) and as a connectivity network imposed onthe brain anatomical picture (Fig. 4C and D). We note that sincethe correlations are calculated between the voxles we present thisillustration upon the EPI image which is the voxels true acquisitionspace.

The results of the FH analysis conducted on a single block (seee.g. Fig. 3), allow to infer in an unsupervised way that the subjectmoved his right hand. In comparison, we conducted a GLM analy-sis upon a single block by creating a new experiment constructedof the same 12-time point’s EPI images (Fig. 5). The GLM analysiswas conducted using Brain Voyager QX Version 2.1 (Brain Innova-tion, Maastricht, The Netherlands). A predictor was constructed fora right hand movement condition. As one can see, opposed to theFH analysis which resulted in two clear significant clusters, GLManalysis of this short time course data resulted in no significantactivations at all, activated voxels appear only with a very highthreshold (p = 0.62). The ROI GLM analysis was constructed uponthe same ROIs as in the FH analysis (illustrated in green squaresin Fig. 5). The predictor of right hand movement condition didnot result with significant contribution to BOLD signal in the ROIs(b = 0.21, p = 0.43).

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Fig. 5. A comparison between the FH and GLM methods. The analysis in both methods was conducted upon the same block time course consisting the same 12-time point’sEPI images. The green squares represent the ROIs in which the analysis was conducted. The FH analysis resulted in two significant clusters (p < 0.05), while the GLM analysisfor this short time course resulted in no significant activations for whole brain analysis (p < 0.63). Also, the predictor of right hand movement condition did not result withsignificant contribution to BOLD signal in the ROIs (b = 0.21, p = 0.43) (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of the article.).

3.2. The effect of the order of motor tasks

The sequence of hand movements, e.g. right after left (LR) orright after right (RR), affected the hemodynamic response withinand between the hemispheres.

Figs. 3, 4 and 6A depict examples from alternating sequences,while Fig. 6B and C shows two examples for a repetitive sequence.

Table 1 summarizes the different observations obtained for thealternating and repetitive hand movements.

The hemodynamic response in the alternating sequence (Case I)always exhibited two clusters: one located mainly in the contralat-eral M1 representing activation, and the other located mainly inthe ipsilateral M1 representing deactivation (Fig. 6A). The hemody-namic response of the repetitive sequence (Case II) resulted in two

Fig. 6. A comparison between Case I and Case II sequences results. (A) Example of hemodynamic response for Case I (RL). The results of a right-handed subject (TARI) forblock #5, which is a left hand movement block that followed a right hand movement block. The correlation matrix shows a pattern of two dominant distinguished clusters.The right hemisphere cluster (colored pink) was located in the M1 region and its averaged BOLD signal demonstrated clear activation, as expected. We also found a secondcluster (colored light green) in the M1 region of the ipsihemisphere. Its averaged BOLD signal showed deactivation with negative correlations to the signal in the pink cluster.This fall below the baseline of the resting state resulted from the preceding activation because of long relaxation time that causes the preceding resting state value to be high.(B) Example of hemodynamic response for Case II (RR). The results of a right-handed subject (BEDA) for right hand movement block that followed a right hand movementblock (block #2) with Case I like hemodynamic response. In this example, the two BOLD signals exhibit activated and deactivated behaviors similar to that of Case I shown in(A). In addition, the two clusters are well located within the two hemispheres. (C) Another example of Case II (RR) response for another subject (KAMI). The special featuresof the hemodynamic response in this example are that one cluster (the green) is scattered in the brain slice image, and its corresponding averaged BOLD signal varies aroundthe baseline resembling a noise signal. Therefore, this cluster is regarded as non-significant. In all the holographic presentations voxels with correlation coefficients above0.8 are linked, and in all presentations on the brain slice images voxels with correlations in the range 0.95–0.99 are connected. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of the article.)

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Table 1The motor tasks four different conditions.

Subject Left/right handed Case 1 Case 1

LR RL RR LL

ADRO R + + + −BEDA R + + + −BEES R + + + −BEZO R + + + −DOIT R + + + −GOIL R + + − +KAMI R + + − −TARI R + + + −ULTI R + + − −BEMA L + + − +BIGI L + + + −GRDA L + + − +MOTA L + + − −PEEI L + + − +RAAM L + + − −

The hemodynamic response of all the subjects for the two cases: alternating handmovements (Case I) and repetitive hand movements (Case II). The first pattern oftwo major distinguished clusters is symbolized with (+). The second pattern of onemajor functional cluster and one very small noise cluster is marked with (−).

different patterns. One pattern of the repetitive hand movementwas similar to the response obtained for the alternating movement(Fig. 6B). The other pattern consisted of one cluster mainly locatedin the contralateral hemisphere, and a second cluster containingweakly correlated voxels scattered between the two hemispheres,with signals fluctuating around the baseline (Fig. 6C). Only the firstcluster was significant according to the spatial proportions test.Most of the subjects (73.3%) showed the former pattern (exhibit twoclusters) in one of the sequences of Case II (RR or LL), whereas therest of the subjects (26.67%) showed the latter pattern (exhibit onecluster) for them both. There was a clear bias towards the appear-ance of the former pattern of two clusters for the subject’s dominanthand sequence. Six out of the nine right-handed subjects showedtwo clusters in the RR sequence, whereas three out of the six left-handed subjects showed the two clusters pattern in the LL sequence(see Table 1).

3.3. Entropy of hemodynamic response clusters

Eigenvalue entropies were used to quantify the functional rela-tions between the hemodynamic responses of the two clusters inthe affinity space (see Section 2.2.4) associated with dominant andnon-dominant hands.

Fig. 7 represents the comparison between right and left hemi-sphere cluster entropies. For each subject, we calculated thedifference between right and left hemisphere averaged entropies.Except for one right-handed subject, all subjects showed lowerentropy for their motor-dominant hemisphere clusters. In a pairedsamples t-test, the left-handed subjects group was found signifi-cant (p = 0.021). The right-handed subjects group was found to benot significant, with the observation of a slight trend (p = 0.083).

We note that when we conducted the t-test analysis on theright-handed group of subjects, while excluding the individualright-handed subject that show the opposite behavior (display-ing higher eigenvalue entropy in the non-dominant hemisphereclusters), the difference between the two hemispheres was foundsignificant (p = 0.039).

4. Discussion

We introduce a new method for the analysis of fMRI data,based on FH, a method originally developed for the analysis ofsubdural EEG recordings of epileptic brain activity (Baruchi et al.,2006). Voxel–voxel correlation matrices were evaluated and inves-

Fig. 7. A comparison between right and left hand movement clusters’ quantifica-tion. This graph displays for each subject the subtraction of the averaged eigenvalueentropy measure of the clusters located in the right hemisphere from that of theleft hemisphere. Except for one right-handed subject, all subjects showed lowerentropy for their motor-dominant hemisphere clusters. The difference between thetwo hemispheres entropies in the left-handed subjects group was found significant(p = 0.021), though it was not the case for the right-handed group (p = 0.083).

tigated first by means of clustering algorithms, then reduced to athree-dimensional space using PCA. This analysis yielded a holo-graphic representation of a functional network related to handaction. Importantly, a straightforward way to quantify the infor-mation embedded in clusters was established and used to estimatehemisphere motor dominance.

4.1. Methodological issues

Our results show that FH analysis needed very few time points todetermine, in high percentage of trials (∼8/10 trials per subject), notonly the exact location of the activation, but also whether subjectsmoved their right or left hand. Uniquely our algorithm deals witha single trial resulting in two statistically distinct clusters of voxelswhich can be regarded as highly connected subgroups (i.e. cliques).One of the cliques consisted mainly of voxels located in the M1contralateral to the movement, with increased BOLD signal, andthe other of voxels mainly located in the ipsilateral M1 region withopposite almost mirror-reversed behavior (see Fig. 3).

Other data-driven and clustering methods, e.g. ICA (McKeownet al., 1998), fuzzy clustering analysis (FCA) (Windischberger et al.,2003), and temporal clustering analysis (TCA) (Zhao et al., 2007),generally require more time points to localize activations. Thus,the FH method may contribute to the development of classifica-tion algorithms for blind identification of different conditions inextremely short time series. Further, the use made by FH of corre-lations between all voxel pairs in the ROIs yields a good separationof different hemodynamic behaviors and captures subtle dynam-ical features that might be overlooked due to prior assumptionsby hypothesis-driven methods. Notably, the FH method capturessensitive hemodynamic variations at a single trial level, withoutthe need for averaging and without the need for contrasts betweenexperimental conditions.

Another key advantage of the FH algorithm lies in its ability todeal with the multivariate nature of fMRI data and to simplify itscomplexity. It does so by reducing the dimensionality of the dataand then connecting voxels in the reduced space. Thus patternscan be visualized in a new space, preserving N-dimensional infor-mation in the correlation matrix (see Fig. 4). It is important to notethat contrary to other approaches which used PCA for the voxelselection (Andersen et al., 1999; Friston et al., 1993; Thirion andFaugeras, 2003), in our method, PCA is not used to identify activa-tions, but simply to visualize patterns in previously found clustersin a 3D space. In this sense, the FH analysis may be regarded as a

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system-level analysis that produces a more complete (holographic)representation of the brain’s BOLD activation.

We note that here we tested the effectiveness of the FH analy-sis as applied to block design experiments. In future work we willinvestigate its applicability to event-related designs). In AppendixC we address this issue using simulated signals that mimic event-related experiments. We show that the method is expected to beeffective even in the case of overlapping signals that mimic rapidevent-related designs, but de-convolution of the BOLD signals isrequired prior to FH analysis.

4.2. Neuronal activations

Our system-based FH analysis captured some neuronal mean-ingful aspects of movement-related activity in M1 region. Likeothers (Cramer et al., 1999; Hanakawa et al., 2005; Horenstein etal., 2009; Kim et al., 1993; Verstynen et al., 2005), we also foundactivated voxels in the ipsilateral hemisphere.

The “mirror” representation of clusters, with increased BOLDsignal in the M1 contralateral to the movement, and the otherof voxels mainly located in the ipsilateral M1 region with oppo-site (almost mirror-reversed) behavior across hemispheres mightbe related to the post-stimulus undershoot of the hemodynamicresponse function (HRF). Since when subjects alternate between aright and a left movement, the contralateral hemisphere for onemovement coincides with the ipsilateral one for the following, it ispossible that the blood flow in the ipsilateral hemisphere may nothave reverted to baseline after the previous movement.

The hemodynamic response of all the subjects in at least oneof the repetitive sequence blocks (Case II) was consistent withthe first explanation of the HRF post-stimulus undershoot: theresponse showed one cluster of highly correlated activated voxels,mainly located in the contralateral hemisphere, while the secondcluster was found to be non-significant and scattered across thehemispheres; while no ipsilateral cluster was found in these cases.However, more than half of the subjects (66.7%) exhibited the firstpattern of two significant clusters (one in the contralateral andone in the ipsilateral hemispheres) in at least one of the repeti-tive sequence blocks. HRF post-stimuli undershoot cannot explainthis behavior. A possible different mechanism may be representedby inhibition of ipsilateral M1 consistent with previous fMRI stud-ies of unilateral finger movements (Allison et al., 2000; Hamzeiet al., 2002; Newton et al., 2005; Nirkko et al., 2001; Stefanovicet al., 2004). This inhibition would caused by the opposite handmovement, which would favor movements of the preferred handby minimizing interference from the opposite one.

It has previously been reported that deactivations were morepronounced with increasing movement duration, and for dominanthand movements (Newton et al., 2005). Interestingly, our resultsare consistent with this latter effect, as subjects almost alwaysshowed the two-cluster pattern with repetitive movements withdominant hand (see Table 1).

In addition, we used eigenvalue entropy to assess the differencesbetween the two calculated clusters in each trial, and to quan-tify the amount of latent information contained in each cluster’scorrelation matrix. Using this measure, we obtained a good separa-tion between right-handed and left-handed subjects. The relativelower entropy in the motor-dominant hemisphere cluster, demon-strates less variability, and therefore greater latent information, inthe motor-dominant hemisphere cluster’s correlations, suggestinga higher modular organization in specific motor-dominant hemi-sphere (see Fig. 7). However, we note that due to the small numberof subjects, this conclusion needs further validation.

In this paper we chose to demonstrate the new method upon theM1 and SMA regions alone. However, other brain locations such asthe thalamus, putamen, and contralateral superior cerebellum are

also involved in the motor system. Including all the aforementionedareas, thus creating a broader network might provide additionalinteresting information and we intend to study it in greater detailsin the future. See Appendix D for preliminary results of the FH anal-ysis upon the cerebellum ROI, showing the opposite pattern as inthe M1 clusters of neuronal activation in the ipsilateral cerebellum.

5. Conclusions

We propose that our FH approach provides a valuable additionalmethod (to the many and efficient existing ones) for analyzingfMRI data as it is very efficient in revealing subtle, system-leveldynamical features that are harder to detect by other methods.

We demonstrated the efficiency of the FH in finding new fea-tures at single subject level. The method’s sensitivity to differencesin the topological and geometrical features of the connectivity net-works (in the holographic representation in the reduced PCA space)between subjects suggests that it could also effectively be appliedto account for differences across subjects.

We also expect that the FH method may also be applied to delin-eate functional connectivity in behavioral paradigms of greatercomplexity.

Finally, we stress that the principles and implementation of theFH analysis are relatively simple and straightforward. The methodhas already been proven to be very efficient in analyzing EEG (sub-dural) data. Since the outcome of the analysis is a holographicpresentation in an abstract reduced space, it is a potentially idealtool for multi-modal analysis of combined data from EEG–fMRIexperiments, with the fMRI providing a high spatial resolution andthe EEG providing the high temporal resolution.

Acknowledgments

We are most thankful to Asaf Madi, Sharon Bransburg-Zabary,Noga Yaniv, Keren Rosenberg and Andry Zhdanov for fruitful dis-cussions. We especially thank Dror Kenett and David Papo forthoughtful review of this manuscript, and Professor Nathan Intra-tor for constructive comments about the mathematical basis of themethod. This research has been supported in part by the US-IsraelBi-National Science Foundation (BSF), the Tauber Family Founda-tion and the Maguy-Glass Chair in Physics of Complex Systems.

Appendix A.

A whole-slice analysis was conducted prior to the ROIs analysispresented in the paper. First we used GLM analysis to select a rep-resentative motor activity slice, and only then did we apply the FHanalysis to all the voxels in the slice. Fig. 8 demonstrates the anal-ysis on a single subject for the course of the entire experiment. Wefound two large clusters located in the M1 and SMA regions with anaveraged (over the voxels that belong to the clusters) BOLD signalthat corresponds to the paradigm (Fig. 8F). In this specific case, twoadditional small clusters were found that show no correlation withthe paradigm.

Based on this consistency test, in order to make the analysisprocess more efficient, we started the FH analysis on voxels locatedin areas that cover the motor regions instead of on the entire brainslice. Another reason for starting the analysis on ROIs in the motorareas is that in whole slices there are many irrelevant voxels thatact like noise sources to the extent that the noise might mask therelevant features.

Appendix B.

An issue that is always encountered when clustering analysis isperformed is related to the assessment of the statistical significance

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Fig. 8. The FH analysis results for the whole slice of a single subject. (A) All the voxels in the axial slice, marked red, were chosen for the analysis, resulting in a very largecorrelation matrix (B) with a clear majority of noise-level correlation around 0.5. (C) The correlation matrix after focusing on the clusters. As a result of cropping all the noise– level uncorrelated voxels, we obtained 4 clusters, as also seen in the FH PCA space (E). (D) We found two relatively large clusters located in the M1 and SMA regions with anaveraged BOLD signal that show a clear response to the experiment paradigm (F). As seen in the correlation matrix (C) and the holographic representation (E), two additionalsmall clusters are detected. Checking the BOLD signals of these clusters (G) we found that they are not correlated with the paradigm. We note that these two clusters exhibitstrong negative correlation reminiscent of that between the two clusters detected for the autonomous brain response. Hence, they could be associated with the resting statethat, in this case of normal subjects, is weaker in comparison to the paradigm-related clusters. The results in (D) indicate that the functional clusters that are correlated withthe paradigm are located in the motor regions. Therefore, to save time in the pre-analysis, we start the FH analysis on ROIs taken in the motor areas. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of the article.)

of the results. From a mathematical perspective, a dataset contain-ing N elements (voxels, in our case) can be organized in any numberof clusters between 1 and N. Hence, in principle, there are no math-ematically objective criteria to determine the number of relevantclusters or optimal number of clusters. Thus, heuristic criteria are

used and then assessed for significance by some statistical test. Bythe same token, there is no one optimal clustering algorithm or oneoptimal statistical test. Instead, there are many efficient clusteringalgorithms, each developed to capture specific features (Halkidi etal., 2002). For each case, then, it is necessary to select (or develop)

Fig. 9. Assessments of the applicability of the FH analysis to investigate event-related experiments. We analyzed simulated signals of four overlapping HRF responses eachdemonstrate a different stimulus given 2–3 TR apart. The HRF signals (overall of 200 simulated voxels) were generated via computer simulation of two-gamma probabilitydensity function with noise injection. (A) Analysis of the additive signal of the four HRFs. The voxel–voxel correlation matrix revealed one cluster and shows no separationbetween the conditions. (B) Once prior de-convolution of the signal which results in four separated signals was preformed, the FH analysis yielded four distinguished clusterscorresponding to the four conditions. It illustrates that the FH analysis can even be applied in case of overlapping HRFs following prior de-convolution of the signals.

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136 Y. Jacob et al. / Journal of Neuroscience Methods 191 (2010) 126–137

the algorithm that is efficient in revealing the features of interestand to assess the statistical significance of the results (the identifiedclusters) accordingly.

Here, since we are interested in correlations, we tested thesignificance of the clusters obtained by the FH analysis by calcu-lating the inter-cluster and the intra-cluster averaged correlations.We observed high intra-cluster correlation values of around(+0.5 ± 0.18) on a scale of −1 to 1, and inter-cluster low correla-tion values of around (−0.3 ± 0.17). We regard the identification ofthe clusters to be statistically significant when the ratio betweenthe inter-cluster and intra-cluster values, is below zero (negativecorrelations between the clusters). We note that this ratio criterionis similar to the well-established measure of homogeneities andseparation used to determine the validity of clusters (Sharan et al.,2003). The final step is to determine the spatial distribution of theclusters. A cluster is considered significant if more than ∼60% of thevoxels selected in a certain hemisphere belong to that cluster.

Appendix C.

We present a preliminary assessment of the possible applicabil-ity of the FH analysis in the case of event-related designs. In order

to evaluate the conditions under which the FH analysis yields validresults it was used to analyze simulated signals that mimic event-related experiments (Fig. 9). The HRF signals were generated viacomputer simulation of two-gamma probability density function.As could be expected, we found the method to be very effective inanalyzing signals that are not overlapping (results are not shownhere as they are similar to the findings presented here for the blockdesign experiments). The method can also be applicable in the caseof overlapping signals (that mimic rapid even-related conditions),but require de-convolution of the BOLD signal prior to FH analysis(Fig. 9).

Appendix D.

Fig. 10 demonstrates preliminary results of the FH analysisapplied on the cerebellum ROI. The FH method indeed finds clustersof activations. Two of the clusters show as expected the oppositepattern as in the M1 clusters of neuronal activation in the ipsi-lateral cerebellum (Fig. 10E) and deactivation in the contralateralcerebellum (Fig. 10C).

Fig. 10. Preliminary results of the FH analysis applied on the cerebellum ROI. This example shows the results of a right-handed subject (BEZO) for right hand movement block(block #1). (A) The correlation matrix showing four clusters. (B) Presentation of the clusters on the SPGR brain image. The corresponding averaged BOLD signals are shownin (C)–(F). Two of the clusters show as expected the opposite pattern as in the M1 clusters of neuronal activation in the right ipsilateral cerebellum (E) and deactivation inthe left contralateral cerebellum (C).

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