neuroimaging analysis ii: magnetic resonance imaging neuroimaging... · and overlapping brain...

[16:52 26/2/2009 5283-Millsap-Ch29.tex] Job No: 5283 Millsap: The SAGE Handbook of Quantitative Methods in Psychology Page: 697 697–715

29Neuroimaging

Analysis II: MagneticResonance Imaging

Este la Camara, Josep Marco-Pal larés,Thomas F. Münte and Antoni Rodr íguez-Fornel ls

ANALYSIS OF MAGNETICRESONANCE IMAGING

Cognitive processes are widely distributedacross the whole brain, involving interactingand overlapping brain regions. Magneticresonance imaging (MRI) provides a non-invasive, in vivo, quantitative measurement ofpsycho-physiologically relevant parametersthat are related to cognitive operations in thenormal and abnormal brain. The combinationof sophisticated experimental designs andpowerful statistical analysis of MRI signalshas become a powerful tool for cognitiveneuroscience and psychology. Recently,the integration of different MRI techniqueshas been shown to be a promising researchavenue for cognitive neuroscience. Thepresent chapter tries to give an overview ofthe main statistical tools used in structural andfunctional MRI analysis. In addition, we willalso consider the analysis (preprocessing)and treatment of magnetic resonance images.

While the following sections are restrictedto MRI, certain points discussed below canalso be applied to other neuroimagingtechniques like positron emissiontomography (PET).

While the electroencephalography (EEG)section in Chapter 28 is focused on thetemporal properties of the signals, this sectionemphasises the spatial aspects. This is areflection of the two major approaches ofMRI in cognitive neuroscience: (1) structuralimaging is performed to obtain an accuratepicture of the morphological characteristicsof the studied brain; and (2) functional imag-ing provides information about the averagehemodynamic response in each part of thebrain (which is compartmentalized into manysmall volume units, ‘voxels’) when a subjectperforms a specific task. In both approaches,a detailed (structural or functional) spatialimage of the brain is obtained.As the temporalresolution of the hemodynamic response israther low, this has led to a strong bias for


698 SPECIALIZED METHODS

MR analysis to describe spatial aspects of theresponse.

STRUCTURAL MAGNETIC RESONANCEIMAGING

Based on the magnetic properties of dif-ferent tissues, high- resolution structuralMRI can generate three-dimensional imageswith detailed structural definition because ofits high sensitivity to soft-tissue contrasts.Additionally, diffusion tensor imaging (DTI)measures the diffusion of water in thebrain; at the microscopic level, neural tissuestructures have distinct boundaries, includ-ing axon membranes and myelin sheaths,which constrain the diffusional propagationof water molecules and force the latter incertain preferential directions. This allowsresearchers to characterize the micro-structureof the medium studied from differences indiffusional properties in various physiologicaland pathological states. Different statisticaltools have been developed in order to testfor specific regional structural changes atdifferent spatial scales.

Region-of-interest analysis

Traditionally, the simplest approach usedto compare local anatomical differences iscommonly referred to as region-of-interest(ROI) analysis, in which a defined regionis identified and statistical comparisons aremade relating its size or intensity value witha particular task under study. Obviously,the crucial point is to delimit the studiedregion. Then, a mean value of the extensionof this region related to a particular taskcan be extracted and compared betweengroups. For example, Figure 29.1 (left panel)shows a typical ROI analysis in which therelative anisotropy (RA) parameter (a micro-structural index that reflects the integrityof white matter fibers) correlates negativelywith age in the anterior part of the corpuscallosum (CC). ROI analyses are restrictedto the (few) regions selected for analysis,and these regions are usually derived from

a priori hypotheses. Possible bias might beintroduced due to manual or semi-automateddefinitions of the ROIs. The additional aver-aging over a brain region also reduces spatialresolution, and some biologically meaningfuldifferences that might be detected at thevoxel-level might be missed (Virta et al.,1999). Given these concerns, a voxel byvoxel comparison between groups of subjectsmight be an attractive method to investigatelocal changes.

Voxel-based morphometry

Voxel-based morphometry (VBM) permitsresearchers to make statistical inferences atthe voxel-level for the whole brain by esti-mating changes in local tissue concentrationsand volumes. This procedure is relativelystraightforward and can be broadly dividedinto three steps (Ashburner and Friston,2000): (1) normalization; (2) segmentation;and (3) smoothing. In addition, some studiesintroduce a modulation step. Figure 29.2(upper panel) summarizes the main stepsof this process. At this point, statisticalanalysis from the subsequent voxel-basedanalysis is directly comparable to the ROIapproach, since each voxel in the smoothedimage contains the average value from thesurrounding voxels (see Figure 29.2B). Thus,regions that differ significantly with respectto a particular effect are commonly revealedbased on the general linear model (GLM)framework. Figure 29.1 shows an examplein which RA values are correlated with agein the whole brain by applying a voxel-basedapproach.

Finally, as with structural diffusion data,it should be noted that some difficultiesmay arise in this procedure if the inten-tion is to examine tissue properties at avoxel level. In this case, identical brain co-ordinates have to be compared across thewhole study population, and even mesoscopicstructural differences are discarded. Thus, themain challenge facing voxel-based diffusionparameter analysis involves meeting therequirement for an optimal matching of thebrains being compared; this is usually quite


NEUROIMAGING ANALYSIS: MAGNETIC RESONANCE IMAGING 699

Figure 29.1 (A) Region of interest (ROI)-based analysis of the anterior and posterior corpuscallosum (CC). The scatter plots depict the relationship between relative anisotropy (RA) andage, and apparent diffusion coefficient (ADC) and age. (B) Normalized and averaged axial RAmap overlays from a sample of 54 healthy volunteers (range 19–71 years) withRA-index-related t-scores. Figures show positive (red) and negative (blue) correlations.

difficult to achieve. Indeed, traditionally,voxel-based analyses of diffusion data haveonly been treated as an exploratory tool.Despite the possible artefacts derived fromthese methodological constraints, however,the development of complementary tools forvoxel-based analysis in diffusion data setshas created very powerful approaches as well(Camara et al., 2007).

FUNCTIONAL MAGNETIC RESONANCEIMAGING

The capacity to map brain functions non-invasively in vivo with functional MRI hasbeen critical for the success of cognitiveneuroscience in the past decade. Blood oxy-genation level dependent (BOLD) contrast isthe main mechanism measured by functional

MRI (Ogawa et al., 1990).Activity dependentchanges in local deoxyhemoglobin levels aretheorized to result from changes in oxygenextraction, blood flow, and blood volumeregulation within the brain. All of theseparameters change during neural activity(Buxton et al., 1998). While the vascularchanges underlying neurovascular couplingare highly correlated with neural changes,the different time constants of the neuraland vascular (and by extension BOLD)phenomena need to be taken into account.Whereas neural activity changes within mil-liseconds, the hemodynamic response hasa long time constant and therefore lowtemporal resolution (see Appendix 1 for themathematical characterization of the hemody-namic response function). The hemodynamicresponse begins with an initial short dip of theBOLD signal and then shows a steep increase,



Figure 29.2 (A) Voxel-based morphometry main preprocessing steps. (B) Four differentGaussian filters are applied to a normalized gray matter segmented image. The effects ofusing different degrees of Gaussian smoothing appear as a blur of the image. FWHM,full-width half maximum.

with a maximum between 4–6 seconds afterthe onset of neural activity. The hemodynamicresponse to a given stimulus can last between20 and 30 seconds until complete return toa baseline level, but this pattern can varybetween regions and subjects (Aguirre et al.,1998; Figure 29.3).

The initial dip in the BOLD responseis thought to reflect a local increase inoxygen consumption that likely reflects anincrease in neural activity. Consequently, thiseffect should be spatially highly specific, butit is unfortunately quite inconsistent across

studies. Such a discrepancy might be due tothe fact that the amplitude of the initial dipis much smaller than the main BOLD signal.It is necessary to use high magnetic fields inorder to enhance the signal-to-noise ratio andobserve the effect (Yacoub et al., 2001). Thesubsequent increase of the BOLD responsehas been related to the adaptive behavior ofneural activity.

A major goal of functional MRI analysisis to mathematically estimate a hemodynamicresponse function that captures the BOLDsignal associated with task-related neural



Figure 29.3 Time-course reconstruction of the hemodynamic response of two differentregions of interest, the inferior frontal gyrus and the left middle frontal gyrus, evoked bydifferent task-related conditions (de Diego et al., 2006). Notice that different time courseresponses are observed between regions and conditions.

activity changes. Nevertheless, differencesbetween hemodynamic response patternsexist from one brain region to anotherand across subjects. The differences ariseprimarily because of variation in the vascula-ture, although non-linearities at the neuronallevel (e.g., the adaptive behavior of neuralactivity) can induce hemodynamic differencesas well (Logothetis et al., 1999; Logothetis,2003). Figure 29.3 shows different BOLDresponses evoked by different task-relatedconditions (de Diego et al., 2006). Noticethat different time course responses areobserved both between regions and betweenconditions.

Experimental designs

Block and event-related designs (Amaroand Barker, 2006) are the main types ofexperimental designs in functional MRI.

Block designsCarrying on the experimental tradition ofstimulation PET, the first studies using func-tional MRI employed block designs because

of their statistical power and simplicity.A series of trials of one condition are groupedtogether during a period of time (a block),usually about 30 seconds. Therefore, it ispossible to pinpoint brain activity that isrelated to cognitive processing differencesbetween experimental and control blocks. Keyaspects of a block design are the numberof conditions, the block length, and thespacing between blocks (Bandettini and Cox,2000). Introducing more conditions in a blockdesign leads to a decrease of the signal-to-noise ratio (SNR), because the durationof the experiment is limited. In a standardblock design, the number of experimentalconditions is therefore limited to three orfour. The block length depends more on thedemands of the experiment itself, given thatsome processes cannot be modulated overshort or long periods. In principle, the shapeand timing of the hemodynamic response isinsensitive to the change of the length of theblocks. Hence, long periods can be robustlymodulated, but one might encounter varia-tions in task performance or non-stationarityeffects.



Event-related designsBecause of the problems of block designsmentioned above, experimental designs thatuse random sequences of events are desirable.Moreover, some cognitive processes could notbe studied by block designs. For example, inaction monitoring tasks aiming at the delin-eation of error-related activity or in memorytasks trying to differentiate successfully andunsuccessfully remembered items, it is notpossible to predict when an error or memoryhit will be produced; hence, block-designs arenot feasible. Event-related designs generallyconsist of rapid trial sequences comprisedof different event types of brief durationpresented in a random order. The developmentof rapid data acquisition sequences hasbeen crucial, since the reconstruction of thetime course of the hemodynamic responseassociated with an individual event requiresfrequent sampling of the measured signal. Itis important to apply optimized paradigms inwhich it is possible to reduce the effectivesampling rate (Dale, 1999). By randomizingthe stimulus-onset-synchrony from one eventto the next, the time course of the BOLDresponse to a particular class of events can beextracted from the BOLD signal (Dale, 1999;Miezin et al., 2000). Nevertheless, the BOLDresponse associated with a specific event classin a rapid event-related design is so weak thatthe number of repetitions of each event hasto be sufficiently large to permit estimation ofthe response. The cost associated with thesedesigns is the increase in the duration ofthe experiment. Additionally, it is importantto note that every possible combination oftrial sequences should be presented to allowan optimal deconvolution. In sum, a trade-off between the number of conditions andthe number of trials per condition and anoptimal trial spacing are crucial factors inevent-related designs (Birn et al., 2002; Dale,1999).

Mixed-models designsThe mixed designs represent an integration ofclassical block and event-related designs. Inthe mixed design, control blocks are alternatedwith task blocks, during which trials are

presented with different intervals betweenthem. In block designs, the overall activityevoked during a task block is estimated,thereby confounding sustained and transientactivity. Event-related designs, on the otherhand, ignore sustained activity and revealonly activity that is transient in nature andassociated with a concrete event of interest.Indeed, one of the main advantages of mixeddesigns is the ability to dissociate sustainedtask-related processes and transient trial-related processes (Visscher et al., 2003).Mixed designs might therefore appear optimalfor many applications, but they requiresophisticated analysis to separate transientand sustained activities, particularly for casesin which a brain region shows transient andsustained activation.

General linear model

One of the simplest statistical approaches usedto infer differences between two conditionsis to compare their mean intensity value,typically by applying a simple t-test. Becauseof the variability between block transitionsand the low number of blocks per condition,a direct comparison between the respectivemeans is not feasible. Therefore, comparisonsbetween conditions are performed by averag-ing the sustained activity between conditionsand excluding the transitions between blocks.In these cases, a normal distribution of the datacan be assumed and the statistical significanceof the comparison is enhanced. The majorshortcoming of using only direct differencesbetween means is that neither the variabilityin the shape of the hemodynamic response norpossible temporal condition-related variationsare captured by steady-state averaging.

The most widely used mathematicalapproach is the GLM framework(Figure 29.4). It models each experimentalsession as an independent single timeseries decomposed into the sum of separatefactors (conditions) and additive noise. Thisapproach allows researchers to estimate andevaluate a predicted model whose parameterestimates are adjusted based on their best fitwith the experimental data.



Figure 29.4 Flowchart describing the main steps used in the study of functional time seriesembedded in the General lineal model.

Formally, time series of signal intensitiesat each voxel are modeled independently asa linear combination of explanatory variablesand Gaussian noise. In this way, the experi-mental data for each voxel (Yk ) is defined asa linear combination of weighted dissociablefactors (xi) plus an additive error term (ε):

Yk = xk1β1 + ... + xksβs + ... + xkSβS + εk

(1)

In a matrix representation:

Y = Xβ + ε (2)

Indeed, the K × S matrix X (where Kis the number of time points and S thenumber of explanatory factors) represents thedesign matrix, and its complexity dependson the experimental constraints. Given thatthe BOLD response is known, the shapeof the estimated response at each sampletime point for each condition is achieved byconvolving the onset times of every conditionwith the shape of the predicted hemodynamicresponse function (HRF). Therefore, effectsof interest in a block design are modeled

by convolving the canonical HRF with aconstant value in the blocked intervals andzeros in the rest of the experiment (theboxcar regressor function). In contrast, event-related designs are modeled by convolvingeach trial onset with the canonical HRF.In addition, parametrical modulations (i.e.,reaction time or the accuracy of a trial) orspecific regressors that reflect movement-correlated patterns can be included in thedesign matrix as a covariate of the modelfactors. It is also common to introduce otherknown sources of variability not relatedto the experiment (nuisance factors), suchas linear drifts of intensity as a result ofsmall deviations in the scanner magneticfield or physiological parameters like small-scale pulsations in the brain derived from theheartbeat or respiration, as additional modelfactors. The inclusion of nuisance factors inthe experimental design allows researchersto reduce the amount of residual varianceexplained by the error term and consequentlyincreases the significance of the contrasts.Nevertheless, it is important to bear in mindthat more degrees of freedom are availableif more factors are involved, and thereby the



statistical corrections applied become moredifficult.

Given the acquired functional volumes Yand defined design matrix X, it is possible toestimate the combination of β̂ values that bestfits the experimental data Y with the predictedmodel by using a least-squares approach:

β̂ = (X ′X)−1

X′Y (3)

β̂values adjust the weight of each factor modeland define the parameter estimates, indicatinghow much each factor contributes to theoverall signal. Alternative approaches have tobe taken into account when the inverted (X′X)does not exist.

Once the parameter estimates have beenextracted, researchers must evaluate whetherthese parameters can explain the experimentaldata. Squared residuals give a measure ofhow well the model fits the data based onthe estimation of its variance (σ̂ ). The GLMassumes that residuals are independent andnormally distributed after adjusting the model,but extensions of the generalized linear modelaccount for non-normally distributed errors.The residual variance can be assessed asthe residual sum-of-squares divided by thecorresponding degrees of freedom.

σ̂ 2 = ε̂t · ε̂

K − p(4)

p = rank(X) (5)

At this point, it is possible to quantify thesignificance of an effect of interest in a givenvoxel for a particular factor by comparingthe amplitude of the parameter estimateswith the distribution of the error measure.Thus, by applying a t-statistic, differences inthe means between one-dimensional contrastsand the null hypothesis ( β1 = 0) can be tested.Afterwards, the significance of the effect(P-value) can be computed by comparingthe t-value with a Student’s t-distribution asa function of the available K-p degrees offreedom.

At this point, it is important to notethat we have been treating each time series

as independent, but time series are tem-porally correlated. Therefore, scans are notindependent measures and their associatederror at a given scan is correlated with itstemporal neighbors. Accordingly, the numberof degrees of freedom available is lower thanthe number of scans considered. Differentapproaches have been developed in orderto deal with short serial correlations. Asa specific example, statistical parametricmapping (SPM) uses an autoregressive pluswhite noise model (AR(1) + wn) to track thetemporal covariance, making the assumptionthat the pattern of error correlations is thesame over all the voxels but that its amplitudediffers. Serial correlations are mainly causedby cardiac, respiratory, or vasomotor sources(Mitra et al., 1997).

Additionally, the combination of severalbeta parameters (a subset of the model)is of interest in some experiments. Here,inferences can be made based on an F-statisticapproach by assuming identically distributederrors. Thus, under the null hypothesis, nolinear combination of the effects accounts forsignificant variance.

Evidence for an effect is provided by sig-nificance thresholding. Classical thresholdingmethods are based on the rejection of the nullhypothesis, which states that the distributionis the same for both conditions apart fromdifferences due to random noise. However,testing for significant differences over thewhole brain involves a large number of simul-taneous statistical comparisons (i.e., one forevery voxel). In this sense, significance valuesneed to be lowered to account for possiblefalse positive effects. Several approaches havebeen described for adjusting the statisticalthreshold to multiple comparisons.

Multiple-comparison correction

The simplest and more conservative approachused in order to circumvent the problem of thelarge number of simultaneous statistical testsis the Bonferroni correction. Information fromnearby voxels tends to be correlated, how-ever, since typically data has been spatiallysmoothed and it is difficult to determine the



number of independent measures that exist.In this regard, this correction is too restrictiveand introduces the risk of increasing the falsenegative rate.

As a response to this problem of theBonferroni-style correction, random fieldtheory has been applied to functional data(Birn et al., 2002; Dale, 1999; Worsleyet al., 1992) in order to generate an accuratetheoretical significance threshold for spatiallysmoothed data. Smoothness can be assessedfrom the observed spatial correlation inthe image, but is usually known fromthe width of the smoothing kernel (full-width half maximum; FWHM) previouslyapplied. Based on this value, the numberof independent comparisons, called resells(r), can be approximated as the quotientbetween the total number of voxels thatbelong to the brain and the number of voxelscontained in the volume defined by FWHMspatial filter applied. Additionally, due to theintrinsic spatial correlation, clustered blobsare expected instead of individual significantvoxels. The Euler characteristic functionpredicts at high thresholds the number of blobsthat should be found by chance in a randomimage at a given statistical threshold.

EC = r · (4 · ln 2) · (2 · π )−32 · Zt · e− z2t

2

(6)

Thus, given a z-score (Zt), the Euler char-acteristic function returns the significancethreshold from which it is possible to concludethat any cluster that remains after thresholdinghas occurred by chance. This approach onlyaccounts for the number of clusters, however,and it does not consider the size of the blob.Usually, regional differences (or activatedvoxels) are not expected to be found in isolatedvoxels. Thus, it is less likely to find a setof contiguous voxels that are all active bychance than it is to find a single voxel active bychance. Cluster-size thresholding or Family-wise approaches face this issue.

A family-wise null hypothesis can be testedin several ways, such as by selecting statisticalvalues larger than the expected in the case ofa null distribution or considering significant

effects that survive at the cluster level afterthresholding for the spatial extension of thecluster. Typically, cluster-size thresholds forimaging data are set around 10–20 voxels.However, these values depend on manyparameters, such as the value of the thresholdselected or the number of voxels in theimaging data.

Given these concerns, multiple compar-isons approaches allow researchers to selectthe proper significance threshold for avoxel-based (functional) magnetic resonanceimaging (fMRI) analysis. While an increaseof sensibility is obtained applying suchcorrections, however, the localizing power isreduced since only clusters or sets of clusterscan become significant. Such approachesmight not be sensitive enough to detectbiologically meaningful differences, whichcould be otherwise detected at the voxel level.Additionally, when a particular hypothesisexists about an anatomical region involvedin a concrete process, the number of voxelstested is largely reduced and minimizes theneed for correction for multiple comparisons.ROI approaches have several advantageswhen a priori expectations exist for a specificregion.

Fixed-effect analysis versusrandom-effect analysis

Up to this point, we have focused on singlesubject analysis. The combination of dataacross subjects allows researchers to testexperimental hypotheses in a more traditionalway. In early studies of neuroimaging, afixed-effect approach was employed. Fixed-effects analysis assumes that experimentalmanipulations have the same effect in termsof functional activation across all subjectsaside from random noise. Based on thisassumption, data from different subjects isconcatenated and treated as if it came from asingle-subject analysis. Fixed-effect analysisenhances the sensitivity of the statisticalanalysis due to the averaging across subjects.Inter-subject variability is not taken intoaccount, however, restricting the analysisto the sample collected. Nevertheless, in



neuroimaging studies, we usually want tomake inferences at a more general levelabout the population from which the subjectsare drawn. Random-effect approaches dealwith inter-subject differences by consideringthat the experimental manipulation has adistribution of the effect of interest acrosssubjects. Random-effect analysis is imple-mented by a two-level model. At the firstlevel, standard statistical maps are estimatedfor each subject for the comparison of interest.Then, in a second level, the distribution ofindividual statistical patterns is tested forsignificance. Therefore, both within-subjectsand between-subjects variance sources arecontrolled. Random-effect analysis is the mostcommon approach implemented in functionalanalysis to make interferences about thepopulation from which the subjects are drawn.

We have been describing the establishedstandards in fMRI analysis, however, sev-eral limitations of this approach shouldbe considered. Classical approaches aremainly mass-univariate analyses, meaning theanalysis is constrained to the voxel-basedlevel. Such analyses basically result in astatic picture of the brain in action. Bycontrast, there is increasing evidence thatbrain functions are supported by integratedbrain networks co-operating flexibly duringtask execution. Voxel-based inferences mightnot be the most appropriate approach toassess such dynamic interactions, since thewhole brain information is not exploited.Multivariate approaches can be considered asan alternative. Pattern recognition methods,for instance, have been used to analyzefMRI data with the goal of decoding theinformation represented in the whole brainat a particular time (Haynes and Rees, 2006)by using sophisticated classifiers, such assupport vector machine (SVM) approaches orartificial neural networks among many others.Principal component analysis (PCA) or singlevalue decomposition (SVD) approaches havealso been incorporated in functional analysis,but their main power still remains in de noisingthe data before preprocessing.

Additionally, classical inferences aredefined in a standard parametric domain.

However, recently an increasing number ofstudies have applied non-parametric tests,such as permutation tests and Bayesianframeworks, in order to enhance thesensitivity of the analyses. Nevertheless,classical parametric models combined withbrain connectivity analyses still persist as themost common approaches used to study brainfunctional specialization and dynamics.

BRAIN CONNECTIVITY: EFFECTIVEAND FUNCTIONAL CONNECTIVITY

In many brain imaging experiments, multipleareas are found to be activated in a given task.Apressing question in cognitive neuroscience,therefore, is to determine how such areasact in concert as functional networks. Ifdifferent regions are involved in a network,they should have strongly correlated activitypatterns. Indeed, the inherently multivariatenature of functional MRI allows researchersto investigate how anatomically distant brainregions interact during a specific cognitivetask. The concepts of functional and effectiveconnectivity were introduced by Friston et al.(1993) to identify functional networks. Bothconcepts assume that temporal correlations inthe BOLD signal reflect synchronous neuralfiring in the interacting regions. In particular,functional connectivity measures the temporalcorrelation between spatially remote neuro-physiological events. In contrast, effectiveconnectivity is defined as the influence of oneneural system on another. Therefore, func-tional connectivity is operationally defined,whereas effective connectivity depends on aspecific model.

Functional connectivity

Correlational analysisFunctional connectivity basically assessesthe correlation between a small number ofpreselected regions or between voxels. Then,the correlation field can be introduced as amatrix of all pair-wise correlation coefficientsthat indicating those regions coactivated witha particular activation pattern. Notice that



voxels within one FWHM should not betaken into account, because high correlationsare introduced by the smoothing process.The autocorrelation matrix can also betransformed and thresholded applying a t-test.

Coherence analysisCoupling between neural components mightresult in phase locking of their BOLDresponse (i.e., increased coherence). Coher-ence provides a measure of the frequency-specific association between two time series,x and y, at a frequency λ (Sun et al., 2004).It is defined as the cross-spectrum of twotime series, fxy (λ), normalized by the powerspectrum fxx (λ) of each time series:

Cohxy(λ) =∣∣fxy(λ)

∣∣2

fxx(λ) · fyy(λ)(7)

where:

fxy(λ) =n∑

u=1

Covxy [u] · e−j·λ·u (8)

fxx(λ) =n∑

u=1

Covxx [u] · e−j·λ·u (9)

where the cross-covariance function is definedbelow for a stationary time series x and y as:

Covxy [u] = E{(x [t] − µx) · (y [t + u] − µy)}(10)

E{}denotes the expected value andµ the meantime series value.

Eigenimages-singular value decompositionSVD seeks to reduce the dimensions ofthe correlation structure to a small numberof weighted orthogonal modes (principalcomponents) (Worsley et al., 2005). SVDtransforms the original time-series (X), whereeach column represents a voxel and each rowa scan of mean-corrected data, into two setsof unitary orthogonal matrices (V, U) and a Sdiagonal matrix of decreasing singular values:

X = USV′ (11)

Columns of V define the spatial components,representing a distributed brain system that

can be displayed as an image (eigenimage).Columns of U represent temporal or scan-order patterns accounting for the time-dependent profiles associated with eacheigenimage (eigenvariate). Eigenvalues arethe singular values squared of the S matrix,estimating the relative amount of varianceaccounted for by each principal component.They permit a qualitative assessment of theimportance of each eigenimage/eigenvariate.Thus, the first eigenimage expresses thepattern over voxels that accounts for thegreatest variability across all the scans,whereas the first eigenvariate is the temporalpattern that reflects the greatest variabilityacross all voxels.

A comparison between correlation-basedand SVD approaches shows that thresholdingcorrelations directly yield those voxel pairsthat are highly correlated whereas SVDdetects connectivity patterns in a morequalitative way. Worsley et al. (2005) showedthat correlations are highly sensitive to detectfocal interactions in practice, whereas SVDare more prone to detecting connectionsbetween more extensive regions. Addition-ally, the main drawback of using SVDlies in selecting the appropriate numberof components that should correspond tothe number of networks under study in aparticular experiment. More components willlead to a more complete but more complexdescription of the system. Therefore, only fewprincipal components are typically selected,and those components that only contributeminimally to the explained variance areneglected.

Effective connectivity

The parameters introduced so far allow us toidentify those regions involved in a particularbrain system by demonstrating high correla-tions between them. The concept of effectiveconnectivity takes this idea a step further,because it seeks to describe and quantify theinfluence of one brain area upon another.In the following section, we will describethree approaches to capture effective connec-tivity: (1) psycho-physiological interactions



(PPI); (2) structural equation modeling; and(3) dynamic causal modeling (DCM).

Psycho-physiological interactionsPPIs can be understood as the modulation thatone cerebral region exerts over another in aspecific experimental context (Friston et al.,1997). In other words, if the activity of oneregion is regressed in terms of another, theslope of this regression reflects the influenceof the second area over the first one. Variationsin the slope of such regressions are related tochanges in experimental cognitive conditions.Thus, regressions are computed for everyvoxel separately for each condition, and theninferences are made between the differentexperimental conditions.

Structural equation modelingSEM is model-dependent, requiring the apriori specification of an aprioristic anatom-ical model that graphically defines thoseanatomical connections considered to befunctionally relevant (Goncalves and Hall,2003). The connectivity model states notonly which regions are connected to eachother but also the direction of the connection.Note that only a small number of regionscan be included in such a model; otherwise,computational problems arise. Thus, SEMestimates the connection strengths (pathcoefficients) that best predict the inter-regional covariances of the functional imag-ing data under the given model. Furtherinformation about SEM can be found inChapters 21 to 25.

Dynamic causal modelingDCM shifts the focus from regionally-specificactivations to inter-regional path-specificactivations using a dynamic deterministicnon-linear model. The basic idea is tomodel the brain as a dynamic system thatis subject to inputs and produces outputsin terms of parameters that represent thecoupling between unobserved brain states.Thus, inputs modulated by the experimentalconditions can induce neuronal responses inspecific anatomical regions, but they also

might change the effective connectivity byinfluencing the coupling between nodes.

The main idea is to submit the systemto different controlled experimental inputsthat directly generate variation in the outputs.By measuring the responses after perturbingthe system, free model parameters can beestimated. Effective connectivity can beexpressed following any non-linear functioncharacterizing the neurophysiological inputsrelated to a brain region relative to otherregions. DCM is also supported with a forwardmodel, which transforms the neural responsesto a measurable hemodynamic response (i.e.,the output). In general, DCM does notrestrict the number of connections that can bemodeled, and consequently a large number offree parameters have to be estimated. Severalconstraints have to be imposed (et al., theneural activity cannot diverge exponentiallyto infinite values). A Bayesian frameworkis an appropriate approach for tacklingsuch an analysis. A complete mathematicalexplanation of DCM can be obtained from(Friston et al., 2003)

Additional approaches for connectivitymapping include multidimensional scaling(Welchew et al., 2002) and hierarchicalclustering (Stanberry et al., 2003). These usedissimilarities rather than partial correlations,and thus they permit the use of complementarymultivariate techniques.

COMMON STATISTICAL PROCEDURESIN ELECTROENCEPHALOGRAPHY ANDFMRI: INDEPENDENT COMPONENTANALYSIS

One of the classical assumptions in the studyof the physiological correlates of behavior isthat the physiological responses (i.e., cerebralelectrical signals registered using EEG or thefMRI BOLD signal) are stationary with prop-erties that are constant over time. For example,the mean of all electrical signals time-lockedto a certain event will give an evoked potentialas a result, and this potential it is supposedto be constant over time. However, several



studies have shown that EEG (Blanco, Garcia,Quiroga, Romanelli, and Rosso, 1995) andfMRI signals (Turner and Twieg, 2005) shownon-stationary behavior. Although traditionalanalysis approaches of brain signals haveprovided relevant information about theimplementation of cognitive functions, otheranalytical techniques not based on stationarya priori are needed. One such technique thathas become increasingly popular over the pastfew years is independent component analysis(ICA). ICA had been developed to solvetechnical problems, such as the separationof multiple human voices in a cocktail partysetting recorded with several microphones. Itturns out that many other problems, includingthe denoising of images, face or speechrecognition, or extracting the componentsof brain activity related to an event, lendthemselves to treatment in a similar way.

Independent component analysisapplied to the study of brain signals

Brain signals recorded at a scalp-electrode(EEG) or from a voxel in the brain (fMRI)can be thought of as representing a mixture ofseveral signals coming from different sources(see Figure 29.5 and 29.6). Three sourceslocated at different places in the brain, eachwith its own temporal evolution, generate asignal at the scalp that is the sum of the threeattenuated signals. Decomposing the signalsthat generate the recorded response is nottrivial. Importantly, each source is associatedto a certain scalp map (see Figure 29.5C),and temporal evolution of each source occursindependently. The goal of ICA applied toEEG data is to find sources (or components)with independent temporal evolutions (andtheir associated scalp maps) based on the timecourse of the activity at the different scalpelectrodes (see Figure 29.5B and D). Thecomponents can then be examined using thestandard logic of ERP research (see Makeiget al., 2002) and applied to fMRI (Espositoet al., 2006).

Another application of ICA to EEG andfMRI signals involves separating the brainsignals related to the performance of a

given task from the signals related toother factors (e.g., noise and movements).Signals produced by movements usually com-prise lower frequencies than signals evokedby a visual simulus and are independent ofthe presentation of a stimulus. Given that thestatistical properties of these different signalsare different, ICA is able to separate brainsignals from other signals, such as ocularmovements, muscular artifacts, and electricalnoise. It can thus be used to denoise EEGdata. Moreover, ICA can also be useful indenoising fMRI data and studying BOLDactivity related to the events presented tothe subject (McKeown et al., 1998). Fora mathematical specification of ICA, seeAppendix 2.

Limitations and problems in the useof independent component analysis

The ICA approach is not without problemsthat should be carefully considered beforeapplying this method. The most importantproblems are:

1. There exists an ambiguity with regard to thesign of the maps (unmixing matrix). Given thatthe original data is recovered by multiplying theunmixing matrix by the activations, this may leadto ambiguous results.

2. Contrary to PCA, there is no equivalent to‘variance explained’ in ICA, and hence it isdifficult to establish an ‘order’ of the differentcomponents. In other words, the specification ofthe ‘component that explains the most of thevariance’ is delicate in the ICA case. To solvethis problem, it has been suggested to define theactivations x(t) as unit variance and suppose thatthe columns of the mixing matrix reflect power ofeach component in the space. Although this canhelp in the determination of the ‘importance’ ofeach component in the decomposition, however,the ambiguity in the determination of an orderbetween components still persists (James andHesse, 2005).

3. ICA can decompose the data maximally intoas many components as there are sensors.This number usually ranges from 19 to 256 inEEG situations, but it can be increased by afactor of two in MEG cases and 1000 in fMRIsituations. Thus, a method for determining the



Figure 29.5 Application of independent component analysis (ICA) to electroencephalography(EEG) data. Three independent sources in the brain (A) generate a scalp signal that is the sumof their respective contributions (B). The ICA procedure finds three independent components(D), each displaying a specific scalp distribution and a temporal evolution that coincides withthe original temporal evolution of the sources. In addition, the application of an inversesolution to the scalp maps of independent components permits the localization of the originalsources (B).

number of components in the solution wouldbe desirable. Although some methods haveproposed to solve this problem (i.e., PCA based,Hyvarinen, Karhunen, and Oja, 2001), however,an ideal solution for this problem has not yetbeen found. In general, the most common methodinvolves including a relatively high number ofcomponents and selecting those that either make

‘physiological’ sense or can be related to somekind of noise.

CONCLUSIONS

In the preceding pages, we have discussedsome of the most important analysis



Figure 29.6 Application of independent component analysis (ICA) in the denoising of fMRIdata. Two independent components account for head movements (left) and ocularmovements (right).

techniques in (f)MRI research. It isimportant to bear in mind that thesetechnical developments are not self-servingbut are necessary prerequisites for thesuccessful application of temporal andspatial neuroimaging techniques in cognitiveneuroscience. In fact, the progress of thisresearch field has resulted largely from therapid development of new analysis tools.Recent years have seen an increase in thespeed of such new developments owing tohuge investments in neuroimaging centers inmany countries throughout the world. Someof the more advanced analysis techniques(such as dynamic causal modeling and relatedmethods) have just begun to penetrate thefield and may significantly change the waycognitive neuroscientists think about thebrain in action.

Because of space limitations, we havenot been able to discuss all of the rel-evant methodological developments in theneuroimaging field. Because of the verydifferent kind of information that EEG-basedand MRI-based techniques deliver, it may

be desirable in some cases to combine thetwo techniques. This maybe achieved ratherinformally by running separate EEG and fMRIexperiments with similar or identical stimulusmaterial in different groups of participants(for example, see the complementary resultsobtained in Bahlmann et al., 2007; Matzkeet al., 2002) or very stringently by recordingEEG and fMRI signals simultaneously in onesession. Recording of EEG signals within theMRI scanner has become possible becauseof sophisticated methods that denoise EEGfrom the gradient artifacts introduced by thescanning procedure (see Laufs et al., 2007 fora review). Such simultaneous measurementscan provide important constraints for sourcemodeling approaches by seeding dipolesources in those regions activated in the fMRIexperiment. It has to be pointed out, however,that such an approach is not without dangers,since one might be inclined to ignore the verydifferent nature of the two signals.

As combined hybrid PET/MRI devices arenow available, other important developmentswill likely result from combining PET and



MRI measurements in the same session. Suchdevices will allow researchers to investi-gate task-related molecular (e.g., receptoravailability/binding) and blood flow changessimultaneously, thus providing an importantlink to the knowledge accruing in molecularneuroscience. All of these methodologicaldevelopments are very exciting. It is importantto bear in mind, however, that sound experi-mental designs based on sophisticated modelsof the cognitive processes under study willremain at the heart of cognitive neuroscience.

NOTE

The figures in this chapter can be found at:http://cid-ea002f4aa27227e8.skydrive.live.com/browse.aspx/Estela%20Book. Black andwhite versions (select only the six figures) areavailable at: http://cid-ea002f4aa27227e8.skydrive.live.com/self.aspx/Book|_black|_white

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APPENDIX 1

Characterization of thehemodynamic response function

Great effort has been placed on characterizingthe hemodynamic response function (HRF).Several approaches have been developed toachieve this goal. The most general approachis the finite impulse response (FIR) model,in which it is possible to capture any shapeof response up to specified time scale from alinear combination of specific basis functions.In this model, the activation of a particularvoxel at time t is defined as the weightedsum of the stimuli (si) at the preceding (n)points (Goutte, Nielsen, and Hansen, 2000).Formally:

yt =n∑

i=1

ai · st−(i−1) + ao + ε (A1.1)

where yt corresponds to the intensity value attime t, ε follows a Gaussian noise distribution,and a0 is a constant parameter. Generally,a least squares approach is used to estimatethose parameters (ai) that minimize theestimated function with the observed data.However, the signal to noise ratio provided bythe hemodynamic response is quite low, andmany parameters have to be modeled. Indeed,the most frequent choice is the canonical HRF.This approach achieves a reasonable goodfit with the impulse response function butreduces the degrees of freedom in the modeland consequently allows powerful statisticaltests. The canonical HRF (or double gammafunction) is defined from the difference of twogamma probability density functions:

H(t) = f (t; 6, 1) − 1

6f (t; 16, 1) (A1.2)

where:

f (t; α, β) = tα−1e−t/β

βα�(α)(A1.3)

The first term models the peak of theHRF, whereas the second is responsible forthe post-stimulus undershoot. Nevertheless,



differences between hemodynamic responsepatterns exist from one brain region to anotherand across subjects. These differences aremostly due to variation in the vasculature,although non-linearities at the neuronal level(e.g., the adaptive behavior of neural activity)can induce hemodynamic differences as well(Logothetis, 2003). This variability can beaccommodated by expanding the HRF interms of temporal basis functions. The useof multiple basis functions provides morevariability in the shape of the hemodynamicresponse, such as differences between thelatency of the peak or in the peak delay, thanthe canonical HRF. However, it is importantto comment that the BOLD response dependsdirectly on the hemodynamic properties ofthe surrounding vasculature. Therefore, alter-ations in vascular dynamics might influenceour ability to attribute BOLD signal changesto alterations in neural activity. In particular,since changes in vasculature are relatedto aging, it is important to be carefulwhen interpreting BOLD changes in elderlypopulations or when vasculature alterationscould be possible (D’Esposito, Deouell, andGazzaley, 2003).

APPENDIX 2

Mathematical bases of theindependent component analysis

The initial point of ICA involves supposingthat signal s(t) is a mixture of statisticallyindependent signals x(t) with the relation:

s = Ax (A2.1)

The goal of ICA is to find the inverse relation:

x = Ws (A2.2)

where W is the so-called unmixing matrix.We can suppose that our signal is com-posed of several elements: increase/decreaseof cerebral activity related to behavioralprocesses of the experiment; noise that canbe random, due to non-cerebral signals as

movements, muscular, or cardiac activity,or due to electrical or magnetic activity;slow processes related to the experiment. Allof these elements will present a temporalevolution xj(t) and behave independently oneeach other, i.e.:

f (x1. . .xm) = f1(x1). . .fm(xm) (A2.3)

where f () is the joint density of signalsand fj() is the marginal density of xj. Inaddition, all these processes can changein time (i.e., evoked cerebral activity bydifferent stimuli can decrease with time dueto habituation), given the above mentionednon-stationary quality of the signal. Evenwhen the electrical activity changes on time,however, we suppose that its sources will notchange. Hence, certain activity xj(t) will beassociated with a fixed map (e.g., a scalppotential distribution in EEG or 3D activitymap in fMRI) that will be represented inmatrix W.,j.

It is important to note that the conceptof ‘statistical independence’ is much morerestrictive than other concepts used in high-order statistics. For example, two variables x1and x2 are not correlated if:

E{x1x2} = E{x1}E{x2} (A2.4)

Non-correlation is a weak form of indepen-dence given that two sources statisticallyindependent are also non-correlated. Theopposite argument is only true when bothdistributions are Gaussian.

The application of ICA to study EEGor fMRI is performed under the followingassumptions:

1. Component maps (activity maps for fMRI orprojection of sources in the scalp in EEG) mustbe constant in time but not in their temporalevolution

2. Temporal activation of sources must be statisti-cally independent

3. The statistical distribution of activations is notGaussian. This condition has to be appliedbecause ICA uses higher order measures thanother methods traditionally used in the analysisof cerebral signals (e.g., Principal Component



Analysis, PCA, and Factor Analysis, FA) that arebased on second order measures (e.g., searchingmaximum variance in PCA),. The former methodsneed x (t) distributions to be Gaussian, whereasx (t) in ICA need to be sub- or super-Gaussian.

In addition, we have to assume a fourthcondition in the EEG case:

4. The signal conduction times are equal (instanta-neous in practice), and the sum of the sources inthe electrodes is linear. This condition is followedin the EEG situation, since it is reasonable to applythe quasistatic approach to frequencies involvedin brain electrical activity (<1 kHz).

The application of ICA to EEG data hasbeen demonstrated to be very useful in

denoising data and studying the componentsinvolved in the generation of an ERP (Marco-Pallares, Grau, and Ruffini, 2005). On theother hand, the use of ICA in the studyof fMRI data is more recent than theirapplication to cerebral electrical activity.However, some studies have demonstratedthat ICA can be useful in denoising aswell as studying bold activity related tothe events presented to the subject. In thissense, McKeown et al., (1998) demonstratedthat the ICA applied to fMRI data revealedseveral statistical independent components,such as activations related to the eventsor blocks presented, slow activations, andactivity related to fast and slow movements(Figure 29.6).

neuroimaging analysis ii: magnetic resonance imaging neuroimaging... · and overlapping brain...

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