Learning and comparing functional connectomes across subjects
aa r X i v : . [ q - b i o . N C ] A p r Learning and comparing functional connectomes across subjects
Ga¨el Varoquaux a,b,c, ∗ , R. Cameron Craddock d,e a Parietal project-team, INRIA Saclay-ˆıle de France b INSERM, U992 c CEA/Neurospin bˆat 145, 91191 Gif-Sur-Yvette d Child Mind Institute, New York, New York e Nathan Kline Institute for Psychiatric Research, Orangeburg, New York
Abstract
Functional connectomes capture brain interactions via synchronized fluctuations in the functional magnetic resonanceimaging signal. If measured during rest, they map the intrinsic functional architecture of the brain. With task-drivenexperiments they represent integration mechanisms between specialized brain areas. Analyzing their variability acrosssubjects and conditions can reveal markers of brain pathologies and mechanisms underlying cognition. Methods ofestimating functional connectomes from the imaging signal have undergone rapid developments and the literature is fullof diverse strategies for comparing them. This review aims to clarify links across functional-connectivity methods as wellas to expose different steps to perform a group study of functional connectomes.
Keywords:
Functional connectivity, connectome, group study, effective connectivity, fMRI, resting-state
1. Introduction
Functional connectivity reveals the synchronization ofdistant neural systems via correlations in neurophysiolog-ical measures of brain activity [14, 37]. Given that high-level function emerges from the interaction of specializedunits [110], functional connectivity is an essential part ofthe description of brain function, that complements thelocalizationist picture emerging from the systematic map-ping of regions recruited in tasks [101]. However, whilethere exists a well-defined standard analysis frameworkfor activation mapping that enables statistically-controlledcomparisons across subjects [39], group-level analysis offunctional connectivity still face many open methodolog-ical challenges. Deriving a picture of a single subject’sfunctional connectivity is by itself not straightforward, asthe brain comprises a myriad of interacting subsystemsand its connectivity must be decomposed into simplifiedand synthetic representations. An important view of brainconnectivity is that of distributed functional networks de-picted by their spatial maps [31]. Another no less impor-tant and complementary view is that of connections link-ing localized functional modules depicted as a graph [17].This representation of brain connectivity is often calledthe functional connectome [102] and is the focus of intenseworldwide research efforts as it holds promises of new in-sights in cognition and pathologies [13, 30, 45].The purpose of this paper is to review methodologicalprogress in the estimation of functional connectomes from ∗ Corresponding author blood oxygenation level dependent (BOLD) based func-tional magnetic resonance imaging (fMRI) data and theircomparisons across individuals. It does not attempt to beexhaustive, as the field is wide and moving rapidly, but de-tails specific tools and guidelines that, in the experience ofthe authors, lead to controlled and powerful inter-subjectcomparisons. The paper is focused on functional connec-tomes in contrast to structural connectomes, as the infer-ence of functional connectivity requires important statisti-cal modeling considerations that are vastly different fromthe complications involved with estimating structural con-nectivity. While the notion of functional connectomics isoften associated with the study of resting state [13], themethods presented in this paper are also relevant for task-based studies. On the other hand, the paper has a focus onfMRI; although the core concepts presented can be appliedto magnetoencephalography (MEG) or electroencephalog-raphy (EEG) [103], additional specific problems such assource reconstruction must be considered [93].“Functional connectivity” is defined as a measure ofsynchronization in brain signals [35]. More generally, itis interesting as a window on underlying synchrony onneural processes [63]. By “functional connectome”, herewe specifically denote a graph representing functional in-teractions in the brain, where the term “graph” is takenin its mathematical sense: a set of nodes connected to-gether by edges . Graph nodes (brain regions) correspondto spatially-contiguous and functionally-coherent patchesof gray matter and edges describe long-range synchroniza-tions between nodes that are putatively subtended by largefiber pathways [68]. A graph can be weighted or not,and is completely equivalent to its adjacency matrix , a
Preprint submitted to Elsevier April 16, 2013 ymmetric matrix tabulating the connection weights be-tween each pair of nodes. Functional-connectivity graphsare used to represent evoked activity, as in task-responsestudies [72], as well as ongoing activity, present in the ab-sence of specific tasks or in the background during taskand often studied in so-called resting state experiments[83]. Another important notion that arises from the studyof distributed modes of brain function is that of specializedfunctional networks [31]. With our definition of the func-tional connectome, functional networks are not directlybuilding blocks of the connectome but appear as a conse-quence of the graphical structure [116, 117].The paper is organized as follows. First we discuss es-timation of functional connectomes. This part, akin to afirst-level analysis in standard activation mapping method-ology, is not in itself a group-level operation, but it is acritical step for inter-subject comparison. In a followingsection, we discuss several strategies for comparing con-nectomes across subjects. Finally we discuss the links be-tween the representation of brain connectivity as graphsof functional connectivity and more complex models, suchas effective-connectivity models.
2. Estimating functional connectomes
Here we discuss the inference of connectomes fromfunctional brain imaging data. We start with preprocess-ing considerations, followed by the choice of nodes i.e. re-gions, signal extraction, and the estimation of graphs.
In addition to standard preprocessing performedfor task-based analysis (slice-timing correction, realign-ment, spatial normalization, and possibly smoothing),connectivity-based analysis require additional denoising toseparate intrinsic activity from confounding signals. Thisprocess involves regressing time series capturing sourcesof structured noise from the fMRI data. Physiologicalnoise due to cardiac and respiration are two importantnoise signals [11, 12, 53, 67] that are difficult to controlfor and as a result are not commonly regressed out. In-stead the mean signal from white matter (WM) and cere-brospinal fluid (CSF) are used as surrogates to measurethese sources of noise as well as other scanner induced sig-nal fluctuations [31, 67]. More complex models account forspatial variation in noise by incorporating voxel-specific re-gressors of neighboring WM (ANATICOR [55]) or the topcomponents from a principal components analysis of high-variance signals (CompCor [7]). Head motion induced sig-nal fluctuations are accounted for by incorporating move-ment parameters [31, 41, 67]. The global mean time series In neuroimaging, the term network is sometimes used to denotea graph of brain function. To disambiguate the notion of segregatedspatial mode [31] from that of connectivity graphs, we will purposelyrestrict its usage in this paper. has been proposed as an additional noise regressor thatappears to improve the spatial specificity of connectivityresults [31, 32]. This practice has become controversialsince the global signal regression introduces negative corre-lations [19, 77, 90]. Removing these sources of nuisance inaddition to linear trends results in more contrasted corre-lation matrices that improve the delineation of functionalstructures (fig. 2).Filtering to remove high frequencies is often performed,based on the initial observation that fluctuations impli-cated in resting-state functional connectivity are predom-inately slower than 0.1 Hz [14, 23]. While high-pass andlow-pass filtering decrease the impact of some confounds,recent studies have shown that connectivity is presentacross the full spectrum of observed frequencies [99, 113].Regressing out a good choice of confound signals is morespecific than frequency filtering, and in our experiencegives more contrasted correlation matrices . In addition,the recent developments of very rapid acquisition proto-cols prevent aliasing of the physiological noise with theneural signal and give access to more specific noise con-founds than traditional low-TR sequences [16].It is important to keep in mind that the proposed cor-rection strategies are approximate and not definitive tech-niques. This has become particularly apparent for headmotion with reports that micromovements on the scale of ≤ . The choice of regions of interests (ROIs) that definethe nodes of the graphs can be very important both inthe estimation of connectomes and for group comparison[119]. Unsurprisingly, simulations have shown that ex-tracting signal from ROIs that did not match functionalunits would lead to erronous graph estimation [100]. Dif-ferent strategies to define suitable ROIs coexist. Whiledense parcellation approaches cover a large fraction of thebrain [1, 8, 25, 116, 119], this coverage can be traded off tofocus on some specific regions, in favor of increased func-tional specificity and thus better differentiation across net-works [28, 46, 114]. In addition, while ROIs are most oftendefined as a hard selection of voxels, it is also possible touse a soft definition, attributing weights as with proba-bilistic atlases, or spatial maps of functional networks ex-tracted from techniques such as independent componentanalysis (ICA) [57, 99].
Regions from atlases.
Atlases can be used to define full-brain parcellations. Popular choices are the AutomaticAnatomic Labeling (AAL) atlas [111], which benefits from Note that naive use of filtering can induce spurious correlations[26].
2n SPM toolbox, or the ubiquitous Talaraich-Tournoux at-las [107]. However, these atlases suffer from major short-comings; namely i) they were defined on a single subjectand thus do not reflect inter-subject variability, and ii) they focus on labeling large anatomical structures and donot match functional layout –for instance only two re-gions describe the medial part of the frontal lobe in theAAL atlas. Multi-subject probabilistic altases such asthe Harvard-Oxford atlas distributed with FSL [98] or thesulci-based structural atlas used in [116] mitigate the firstproblem, and the high number of regions defined usingsulci also somewhat circumvent the second problem (seefig. 1). Defining regions from the literature.
Regions can be de-fined from previous studies, informally or with system-atic meta-analysis. This strategy is used to define themain resting-state networks, such as the default mode net-work, but may also be useful to study connectivity in task-specific networks [14, 28, 47, 86]. The common practice isto place balls of a given radius, 5 or 10 mm, centered at thecoordinates of interest. Given that functional networks aretightly interleaved in some parts of the cortex, such as theparietal lobe, care must be taken not to define too manyregions that would overlap and lead to mixing of the signal.
FMRI-based function definition.
Defining regions directlyfrom the fMRI signal brings many benefits. First, it cancapture subject-specific functional information. Second,it adapts to the signal at hand and its limitations, suchas image distortions or vascular and movement artifactsthat are isolated in ICA-like approaches. Lastly, incorpo-rating functional information into regional definition willresult in more homogenous regions that better representconnectivity present at the voxel level than anatomically-defined atlases such as AAL or Harvard-Oxford [25]. Thesimplest approach to define task-specific regions is to useactivation maps derived from standard GLM-based anal-ysis in a task-driven study (see for instance [81]). Re-gions are extracted by thresholding the maps, or usingballs around the activation peaks. For resting-state stud-ies, unsupervised multivariate analysis techniques are nec-essary. Clustering approaches extract full-brain parcella-tions [9, 25, 109, 121], and have been shown to segmentwell-known functional structures from rest data. Alterna-tively, decomposition methods, such as ICA [6], can unmixlinear combinations of multiple effects and separate outpartially-overlapping spatial maps that capture functionalnetworks or confounding effects, as for instance with thepresence of vascular structure in functional networks. Athigh model order, ICA maps define a functional parcel-lation [57]. Extracting regions from these maps requiresadditional effort as they can display fragmented spatialfeatures and structured background noise, but incorporat-ing sparsity and spatial constraints in the decompositiontechniques leads to contrasted maps that outline many dif-ferent structures [117] (see fig. 1).
Figure 1: Different full-brain parcellations: the AAL atlas [111], theHarvard-Oxford atlas, the sulci atlas used in [116], regions extractedby Ncuts [25], the resting-state networks extracted in [97] by ICA,and in [115] by sparse dictionary learning.
Optimal number of regions.
Defining an optimal numberof regions to use for whole-brain connectivity analysisbears careful consideration. On one hand we desire a suf-ficiently large number of regions to guarantee that theyare functionally homogeneous regions and adequately rep-resent the connectivity information present in the data.On the other hand too many regions will render statisticalinference challenging, result in an explosion in computa-tional complexity, and interfere with the interpretability ofobserved connections. For functional parcellation, cross-validation methods can be employed to estimate an opti-mal number of regions based on homogeneity, the abilityto reproduce connectivity information present at the voxelscale, and the ability to obtain the same parcellations fromindependent data [15, 25]. In general these metrics do notresult in an obvious peak at a “best” number of regions,but instead offer a range over which the number of regionscan be chosen based on the needs of the analysis at hand.Finally, it is important to keep in mind that there is nouniversally better parcellation and associated number ofregions. From a practical standpoint, these choices willdepend on the task at hand, and more fundamentally, agood description of brain function should cover multiplescales. Given that it is not clear that an optimal parcel-lation can be identified from the sample size of a typicalstudy, randomized parcellation, as used in structural con-nectomes [124] or activation mapping [118], may also beconsidered.
The concept of functional connectivity has been calledelusive [51]: it has many mathematical instantiations al-3 o r s P CC L D M NRD M N M ed D M N F r on t D M NR P o s t T e m p RD L P F CR P a r R F r on t po l L P a r L D L P F C L F r on t po l L I PS R I PS L A n tI PS R A n tI PS M o t o r L A ud R A udL S T S R S T S L I n s R I n s C i ng VA CCD A CCR A I n s B a s a l B r o c a R P a r s O p S up F r on t S L T P J R T P J C e r ebLL O CR L O C V i s S t r i a t e O cc po s t D o r s P CC L D M NR D M N M ed D M N F r on t D M NR P o s t T e m p R D L P F CR P a r R F r on t po l L P a r L D L P F C L F r on t po l L I PS R I PS L A n t I PS R A n t I PS M o t o r L A ud R A udL S T S R S T S L I n s R I n s C i ng V A
CCD A CCR A I n s B a s a l B r o c a R P a r s O p S up F r on t S L T P J R T P J C e r ebL L O CR L O C V i s S t r i a t e O cc po s t Without regressing outCompCor, WM and CSF
Regressing outglobal signal
Figure 2: Correlation matrices of rest time-series extracted from the39 main regions of the Varoquaux 2011 [115] parcellation with differ-ent choices of confound regressors –
Left : regressing out CompCorsignals, as well as white matter and CSF average signals and move-ment parameters. The insert shows the connections restricted to afew major nodes. –
Upper right : regressing out only movement pa-rameters. –
Lower right : regressing out movement parameters andglobal signal mean. No frequency filtering was applied here. Whenno confounding brain signals are regressed, all regions are heavily cor-related. Regressing out common signal, in the form of well-identifiedconfounds or a global mean, teases out the structure. though in essence they all strive to extract simple statisticsfrom functional imaging in order to characterize synchronyand communication between large ensembles of neurons.Here we choose to focus on second order statistics thatcan be related to Gaussian models, the simplest of whichbeing the correlation matrix of the signals of the differentROIs.
Signal extraction.
Given a set of graph nodes, the nextstep is to extract a representative time series for each node.To study intrinsic activity, e.g. with rest data, signal ex-traction can be achieved by either averaging the fMRI timeseries across the voxels in a region, or by taking the firsteigenvariate from a principle components analysis of thetime series [40]. Comparisons of these methods has shownthat the eigenvariate method is more sensitive to functioninhomogeneity [25] and exhibits worse test-retest reliabil-ity than averaging time series [128]. In addition, improvedspecificity to BOLD signal can be enforced by using onlysignal in voxels near gray-matter tissues. For this pur-pose, we suggest summarizing the signal in an ROI bya mean of the different voxels weighted by the subject-specific gray matter probabilistic segmentation, as outputby e.g.
SPM’s segmentation tool [4] or FSL’s FAST pro-gram [126].Studying connectivity from evoked activity with task-driven studies requires disambiguating task-specific con-nectivity effects from intrinsic connectivity mediated byshared neuromodulatory/task inputs, anatomical path-ways, etc . In this regard, it can be beneficial to run aGLM-based first-level analysis, enforcing specificity of themeasure extracted to the task. With slow event-related D o r s P CC L D M NRD M N M ed D M N F r on t D M NR P o s t T e m p RD L P F CR P a r R F r on t po l L P a r L D L P F C L F r on t po l L I PS R I PS L A n tI PS R A n tI PS M o t o r L A ud R A udL S T S R S T S L I n s R I n s C i ng VA CCD A CCR A I n s B a s a l B r o c a R P a r s O p S up F r on t S L T P J R T P J C e r ebLL O CR L O C V i s S t r i a t e O cc po s t D o r s P CC L D M NR D M N M ed D M N F r on t D M NR P o s t T e m p R D L P F CR P a r R F r on t po l L P a r L D L P F C L F r on t po l L I PS R I PS L A n t I PS R A n t I PS M o t o r L A ud R A udL S T S R S T S L I n s R I n s C i ng V A
CCD A CCR A I n s B a s a l B r o c a R P a r s O p S up F r on t S L T P J R T P J C e r ebL L O CR L O C V i s S t r i a t e O cc po s t Without sparsity
GraphLasso estimate
Figure 3: Different inverse-covariance matrices estimates corre-sponding to fig. 2 –
Left : group-sparse estimate using the ℓ es-timator [116]. The insert shows the connections restricted to a fewmajor nodes. – Upper right : non-sparse estimate: inverse of thesample correlation matrix. –
Lower right : sparse estimate usingthe Graph Lasso [34]. designs, task-specific functional connectivity can be cap-tured in trial-to-trial fluctuations in the BOLD response,estimated using a GLM analysis with one regressor pertrial [47, 75, 86]. This approach, known as beta-series re-gression, has been adapted for rapid event-related designs,using multiple GLMs to optimize deconvolution of eachtrial [76].
Correlation and partial correlations.
Given ROIs definingthe nodes of the functional-connectome graph, one needsto estimate the corresponding edges connecting them.Functional connectivity between the ROIs can be mea-sured by computing the correlation matrix of the extractedsignals. An important and often neglected point is that thesample correlation matrix, i.e. the correlation matrix ob-tained by plugging the observed signal in the correlationmatrix formula, is not the population correlation matrix, i.e. the correlation matrix of the data-generating process.If the number of measurements was infinite, the two wouldcoincide, however if this number is not large compared tothe number of connections (that scales as the square of thenumber of ROIs), the sample correlation matrix is a poorestimate of the underlying population correlation matrix.In other words, the sample correlation matrix captures alot of sampling noise, intrinsic randomness that arises inthe estimation of correlations from short time series. Con-clusions drawn from the sample correlation matrix can eas-ily reflect this estimation error. Varoquaux et al. [116] andSmith et al. [100] have shown respectively on rest fMRI andon realistic simulations that a good choice of correlationmatrix estimator could recover the connectivity structure,where the sample correlation matrix would fail. In general,the choice of a better estimate depends on the settings andthe end goals [114, 117], however the Ledoit-Wolf shrink-age estimate [62] is a simple, computationally-efficient, and4 ithout regressing outCompCor, WM and CSF
Regressing outglobal signal
Figure 4: Inverse-covariance matrices for different choice of con-found regressors –
Left : regressing out only movement parameters–
Right : removal of the global mean, instead of the white matter,CSF, and CompCor time courses. parameter-less alternative that performs uniformly betterthan the sample correlation matrix [116, 117] and shouldalways be preferred.For the problem of recovering the functional-connectivity structure , i.e. finding which region is con-nected to which, sparse inverse covariance estimators havebeen found to be efficient [89, 100, 116]. The intuition forrelying on inverse covariance rather than correlation stemsfrom that fact that standard correlation (marginal correla-tion) between two variables a and b also capture the effectsof other variables: strong correlation of a and b with a thirdvariable c will induce a correlation between a and b . Onthe opposite, the inverse covariance matrix (also called precision matrix ) captures partial correlations, removingthe effect of other variables [71]. In the small sample limit,this removal is challenging from the statistical standpoint.This is why an assumption of sparsity, i.e. that only fewvariables need to be considered at a time, is importantto estimate a good inverse covariance. Various estimationstrategies exist for sparse inverse covariance, and have animpact on the resulting networks [116, 117]. The GraphLasso ( ℓ -penalized maximum-likelihood estimator) [34] isin general a good approach for structure recovery. In groupstudies, the ℓ estimator [50, 116] is useful to impose acommon sparsity structure across different subjects andachieve better recovery of this common structure. Simplyput, these approaches are necessary because estimationnoise creates a background structure (see fig. 3); however,unlike in a univariate situation, the parameters are not in-dependent, and the spurious background connections de-grade the estimation of the actual connections. The sparseestimators make a compromise between imposing simplermodels, i.e. with less connections, and providing a goodfit to the data. This compromise is set via a regulariza-tion parameter which controls the sparsity of the estimate.A good procedure to choose this parameter is via cross- Covariance and correlation matrices differ simply by the factthat a covariance matrix captures the amplitude of a signal, via itsvariance, while a correlation matrix is computed on standardized(zero mean, unit variance) signals. validation [116].
Network structure extracted.
The correlation matrices andinverse-covariance matrices that we extract contain a lot ofinformation on the functional structure of the brain. First,the correlation matrix (fig. 2) shows blocks of synchronizedregions that can be interpreted as large-scale functionalnetworks, such as the default mode network. Note that thesplit in networks is not straightforward. Different orderingof the nodes will reveal different networks. Indeed, becauseof the presence of hubs and interleaved networks, the pic-ture in terms of segregated networks is not sufficient to ex-plain full-brain connectivity [117]. Connectivity matrices,correlation matrices and inverse-covariance matrices, canbe represented as graphs: nodes connected by weightededges (inserts on fig. 2 and fig. 3). The inverse-covariancematrix, which captures partial correlations, appears thenas extracting a backbone or core of the graph. While suchstructure has been used as a way to summarize anatomi-cal brain connectivity graphs [49], here it has a clear-cutmeaning with regard to the BOLD signal: it gives the con-ditional independence structure between regions [117]. Inother words, regions a and b are not connected if the sig-nal that they have in common can be explained by a thirdregion c . In this light, the choice of nuisance regressorsto remove confounding common signal is less critical withpartial correlations than with correlations. Indeed, whilewith correlation matrices regressing out the global meanhas a drastic effect (fig. 2 upper right and lower right), oninverse covariance it only changes the resulting matricesvery slightly (fig. 4).There have been debates on whether to regress out cer-tain signals, such as the global mean, as it induces negativecorrelations [19, 32, 77], and these may seem surprising:one network appears as having opposite fluctuations toanother. However, correlation between two signals onlytakes its meaning with the definition of a baseline. A sim-ple picture to explain anti-correlations between two regionsis the presence of a third region, mediating the interac-tions. Using this third region as a baseline would amountto estimate partial correlations in the whole system. Us-ing inverse-covariance matrices or partial correlations tounderstand brain connectivity makes the interpretationin terms of interactions between brain regions easier andmore robust to the choice of confounds.
3. Comparing connectivity
We now turn to the problem of comparing functionalconnectivity across subjects or across conditions.
First, we focus on detecting where the connectivity ma-trices estimated in the previous section differ.5 ass-univariate approaches.
The most natural approachis to apply a linear model to each coefficient of the con-nectivity matrices [47, 64]. This approach is similar to thesecond-level analysis used in mass-univariate brain map-ping, and gives rise to many of the well-known techniquesused in such a context, such as the definition of a second-level design, with possibly the inclusion of confounding ef-fects, and statistical tests (T tests or F tests) on contrastvectors. Importantly, in order to work with Gaussian-distributed variables, it is necessary to apply a Fisher Ztransform to the correlations. Note that in these set-tings, the Ledoit-Wolf estimator [62] is often a good choiceto estimate the correlation matrix, as it is parameter-freeand gives good estimation performance without imposingany restrictions on the data. For hypothesis testing, cor-recting for multiple comparison can severely limit statis-tical power, as the number of tests performed scales asthe square of the number of regions used. Controlling forthe false discovery rate (FDR) mitigates this problem. Al-ternatively, as the assumptions underlying the Benjamini-Hochberg procedure [10] for the FDR can easily be broken,non-parametric permutations-based tests give reliable ap-proaches. In particular, the max-T procedure [42, 79] isinteresting to avoid the drastic Bonferroni correction whencontrolling for multiple comparison in family-wise errorrate. Accounting for distributed variability.
A specific challengeof connectivity analysis is that the connectivity strengthbetween different regions tends to covary. For instance,with resting-state data, functional networks comprisingmany nodes can appear as more or less connected acrosssubjects (see for instance fig. 5, showing variability in acontrol population at rest). In other words, non-specificvariability is distributed across the connectivity graph,and it is structured by the graph itself. This obser-vation brings the natural question of whether second-level analysis should be performed on correlation matri-ces, inverse-covariance matrices, or another parametriza-tion that would disentangle effects and give unstructured(white) residuals. While inverse-covariance matrices showless distributed fluctuations than correlation matrices,they capture a lot of background noise, as partial corre-lations are intrinsically harder to estimate. Preliminarywork [114] suggests performing statistical tests on residu-als of a parametrization intermediate between correlationmatrix and inverse covariance matrix, as it can decoupleeffects and noise.Taking a different stance on distributed variability, the“network-based statistics” approach [122] draws from thehypothesis that if, in a second-level analysis, an effect isdetected on a connection that lies in a network of stronglyconnected nodes, a large sub-network is likely to carry an See http://en.wikipedia.org/wiki/Fisher_transformation or[3] section 4.2.3 for mathematical arguments. -11 a . Correlation matrices -44 b . Z score on difference -11 c . Inverse-covariance matrices -44 d . Z score on inverse-covariance -44 e . Z score on residuals [114] Figure 5: Inter subject variability. Note that this is variability occur-ring in a healthy population at rest, in other words it is non specificvariability – a : single-subject correlation matrices for different sub-jects – b : Corresponding Z-score (effect / standard deviation) of thedifference between a subject and the remaining others – c : single-subject inverse-covariance matrices – d : Corresponding Z-score forthe inverse-covariance matrices – e : Corresponding Z-score for thesubject residuals, as defined in [114]. effect. Thus, they adapt cluster-level inference to connec-tivity analysis, in order to mitigate the curse of multiplecomparisons. Both the multiple comparison issue and the network-level distributed variability are a plague to edge-level com-parison of connectomes. A possible strategy to circumventthese difficulties is to perform comparisons and statisticaltesting at the level of the network, rather than the indi-vidual connection.
Network integration.
Marrelec et al. [69] introduce theuse of entropy and mutual information as a measure ofnetwork-level functional integration . Gaussian entropycan be seen as a simple metric to generalize correlation orvariance to multiple nodes (see [3] § § a , b and c . Their correlationstructure is captured by three correlation coefficients: ρ ab , ρ bc and ρ ac . Summarizing these by their mean, as mightseem natural, discards the relationship between the sig-nals, while using the integration metric, defined as theGaussian entropy, tells us how much two signals can be See [116] for simplified formulas for network integration and mu-tual information. ntegration: Integration:
Figure 6: Two different correlation matrices with the same averagecorrelation, but with very different integration values. Indeed, thematrix on the left was chosen to represent three signals a , b and c asdifferent from each other as possible, given ρ ab + ρ bc + ρ ac = .
35; itthus has a small integration value. On the opposite, for the matrixon the right, signal b can almost be fully recovered by combiningsignals a and c ; the matrix thus has a large integration value. combined to form the third (see fig. 6). Cross-entropy –ormutual information– [69] measures the amount of cross-talk between two systems in a similar way as Gaussianentropy is used to measure the integration of a brain sys-tem. The functional-connectivity structure, or its repre-sentation in the form of a correlation matrix, can thus becharacterized via the integration and cross-talk of someof its sub-systems. This approach gives a simplified rep-resentation with a small number of metrics that can becompared across subjects. Graph-topological metrics.
Functional connectivity graphshave been found to display specific topological prop-erties that are characteristic of small-world networks[1, 17, 91, 103]. These networks display excellent transportproperties: although they have a relatively small numberof connections, any two regions of the brain are well con-nected. Another interesting consequence of their specifictopology is the resilience it gives the system to attacks suchas resulting from brain lesions [1]. This overall structureof functional-connectivity graphs can be summarized bya few metrics, such as the average path length betweenany two nodes, the local clustering coefficients, or thenode degree centrality [87]. Given that pathologies with-out a localized focus, such as schizophrenia, are thoughtto have a global impact on brain connectivity [5, 65], thegraph-topological metrics are promising markers to per-form inter-subject comparison. Such an approach is ap-pealing as it is not subject to multiple comparison issues.However, it has been criticized as giving a fairly unspe-cific characterization of the brain and being fragile to noise[54]. Another caveat is that these properties are not spe-cific to brain function: correlation matrices display small-world properties such as local clustering by construction.Indeed, if two nodes are strongly correlated to a third,they are highly likely to be correlated to each other [123].This observation highlights the need for well defined null-hypothesis [88, 123], but also for controlled recovery ofbrain functional connectivity going beyond empirical cor-relation matrices, as discussed in the previous section. In the neuroscience world, these descriptions are grouped underthe terms of “graph-theoretical approaches”, however graph theoryis an entire division of mathematics and computer science that isconcerned with much more than topology of random graphs.
Predictive modeling is concerned with learning (or fit-ting) a model that is capable of predicting informationfrom unseen data [80]. In the context of connectomes, pre-dictive modeling can extract connectivity-based biomark-ers of disease diagnosis, prognosis, or other phenotypicoutcomes [24, 27]. The accuracy of a predictive modelprovides a measure of the amount of information presentin the connectome about the phenotypic measure beingevaluated [58, 59]. When combined with reproducibility,prediction accuracy provides a metric for evaluating ex-perimental trade-offs for data acquisition, preprocessing,and analysis [60, 106]. Multivariate predictive models areattractive in connectomics because they are sensitive todependencies between features and avoid the need to cor-rect for multiple comparisons since the significance of anentire pattern is evaluated using a single statistical test.Additionally, modern predictive modeling techniques drawfrom the statistical learning literature, which specificallyaddresses high dimensional datasets with few observations.Predictive modeling has been successfully applied to iden-tify connectome-based biomarkers of Alzheimer’s disease[104], depression [24, 125], schizophrenia [18, 94], autism[2], ADHD [127], aging [27], as well as to classify mentaloperations [85, 95]. The growing interest for applying pre-dictive modeling to connectivity analysis was highlightedby the ADHD200 Global Competition, in which the objectwas to identify a connectivity-based biomarker of ADHD[108]. Recent work has illustrated the utility of predictivemodeling for deriving connectivity models at the individ-ual level [22].Technically, predictive modeling is a supervised ma-chine learning problem where a target to be predicted– e.g. age, disease state, cognitive state– is available foreach observation of the data. In the context of comparingconnectomes, features used in the predictive model corre-spond to bivariate measures of connectivity [27, 85, 95],or any of the previously discussed graph summary metrics[18, 29]. The quality of a predictive model is determined byits prediction accuracy (or generalization ability) which ismeasured using one or more iterations of cross-validation.Cross-validation iteratively subdivides available data intoa subset used for training the classifier and a dataset forevaluating classifier performance [80]. The significance ofachieved prediction accuracy can be assessed using permu-tation tests [44]. Predictive modeling approaches typicallyrequire the specification of several parameters, which maybe chosen based on domain specific knowledge or require-ments [21], determined using an analytical approach [20],or optimized using a second-level cross validation proce-dure [33]. Several strategies exist for performing cross-validation and thecommonly used approach of using only a single observation for testing(leave-one-out cross-validation) results in highly variable estimatesof prediction accuracy [33]. Alternative approaches such as (5 or 10)-fold cross-validation, or 0 . + bootstrap should be preferred [33].
4. Beyond correlation, effective connectivity?
All the approaches that we have presented in this re-view are based on second-order statistics of the signal,in other words correlation analysis. Traditionally, theseare defined as functional connectivity , defined as “tempo-ral correlations between remote neurophysiological events”[35], and opposed to effective connectivity , i.e. “the in-fluence one neural system exerts over another” [35]. Toconclude this review, we would like to bridge the gap be-tween these concepts, which in our eyes should be seen asa continuum rather than an opposition (this opinion is alsoexpressed in [73]).A first step to move from purely descriptive statisticsto interaction models with functional connectivity analysisis to consider a correlation matrix as a Gaussian graphi-cal model, i.e. a well-defined probabilistic model that de- scribes observed correlations in terms of an independencestructure and conditional relations [61, 117]. In such set-tings, the inverse covariance graph or the partial correla-tions are a measure of influence from one node to another,albeit undirected. Inferring directionality in a Gaussianmodel is impossible. Linear structural equation models(SEMs) [74] rely on a similar model that consists in speci-fying a candidate directed graphical structure. This struc-ture constraints the covariance matrix of the signals andcan thus be tested on observed data. In fact some forms ofSEMs are known as “covariance structure models”. Thereis thus a strong formal link between correlation analysis inthe framework of graphical models and SEMs: the formeris undirected but fully exploratory, as it does not requirethe specification of candidate structure, while the latter isdirected but confirmatory. This link has been exploited tospecify candidate structures for SEMs using partial cor-relations [70]. More complex models, such as dynamicalcausal models (DCMs) [38] or Granger causality [43] re-quire additional hypotheses such as non-linear couplingsor time lags.Most importantly, more complex models can only beused to model interactions between a small number ofnodes. This is not only due to a computational difficulty,but also to fundamental roadblocks in statistics: the com-plexity of the model must match the richness of the data.While injecting prior information can help model estima-tion, the more informative this prior is, the more fragilethe inference becomes. The ongoing debate on the impactof hemodynamic lag on Granger-causality inference [96]is an example of such fragility. Note that although mostof the theory underpinning correlation analysis (Gaussiangraphical models) is based on a Gaussian assumption, thecore results are robust to violations of this assumption [84].It is tempting to favor more neurobiologically-inspiredmodels that give descriptions close to our knowledge of thebrain basic mechanisms, however, as George Box famouslysaid, “all models are wrong; some models are useful”. De-pending on the question and the data at hand, a trade-offshould be chosen between complex models based on a bio-physical description, and simple phenomenological modelssuch as correlation matrices. In particular to model inter-actions between a large number of regions, as in full-brainanalysis, and learn a large connectome , simple models areto be preferred. For more hypothesis-driven studies, suchas the analysis of the mechanisms underlying a specifictask, more complex models can be preferred, if rich datais available. Automatic choice of model is a difficult prob-lem, however, cross-validation (as used in [25, 105, 116])is a useful tool. The central principle of cross-validation isto test a model on different data than the data used to fitthe model. Models too complex for the data available willfit noise in the data, and thus generalize poorly. The mainbenefit of cross-validation is that it is a non-parametricmethod which does not rely strongly on modeling assump-8ions .
5. Conclusion
Horwitz el al. [52] claimed almost 20 years ago that“the crucial concept needed for network analysis is covari-ance”. In our eyes, this still holds today. Estimation func-tional connectomes relies largely on fitting covariance mod-els. Their comparison requires understanding how thesecovariances vary and finding metrics to capture this vari-ability. The additional secret ingredient may be using con-founds regressors in all statistical steps. A good choice ofa small number of relevant regions facilitates connectomecomparison. However, such a choice cannot yet be fullyfactored out via methods and must rely on neuroscientificexpertise.Methodological challenges to functional-connectome-based group studies arise from the dimensionality andthe variability of the connectome. With the currenttools, inter-subject comparison of connectomes compris-ing many nodes is limited by the difficulty of estimatinghigh-dimensional covariance matrices and the loss of sta-tistical power due to multiple comparisons. Better algo-rithms integrating powerful a priori information are re-quired to push the limits of covariance estimation. Bettercharacterization of inter-subject variability of connectomes[56] will help choosing parameterizations and invariants toavoid testing each edge for a difference, as this strategyinevitably leads to a needle in a haystack problem.Reviewing methodological options to learn and com-pare connectomes highlights that there is currently nounique solution, but a spectrum of related methods andanalytical strategies. More empirical results are requiredto guide the choices. However this diversity is probablyunavoidable: a diffuse disease like schizophrenia will notlead to the same connectome modifications as a focal le-sion. In statistical learning, “no free lunch” theorems [120]tell us that no strategy can perform uniformly better in allsituations. In practice, the key to a successful analysis isto understand well the assumptions and interpretation ofeach option, in order to match the method to the question.Similarly, the idealized notion of an unique functional con-nectome to describe connections in brain function is prob-ably an utopia, and various connectomes should be con-sidered in different settings, such as the study of varying This is to be contrasted to Bayesian model comparison, whichwill give well-controlled results only if the true generative model is inthe list of models compared. [36] argues that, based on the Neyman-Pearson lemma, cross-validation is less powerful than likelihood ratiotests using the full dataset. However, it is important to keep in mindthat these approaches only test for self-consistence, as the Neyman-Pearson lemma is established under the hypothesis that the modelused to define the test is indeed the data-generating process [78],while in practice it is often the case that this model gives poor fits tothe data [66]. Applying test procedures on different data than thatused to fit the model, as in cross-validation, is much more resilientto modeling errors. phenotypic conditions, or that of on-going activity versusactivity related to specific tasks.
Acknowledgments
GV acknowledges funding from the NiConnect grantand the Dynamic Diaschisis project DEQ20100318254from
Fondation pour la Recherche M´edicale , as well asmany insightful discussions with Andreas Kleinschmidt onon-going activity and Bertrand Thirion on statistical dataprocessing. RCC would like to acknowledge support bya NARSAD Young Investigator Grant from the Brain &Behavior Research Foundation. The authors would liketo thank the anonymous reviewers for their suggestions,which improved the manuscript.
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