Functional Topography: MultidimensionalScaling and Functional Connectivity inthe Brain

K. J. Friston, C. D. Frith, P Fletcher, P. F. Iiddle, and R. S. J.Frackowiak

Wellcome Department of Cognitive Neurology, Institute ofNeurology, London, United Kingdom

In neuroimaging, functional mapping usually implies mapping functioninto an anatomical space, for example, using statistical parametricmapping to identify activation foci, or the characterization of distrib-uted changes with spatial modes (eigenimages or principal compo-nents) (Friston et al., 1993a). This article is about a complementaryapproach, namely, mapping anatomy into a functional space. We de-scribe a simple variant of multidimensional scaling (principal coor-dinates analysis; Gower, 1966) that uses functional connectivity as itsmetric. The scaling transformation maps anatomy into a functionalspace. The topography, or proximity relationships, in this space em-body the functional connectivity among brain regions. The higher thefunctional connectivity, the closer the regions. Functional connectivityis defined here as the correlation between remote neurophysiologicalevents. The technique represents a descriptive characterization of an-atomically distributed changes in the brain that reveals the structureof corticocortical interactions in terms of functional correlations. Toillustrate the approach we have analyzed data from normal subjectsand schizophrenic patients obtained with PET during the performanceof word generation tasks. In particular, we focus on prefrontotemporalintegration in normal subjects and show that in schizophrenia, theleft temporal regions and prefrontal cortex evidence abnormal func-tional connectivity.

This article is about the topography of functional brain spacesand corticocortical interactions. We present a descriptivemethod for characterizing the interrelationships of corticalareas in terms of functional connectivity. The method em-ploys metric multidimensional scaling with functional con-nectivity as the metric, or measure, that determines the prox-imity between cortical areas. The objective is to transformanatomical space so that the distance between cortical areasis directly related to their functional connectivity. This trans-formation defines a new space whose topography is purelyfunctional in nature.

Functional ConnectivityIn the analysis of neuroimaging, time series functional con-nectivity is defined as the temporal correlations betweenspatially remote neurophysiological events (Friston et al.,1993a). This definition is operational and provides a simplecharacterization of functional interactions. The alternative isto refer explicitly to effective connectivity (i.e., the influenceone neural system exerts over another) (Friston et al.,1993b). These sorts of concepts were originated in the anal-ysis of separable spike trains obtained from multiunit elec-trode recordings (e.g., Gerstein and Perkel, 1969; Gerstein etal., 1989; Aertsen and Preissl 1991; Gochin et al., 1991). Inelectrophysiology, it is often necessary to remove the con-founding effects of stimulus-locked transients (which intro-duce correlations that are not causally mediated by directneural interactions) in order to reveal the underlying effectiveconnectivity. The confounding effect of stimulus-locked tran-sients is less problematic in neuroimaging because the pro-mulgation of dynamics from primary sensory areas onward ismediated by neural connections (usually reciprocal and inter-

connecting). However, it should be remembered that func-tional connectivity is not necessarily due to effective connec-tivity and, where it is, effective influences may be indirect.Because functional connectivity (as defined here) is simply acomment on observed correlations, it cannot be used to infercausal relationships in any rich way; however, it does providea very useful phenomenological characterization of corticalinteractions at any scale.

Clearly, the biological nature of functional connectivity inneuroimaging is different from functional connectivity inelectrophysiology. The neural networks that might be identi-fied on the basis of phase-locked interactions (using multiunitelectrode recordings) in a particular and transient brain stateare not the same as macroscopic systems identified on thebasis of correlated blood flow observed with neuroimagingover a variety of brain states. However, in both instances thedistributed and coordinated physiological changes can beused to infer something about functional interactions eitherat the level of neural dynamics and phase-locked cohorts orat the level of hemodynamics and cortical coactivations.

Consider two times-series of K hemodynamic measure-ments, from voxels i and j in the brain. Let m'k denoted thefeth measurement from voxel i. The functional connectivitybetween i and j can be defined as

= 2 mvm'k, (1)

where the time series have been normalized to zero meanand unit sum of squares (Euclidean normalized, i.e.,1(mlj2 =1). pM is also known as the scalar or dot product of vectorsm' and m1. Patterns of functional connections, or correlations,define distributed brain systems. These systems are identifiedusing principal component analysis (PCA) or singular valuedecomposition (SVD) of the functional connectivity matrix.The distributed systems that ensue are called eigenimages orspatial modes, and have been used to characterize the spa-tiotemporal dynamics of physiological time series from sev-eral modalities, including multiunit electrode recordings (May-er-Kress et al., 1991), EEG (Friedrich et al., 1991), MEG (Fuchset al., 1992), PET (Friston et al., 1993a), and functional MRI(Friston et al., 1993c).

In the present application, functional connectivity is usedin a different way, namely, to constrain the proximity of twocortical areas in some functional space. This application cap-italizes on the fact that the functional connectivity betweeni and> is the same as between j and i. This symmetry meansfunctional connectivity can support a measure of distance ina Euclidean sense (a metric). The space on which this mea-sure is made is constructed using multidimensional scaling.

Multidimensional ScalingMultidimensional scaling is a descriptive method for repre-senting the structure of a system, on the basis of pairwisemeasures of similarity or confusability (Torgerson, 1958; Shep-ard, 1980). The resulting multidimensional spatial configura-tion of the system's elements embodies (in its proximity re-

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lationships) the comparative similarities. The technique wasdeveloped primarily in the analysis of perceptual spaces. Theproposal that stimuli be modeled by points in space, so thatperceived similarity is represented by spatial distances, goesback to the days of Isaac Newton (1704). The implementationof this idea is, however, relatively new (Kruskal, 1964; Gower,1966; Shepard, 1980). In this article we focus on classical ormetric scaling (see Chatfield and Collins, 1980). The input toa scaling analysis is a (w X ri) square symmetric matrix ofsimilarities, and the output is an (w X r) matrix of coordinatesof n point in r dimensions. A typical model underlying clas-sical scaling can be summarized by

dM = (2)

where Fmon(-) is a decreasing monotonic function. 8,, is themeasure of similarity between elements i and j . di( is the dis-tance between them in a Euclidean space, x*, is the projectionof the fth point onto the fth dimension (= means equal, ex-cept for unspecified error terms). The points are usually plot-ted in a subspace of this Euclidean space spanned by the reigenvectors (of the matrix of dot products of the point lo-cations) with "large" eigenvalues (Carroll and Wish, 1974;Chatfield and Collins, 1980; Shepard, 1980) (see below). Theresulting distribution of points in the new r-dimensional sub-space will capture, in a parsimonious way, the structure ofthe comparative similarities.

Multidimensional Scaling with Functional ConnectivityIn this section we observe that if the correlation or functionalconnectivity is used as the measure of similarity betweenbrain regions, then there is a very simple way to compute thedistances d£j above to construct a functional (multidimension-al scaling) space. The approach is equivalent to a principalcoordinates analysis (Gower, 1966) of the imaging time series. .

One normally considers K measurements at n voxels as Kpoints in an n-dimensional space (w-space). However, thereis an entirely equivalent representation of n points in a K-space. The distance between points in this X'-space can beused directly as a measurement of diy This is the same as usingthe functional connectivity (p,,) as the measure of similarityCStj = p,,), where the function relating similarity and distanceis given by

= dl( = V 2 V l - Pl).- P.i- (3)

The points in Jf-space are simply rotated to reveal thegreatest structure using the eigenvectors of the K X K dotproduct matrix of their locations. This rotation brings the"principal coordinates" of the distribution into view. The ve-racity of Equation 3 is demonstrated by noting that orthogo-nal rotation does not change Euclidean distances, and so

(x\ - (m'k -

C4)

where m't and x1, are the coordinates of the points before andafter rotation. This approach to identifying the coordinates x1,is called a principal coordinates analysis (Gower, 1966), al-though the term classical scaling is preferred to avoid con-fusion with PCA (Chatfield and Collins, 1980).

Although care has been taken to relate this characteriza-tion of functional topography to classical scaling, principalcoordinates analysis, and metric multidimensional scaling, theunderlying idea is very simple: imagine K measures from n

voxels plotted as n points in a X-space. Because they havebeen normalized to zero mean and unit sum of squares, thesepoints will fall on an K — 1 dimensional hypersphere. Thecloser any two points are to each other, then the greater theircorrelation or functional connectivity (in fact, the correlationis the cosine of the angle subtended at the origin). The dis-tribution of these points embodies the functional topography.A view of this distribution, that reveals the greatest structure,is simply obtained by rotating the points to maximize theirapparent dispersion (variance). In other words, one looks atthe subspace with the largest "volume" (spanned by the ei-genvectors with the largest eigenvalues). Note that in thisview (or projection) the distances seen will not be the actualdistances in the K — 1 dimensional space. One can eitherregard this discrepancy as being attributable to "noise"(where the variance in the remaining dimensions is sufficient-ly small to be ignored and the equality in Eq. 3 becomes =),or acknowledge explicitly that one is looking at a high di-mensional space "from the side."

Mathematically, this rotation can be implemented usingSVD. Let M = [m1 . . . mn] be a matrix of the normalized data(one column vector per voxel time series), and X = [x1 . . .xr]T be the matrix of desired coordinates C denotes transpo-sition). Using SVD, M can be factorized (Golub and Van Loan,1991):

such that

[usv] = SVD{M}

M = u s vT, (5)

where u and v are unitary orthogonal matrices and s is adiagonal matrix. The principal axes of the n points in Af-spaceare given by the eigenvectors of M • W, that is, u:

where X. = s2 and

M MT = u \ uT,

X = MTu. (6)

Voxels that have a correlation of unity will occupy thesame point in the new space. Voxels that have independentdynamics <pi( = 0) will be V2 apart. Voxels that are negativelybut totally correlated (p,, = — 1) will be maximally separated(by a distance of 2). Profound negative correlations denote afunctional association that are modeled in the functionalspace as diametrically opposed locations on the hypersphere.In other words, two regions with profound negative correla-tions will form two "poles" in functional space.

There is an interesting aspect of this application of classi-cal scaling to neuroimaging data. Normally, the data used inmultidimensional scaling represent similarities between dis-crete elements (e.g., voxels). However, neuroimaging data canalso be thought, of as a good lattice representation of a con-tinuous and smooth process in anatomical space. This meansthat the scaling transformation represents a mapping (or dis-tortion) of one volume into another. In other words an ana-tomical region (e.g., the superior temporal gyms) has a con-tinuous and distributed representation in the functional spacedefined by the scaling procedure. The location and shape ofthis new volume will, of course, be completely different fromthe anatomical volume, but local contiguity relationships willbe preserved. This preservation is due to high local autocor-relations (smoothness) in the underlying process (that is as-sumed to have a twice differentiable autocorrelation functionat zero). Consider two points in the image process separatedby dx. As dx tends to zero the correlation between the twopoints will tend to unity (because of the assumption aboutthe autocorrelation function) and the distance in functionalspace will tend to zero by Equation 3. In other words, prox-

Cerebra] Cortex Mar/Apr 1996, V 6 N 2 157

Figure 1. Experimentally introduced vari-ance. Left, Statistical parametric map[SPMj of the F ratio following an AN-COVA of the six-subject 12-conditionverbal fluency study. The display formatis standard and provides three views ofthe brain in the stereotactic space of Ta-lairach and Tournoux (1988) (from theback, from the right and from the top).Right, Eigenvalues (singular valuessquared) of the functional connectivitymatrix reflecting the relative amounts ofvariance accounted for by the 11 dimen-sions of the functional space. Only twoeigenvalues are greater than unity andto all intents and purposes the space de-fined by classical scaling can be consid-ered two dimensional.

] \ ^ " /

SPMprojections

experimental variance10

dimension

imate points in anatomical and functional spaces both tendto zero in the limit of small separations. Contiguity of this sortimplies that bounded regions in anatomical space remain con-nected in functional space (however tenuously); however,these regions may be intersect themselves in a highly com-plicated way and two anatomical regions can occupy thesame functional space. Clearly for real (voxel) data this con-tiguity preservation depends on voxel sizes being "small" rel-ative to the width of the autocorrelation function. For PETdata this is assured but in other modalities (e.g., functionalMRI) this may not be the case.

In what follows, anatomical regions are represented as con-tinuous distributions in functional space with varying density.This density is simply the density of points corresponding tovoxels in the original anatomical volume.

The Functional Topography of Word GenerationIn this section we apply the scaling transformation to a PETtime series from a verbal fluency activation study. These dataare the same used to illustrate the identification of spatialmodes using PCA in Friston et al. (1993a). In brief, the datawere obtained from six subjects scanned 12 times (every 8min) while performing one of two verbal tasks. Scans wereobtained with a CTI PET camera (model 953B, CTI, Knoxville,TN). "O was administered intravenously as radiolabeled waterinfused over 2 min. Total counts per voxel during the buildupphase of radioactivity served as an estimate of regional cere-bral blood flow (rCBF) (Fox and Mintun, 1989). Subjects per-formed two tasks in alternation. The first task involved re-peating a letter presented aurally at one per 2 sec (word shad-owing). The second was a paced verbal fluency task, wherethe subjects responded with a word that began with the letterpresented (intrinsic word generation). To facilitate intersub-ject pooling, data were stereotactically normalized (Friston etal., 1990) and whole brain differences were removed usingANCOVA (Friston et al., 1991). Although the scaling transfor-mation can be applied to single subjects, we used the averagevoxel rCBF over all the subjects for the same reasons givenin Friston et al. (1993a).

A subset of voxels was selected in which a significantamount of variance, due to the 12 conditions, was observed[ANCOVA F(l 1,54) > 2.6,p < 0.05]. This subset is shown ina statistical parametric map (Friston et al., 1991) of the F ratioin Figure 1 (left). The time series from each of these voxelsformed the data matrix M with 12 rows (one for each con-dition) and 6477 columns (one for each voxel). Followingnormalization (to zero mean and unit sum of squares overeach column), M was subject to singular value decomposition

according to Equation 5 and the coordinates X of the voxelsin the functional space computed as in Equation 6.

This space was essentially two dimensional (only two ei-genvalues were greater than unity; see Fig. 1, right). The lo-cation of voxels in this two-dimensional subspace is shownin Figure 2 by rendering voxels from different regions in dif-ferent colors. The anatomical regions corresponding to thedifferent colors are shown in the top row. Anatomical regionswere selected to include those parts of the brain that showedthe greatest variance during the 12 conditions (Fig. 1, left).Anterior regions (Fig. 2, right) included the mediodorsal thal-amus (blue), the dorsolateral prefrontal cortex (DLPFC) andBroca's area (red), and the anterior cingulate (green). Poste-rior regions (Fig. 2, left) included the superior temporalregions (red), the posterior superior temporal regions (blue),and the posterior cingulate (green). The voxels constitutingthese regions were within 20 mm of appropriate centers se-lected from the atlas of Talairach and Tournoux (1988) (seeTable 1). The reason that anterior and posterior regions arepresented separately is simply due to the fact that there areonly three primary colors to play with, but there are morethan three regions of interest.

The corresponding functional space (Fig. 2, lower row)reveals a number of things about the functional topographyelicited by this set of activation tasks. First, each anatomicalregion maps into a relatively localized portion of functionalspace. This preservation of local contiguity reflects the highcorrelations within anatomical regions, due, in part, tosmoothness in the original data and to high degrees of intrare-gional functional connectivity. Second, the anterior regionsare almost in juxtaposition, as are the posterior regions; how-ever, the confluence of anterior and posterior regions formtwo diametrically opposing poles (or one axis). This config-uration suggests an anterior-posterior axis with prefronto-temporal and cingulocingulate components. Third, within theanterior and posterior sets of regions certain generic featuresare evident. The most striking is particular ordering of func-tional interactions. For example, the functional connectivitybetween the posterior cingulate (green) and superior tem-poral regions (red) is high and similarly for the superior tem-poral (red) and posterior temporal regions (blue), yet the pos-terior cingulate and posterior temporal regions show very lit-tle functional connectivity (they are V2 apart or equivalentlysubtend 90° at the origin). Finally, within the main anteropos-terior axis there appear to be two subordinate axes. The firstis a prefrontotemporal axis (red/blue-red), and the second isan anterior-posterior cingulate axis (green-green). These twoaxes are closely aligned but are not completely confounded.

158 Functional Topography • Friston et al.

- 1 -0.5 0 0.5 1Functional space

Figure 2. Scaling analysis of the func-tional topography of intrinsic word gen-eration in normal subjects. Top, Anatom-ical regions categorized according totheir color. The location of these regionsand their designation are given in Table1. Bottom, Regions plotted in a functionalspace following the scaling transforma-tion. In this space the proximity relation-ships reflect the functional connectivitybetween regions. The color of each vox-el corresponds to the anatomical regionit belongs to. The brightness reflects thelocal density of points corresponding tovoxels in anatomical space. This densitywas estimated by binning the number ofvoxels in 0.02 "boxes" and smoothingwith a Gaussian kernel of full width athalf maximum of three boxes. Each colorwas scaled to its maximum brightness.

-1 -0.5 0 0.5Functimal space

These results are consistent with known anatomical con-nections. For example, DLPFC-anterior cingulate connections,DLPFC-temporal connections, bitemporal commissural con-nections, and mediodorsal thalamic-DLPFC projections haveall been demonstrated in nonhuman primates (e.g., Goldman-Rakic, 1986, 1988). The mediodorsal thalamic region andDLPFC are so correlated that one is embedded within theother (purple area). This is pleasing, given the known thala-mocortical projections to the DLPFC.

Interpretation of the Functional SpaceAt this point, one might ask if absolute position in this func-tional space has any meaning. For example, is the fact thatthe prefrontotemporal axis is horizontal (as opposed to ver-tical) important. The answer is yes. The dimensions of thetransformed space have specific functional attributions thatdepend on the tasks employed to elicit the functional inter-actions. Because the dimensions of the functional space are

Table 1Location of anatomical regions in Talairach and Tournoux stereotaxic space

Name

Mediodorsal thalamusLeft DLPFCBroca's areaAnterior cingulatePosterior cingulateSuperior temporal gyrusPosterior middle temporal gyms

Location x,y,z(mm)

0 - 1 2 4- 4 8 32 12- 5 8 16 24- 1 2 24 24

- 8 - 4 8 24±56 8 4±54 - 5 6 0

PutativeBrodmann'sarea

464432322122

Color

blueredredgrengreenredblue

Regions chosen for the analysis of the 12-condition word generation study of normal subjects. Allvoxels that reached criteria following ANC0VA and fell within 20 mm of the above location constituteda "region." See Figure 2 (left) for a graphical presentation of this anatomical parcellalion.

defined by unit vectors in a Kspace of tasks, each dimensionis associated with a particular profile of the experimentalconditions. For example, the first dimension points in thedirection of all the intrinsic word generation tasks and awayfrom the baseline word-shadowing tasks. Conversely, the sec-ond dimension points toward the first scans and away fromthe last scans. The vectors denning these directions are simplythe first two columns of u and are shown in Figure 3 (left).On the basis of these task-dependent profiles one could des-ignate the first dimension of the functional space as inten-tional (corresponding to the intentional or intrinsic genera-tion of words) and the second as attentional (attentionalchanges or changes in perceptual set as the experiment pro-ceeds).

This perspective provides a slightly richer interpretationof the functional space in the following way: functional con-nectivity (distance) between two regions can be partitionedinto intentional (horizontal) and attentional (vertical) com-ponents. For example, the horizontal proximity of the DLPFC(red) and anterior cingulate (green) is greater than their ver-tical proximity. In other words, the functional connectivitybetween the DLPFC and anterior cingulate is dominated bythe intentional aspects of the tasks used to elicit the func-tional interactions. Similarly, the (horizontal) prefrontotem-poral axis is almost entirely intentional, whereas the (oblique)anteroposterior cingulate axis suggests both intentional andattentional components. This interpretation will be importantbelow in examining the functional topography of schizophre-nia.

The Relationship between the Functional Space and the Spatial Modes(Eigenimages) of the Time SeriesThe last part of this section comments on the intimate rela-tionship between the dimensions of the functional space andthe eigenimages or spatial modes associated with the time

Cerebral Cortex Mar/Apr 1996, V 6 N 2 159

Figure 3. Functional attribution of thefunctional space. Left, Eigenvectors ofthe distribution of points in the functionalspace, for instance, eigenvectors ofMM7 . These eigenvectors (or singularvectors) are unit vectors that define adirection in functional space. The attri-bution of this direction or dimension de-pends on relating this vector to the tasksemployed during the activation. The firsteigenvector (fop) is clearly related to thedifference between word generation(even-numbered conditions) and wordshadowing (odd-numbered scans). Thisdifference is the intentional or intrinsicgeneration of word representations. Thesecond eigenvector {bottom) corre-sponds to some largely monotonic timeeffect we have labeled attentional. RightThe eigenimages corresponding to thefirst two eigenvectors of the functionalconnectivity matrix. These eigenimages(or spatial modes) are the eigenvectorsof M U l The eigenimages are displayedas a maximum intensity projection instandard SPM format The color scale isarbitrary, and each SPM is scaled to itsmaximum.

sagittal

5 10singular vectors

0.6

0.4

0.2

0

-0.2

-0.4

spatial modes {eigenimages}

sagittal

SPMprojections

5 10conditions

series. The relationship is, in fact, very simple (see Chatneldand Collins, 1980, p. 200). The time-dependent expression ofthe eigenimages are the same as the vectors describing thedimensions in the functional ^T-space. Figure 3 (right) showthe eigenimages that correspond to the two dimensions used

• in the scaling transformation. They are images of the first twocolumns of v in Equation 5 (see Friston et al., 1993a, for afuller discussion of how one interprets these eigenimages). Inbrief, they represent the distributed systems that best accountfor the observed variance-covariance structure exhibited bya neurophysiological time series (it should be noted that theeigenimages presented here are not exactly the same as thosepresented in Friston et al. (1993a), because the current eigen-images are images of the eigenvectors of the correlation ma-trix, as opposed to the covariance matrix that is usually used).Consider again the singular value decomposition of M:

and

M = u-s-vT

MTM = p = v - \ v T

Therefore, v is a matrix whose columns correspond to theeigenimages of M. The rotation implicit in our scaling ap-proach is effected by

X = MT u = v s .

X is a matrix of the eigenvectors v scaled by their singularvalues. Put simply, one can either use the eigenvectors of thefunctional connectivity matrix to (1) generate a series of ei-genimages, or (2) scale them according to tiieir singular val-ues and use them as coordinates to construct a functionalspace. These two analyses (principal coordinates analysis andprincipal components analysis) are entirely equivalent from a

mathematical point of view, but reveal the nature of function-al interactions from different perspectives.

Functional Disintegration in SchizophreniaIn this section we present an analysis of previously publishedPET data examining functional connectivity in schizophrenia(Friston et al., 1996). The notion that schizophrenia repre-sents a disintegration of the psyche is as old as its name, in-troduced by Bleuler (1913) to convey a "splitting" of mentalfaculties. We have investigated the hypothesis that this men-talistic "splitting" has a physiological basis, with a precise andregionally specific character.

Neurodevelopmental (e.g., Weinberger, 1987) and cogni-tive models of schizophrenia (e.g., Frith, 1987) have empha-sized abnormal frontolimbic and prefrontotemporal integra-tion. Structural MR1 studies of schizophrenic brains havefound abnormalities in the temporal cortex and underlyingwhite matter with some consistency (Shenton et al., 1992;Williamson et al., 1992; McCarley et al., 1993). Our previousanalysis of the eigenimages, derived from word-generationPET activation studies, in normal subjects and schizophrenicpatients, pointed to abnormal functional connectivity be-tween the left dorsolateral prefrontal cortex (DLPFQ and theleft superior and middle temporal gyri (Friston et al., 1996).We applied the scaling transformation to the data in the hopeof revealing, in a direct way, the relationship between thetemporal regions and prefrontal areas, in terms of functionalconnectivity.

The details of the experimental design and data acquisitionhave been described elsewhere (Friston et al., 1996) and willbe summarized briefly. Four groups of six subjects werescanned six times during the performance of a series of word-generation tasks (verbal fluency, semantic categorization, and

160 Functional Topography • Friston et al.

word shadowing; each task was preformed twice in balancedorder). The four groups comprised (1) controls, a normalgroup; (2) poor, patients who produced a small number ofwords during FAS verbal fluency; (3) odd, patients who pro-duced odd, inappropriate words; and (4) unimpaired, patientswhose performance was near normal. The patients all metDSMIII-R criteria (American Psychiatric Association, 1987) forschizophrenia.

The data were stereotactically normalized (Friston et al.,1991) and a mean rCBF estimate for each voxel, for eachcondition, for each group, was obtained by averaging oversubjects in each group using the same techniques mentionedin the previous section. A subset of voxels was selected atwhich differences between any of the six scans accountedfor a significant amount of variance [ANCOVA,F(5,24) > 3-9,p < 0.001] in one or more of the four groups. The result wasa large matrix of rCBF estimates (M), each comprising sixrows (one for each condition) and 4802 X 4 columns (onefor each voxel in each group). M was normalized to a meanof zero and unit sum of squares (over each column).

The matrix (M) was subject to the scaling transformation,as described in the above section. Note that all four groupswere entered at the same time. This meant that the functionaldesignation of the dimensions of the functional space was thesame for all groups. The results of these analyses are seen inFigures 4 and 5. Figure 4 (left) shows the anatomical regionsrendered in subsequent figures. They included the left DLPFCand medial prefrontal cortex (red), the left superior temporalregion (green), and the left posterior middle temporal cortex(blue). Table 2 gives the centers of these regions in stereotac-tic coordinates. The two dimensions used in the scaling trans-formation were very similar to the intentional and attentionaldimensions seen in the previous section. The first dimension(Fig. 4, top right) pointed toward the verbal fluency (first andlast conditions) and away from word shadowing (middle con-ditions). It was largely indifferent (orthogonal) to the seman-tic categorization conditions. The second dimension showedmonotonic time effects suggesting physiological adaptationdue to putative attentional changes.

The functional space for the normal subjects and theschizophrenic groups are shown in Figure 5. In the normalsubjects (top left), this set of tasks elicited a prefrontotem-poral axis. The axis is slightly oblique, suggesting some of thisfunctional connectivity is due to systematic time-dependenteffects. The similarity between this configuration and that ofsimilar regions in the previous section is evident. The equiv-alent spaces for the schizophrenic groups are markedly dif-ferent from the normal space. Although the DLPFC (red) hasretained its position, the temporal regions have moved acrossfrom the opposite side to occupy a domain that spans highpositive correlations with the DLPFC to total independence.The migration of the superior temporal regions is remarkablyconsistent across the three schizophrenic groups and is pre-dominantly in a right-left direction, suggesting this abnormal-ity is due to intentional aspects of the tasks employed. Con-versely, the posterior temporal regions are less consistent intheir displacement. The horizontal (intentional) shift is similarin all three groups, but the vertical or attentional componentis different for each of the three groups (the unimpairedgroup showed a pronounced movement of posterior tempo-ral regions in the attentional dimension). This suggests thatthe functional connectivity elicited by intentional aspects ofthe word generation tasks result in an abnormal pattern ofprefrontotemporal integration that is largely invariant over dif-ferent schizophrenic subgroups. However, the (dys)functionalconnectivity elicited by attentional components is specific tothe grdup in question.

Notice that in the poor group the distance between the

left DLPFC (red) and the superior temporal regions (green)suggests an absence of functional connectivity (positive ornegative). This represents a true left prefronto-superior tem-poral disintegration.

This is not the place to embark on a detailed analysis ofthese results in terms of the neuropsychology of schizophre-nia; however, it is worth pointing out that the observed re-versal and/or loss of prefrontotemporal integration is partic-ularly relevant given the signs and experiential symptoms ofschizophrenia (for a fuller discussion, see Frith, 1993; Fristonet al., 1996).

DiscussionWe have presented a simple application of metric multidi-mensional scaling that uses functional connectivity as the un-derlying metric. Functional connectivity is simply the corre-lation between remote neurophysiological events. The tech-nique provides an expedient transformation that maps ana-tomical space into a functional space. The topography of thisfunctional space is such that proximity implies a high degreeof functional connectivity. The nature of this mapping meansthat anatomically distributed systems that are functionallyconnected converge toward the same locus in functionalspace.

Potential applications of the technique have been dem-onstrated in the context of word generation in normal sub-jects and abnormal prefrontotemporal integration in schizo-phrenia. In particular, the negative correlations between pre-frontal and temporal activity normally seen are reversed inschizophrenia and the left superior temporal gyrus appearsto be dissociated from the prefrontal systems implicated inword generation.

The techniques described here are not new. Principal co-ordinates analysis or classical metric scaling was introducedin the 1960s, and other forms of multidimensional scalinghave been used in the context of neuroimaging (see Golden-berg, 1989; Goldenberg et al., 1989). What is new here is thatthe correlations used in the classical scaling are correlationsin neuroimaging time series. These correlations are a simplecharacterization of functional interactions and render thespace defined by the scaling technique meaningful in termsof functional connectivity. The second novel aspect of theproposed (voxel-based) application is that the transformationcan be thought of as being applied to continuous volumes (ifthe voxel data are a good lattice representation of a smoothcontinuous processes).

The Relationship between Eigenimages, Spatial Modes, andFunctional TopographyThere is a pleasing and complementary relationship betweenfunctional topography defined using the scaling transforma-tion arid the use of the eigenvector solution of the functionalconnectivity matrix to identify spatial modes (e.g., Friedrichet al., 1991; Friston et al., 1993a). In the latter approach, thedata are considered as K points in an w-dimensional space.These points define a trajectory in a space who dimensionsare voxels. The principal axes (eigenvectors) of the distribu-tion traced out by the trajectory correspond to the spatialmodes embedded within the data. An image of these eigen-vectors is called an eigenimage. Eigenimages, or spatial modesrepresent a simple and powerful way of mapping functioninto anatomical space.

In defining a functional space one considers the data as npoints in an iC-dimensional space. The principal axes (eigen-vectors) of this distribution are used to rotate the points toreveal the greatest structure in their interrelationships. A sub-space of the rotated points represents a mapping of anatomyinto a functional space.

Cerebral Cortex Mar/Apr 1996, V 6 N 2 161

Figure 4. The functional topography ofnormal subjects and schizophrenic pa-tients. Left, Anatomical regions detailedin Table 2. Right The first two eigenvec-tors of the distribution in the functionalspace showing that the first {top) vectoris associated with the difference be-tween the first and last conditions (in-trinsic word generations) and the middletwo conditions (word shadowing). Thesecond vector [beloW) corresponds to amonotonic time effect

-0.5

Anatomical regions

-0.5

Figure 5. The functional topography ofnormal subjects and schizophrenic pa-tients. Top left. Functional space of thenormal group demonstrating the markedprefrontotemporal axis that characteriz-es normal functional connectivity. Topright The equivalent space for the poorgroup of schizophrenic subjects in whichall the temporal regions have migratedfrom the left-hand pole to the bottomright quadrant This corresponds to aloss and reversal of normal negative pre-frontotemporal functional connectivity.Bottom left Functional space for the oddgroup. Bottom right Functional space forthe unimpaired group.

- 1 -0.5 0 05 1controls

-1 -0.5 0 0.5 1poverty

-1 -0.5 0 05 1 -1 -0.5 0 0.5 1unimpaired

162 Functional Topography • Friston et al.

Table 2Location of anatomical regions in Talairach and Tournous stereotaxic space

Name

LeftDLPFCBraca's areaMedial PFCSuperior temporal gymsPosterior middle temporal gyms

Location x,y,z(mm)

- 4 8 36 12- 5 8 16 24- 1 2 46 24- 5 6 - 8 4- 4 0 - 5 8 - 8

PutativeBrodmann'sarea

46449

2221

Color

redredredgreenblue

Regions chosen for the analysis of the six-condition word generation study of normal subjects andschizophrenic patients. All voxels that reached criteria following ANCOVA and fell within 20 mm of theabove location constituted a "region." See Figure 4 (left) for a graphical presentation.

As with eigenimages, the functional spaces created usingclassical scaling will change fundamentally with differentbrain states and are, as a consequence, experiment and timedependent.

There is a parallel between the present work, using func-tional connectivity and that of Young (1992) who used ameta-analysis of anatomical connectivity and nonmetric mul-tidimensional scaling. This analysis allowed the authors tocomment on the segregation of dorsal and ventral processingstreams and reconvergence in the DLPFC and the superiortemporal area. Although we have chosen to illustrate the tech-nique with an (important) example of abnormal functionaltopography in schizophrenia, there are clearly many applica-tions to normal functional anatomy. It would be interestingto examine the issues addressed by Young (1992) to providea complementary functional perspective. The technique ap-plied in this article uses metric multidimensional scaling asopposed to nonmetric scaling used by Young (1992). Therehave been some concerns expressed about the application ofnonmetric scaling to connectivity data (Simmen et al., 1994;Young et al., 1994). These concerns are avoided with metricscaling. In this sense, the current application of metric scalingcould prove very useful in resolving important questionsabout large scale connectivity and functional organization inthe brain.

At the present time, it is not easy to make statistical infer-ences about the topographic features or changes in these fea-tures obtained with multidimensional scaling (Chatfield andCollins, 1980); however, this does not detract from the pro-posed application as a powerful descriptive approach to neu-roimaging data.

NotesThis work was based on an original idea of Stephen Kosslyn. We areindebted to him for his inspirational comments and ensuing persis-tence. We also thank our colleagues at the MRC Cyclotron Unit, with-out whom these studies would not have been possible.

Address correspondence to Wellcome Department of CognitiveNeurology, Institute of Neurology, Queen Square, London WCIN 3BG,UK.

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