{"title": "Infinite Relational Modeling of Functional Connectivity in Resting State fMRI", "book": "Advances in Neural Information Processing Systems", "page_first": 1750, "page_last": 1758, "abstract": "Functional magnetic resonance imaging (fMRI) can be applied to study the functional connectivity of the neural elements which form complex network at a whole brain level. Most analyses of functional resting state networks (RSN) have been based on the analysis of correlation between the temporal dynamics of various regions of the brain. While these models can identify coherently behaving groups in terms of correlation they give little insight into how these groups interact. In this paper we take a different view on the analysis of functional resting state networks. Starting from the definition of resting state as functional coherent groups we search for functional units of the brain that communicate with other parts of the brain in a coherent manner as measured by mutual information. We use the infinite relational model (IRM) to quantify functional coherent groups of resting state networks and demonstrate how the extracted component interactions can be used to discriminate between functional resting state activity in multiple sclerosis and normal subjects.", "full_text": "In\ufb01nite Relational Modeling of Functional\n\nConnectivity in Resting State fMRI\n\nMorten M\u00f8rup\n\nSection for Cognitive Systems\n\nDTU Informatics\n\nTechnical University of Denmark\n\nmm@imm.dtu.dk\n\nKristoffer Hougaard Madsen\n\nDanish Research Centre for Magnetic Resonance\n\nCopenhagen University Hospital Hvidovre\n\nkhm@drcmr.dk\n\nAnne Marie Dogonowski\n\nDanish Research Centre for Magnetic Resonance\n\nCopenhagen University Hospital Hvidovre\n\nannemd@drcmr.dk\n\nHartwig Siebner\n\nDanish Research Centre for Magnetic Resonance\n\nCopenhagen University Hospital Hvidovre\n\nhartwig.siebner@drcmr.dk\n\nLars Kai Hansen\n\nSection for Cognitive Systems\n\nDTU Informatics\n\nTechnical University of Denmark\n\nlkh@imm.dtu.dk\n\nAbstract\n\nFunctional magnetic resonance imaging (fMRI) can be applied to study the func-\ntional connectivity of the neural elements which form complex network at a whole\nbrain level. Most analyses of functional resting state networks (RSN) have been\nbased on the analysis of correlation between the temporal dynamics of various\nregions of the brain. While these models can identify coherently behaving groups\nin terms of correlation they give little insight into how these groups interact. In\nthis paper we take a different view on the analysis of functional resting state net-\nworks. Starting from the de\ufb01nition of resting state as functional coherent groups\nwe search for functional units of the brain that communicate with other parts of\nthe brain in a coherent manner as measured by mutual information. We use the\nin\ufb01nite relational model (IRM) to quantify functional coherent groups of resting\nstate networks and demonstrate how the extracted component interactions can be\nused to discriminate between functional resting state activity in multiple sclerosis\nand normal subjects.\n\n1\n\nIntroduction\n\nNeuronal elements of the brain constitute an intriguing complex network [4]. Functional magnetic\nresonance imaging (fMRI) can be applied to study the functional connectivity of the neural elements\nwhich form this complex network at a whole brain level. It has been suggested that \ufb02uctuations in\nthe blood oxygenation level-dependent (BOLD) signal during rest re\ufb02ecting the neuronal baseline\nactivity of the brain correspond to functionally relevant networks [9, 3, 19].\nMost analysis of functional resting state networks (RSN) have been based on the analysis of corre-\nlation between the temporal dynamics of various regions of the brain either assessed by how well\nvoxels correlate with the signal from prede\ufb01ned regions (so-called) seeds [3, 24] or through un-\nsupervised multivariate approaches such as independent component analysis (ICA) [10, 9]. While\n\n1\n\n\fFigure 1: The proposed framework. All pairwise mutual information (MI) are calculated between\nthe 2x2x2 group of voxels for each subjects resting state fMRI activity. The graph of pairwise\nmutual information is thresholded such that the top 100,000 un-directed links are kept. The graphs\nare analyzed by the in\ufb01nite relational model (IRM) assuming the functional units Z are the same\nfor all subjects but their interactions \u03c1(n) are individual. We will use these extracted interactions to\ncharacterize the individuals.\n\nthese models identify coherently behaving groups in terms of correlation they give limited insight\ninto how these groups interact. Furthermore, while correlation is optimal for extracting second order\nstatistics it easily fails in establishing higher order interactions between regions of the brain [22, 7].\nIn this paper we take a different view on the analysis of functional resting state networks. Starting\nfrom the de\ufb01nition of resting state as functional coherent groups we search for functional units of the\nbrain that communicate with other parts of the brain in a coherent manner. Consequently, what de-\n\ufb01ne functional units are the way in which they interact with the remaining parts of the network. We\nwill consider functional connectivity between regions as measured by mutual information. Mutual\ninformation (MI) is well rooted in information theory and given enough data MI can detect func-\ntional relations between regions regardless of the order of the interaction [22, 7]. Thereby, resting\nstate fMRI can be represented as a mutual information graph of pairwise relations between voxels\nconstituting a complex network. Numerous studies have analyzed these graphs borrowing on ideas\nfrom the study of complex networks [4]. Here common procedures have been to extract various\nsummary statistics of the networks and compare them to those of random networks and these anal-\nyses have demonstrated that fMRI derived graphs behave far from random [11, 1, 4]. In this paper\nwe propose to use relational modeling [17, 16, 27] in order to quantify functional coherent groups\nof resting state networks. In particular, we investigate how this line of modeling can be used to\ndiscriminate patients with multiple sclerosis from healthy individuals.\nMultiple Sclerosis (MS) is an in\ufb02ammatory disease resulting in widespread demyelinization of the\nsubcortical and spinal white matter. Focal axonal demyelinization and secondary axonal degenera-\ntion results in variable delays or even in disruption of signal transmission along cortico-cortical and\ncortico-subcortical connections [21, 26]. In addition to the characteristic macroscopic white-matter\nlesions seen on structural magnetic resonance imaging (MRI), pathology- and advanced MRI-studies\nhave shown demyelinated lesions in cortical gray-matter as well as in white-matter that appear nor-\nmal on structural MRI [18, 12]. These \ufb01ndings show that demyelination is disseminated throughout\nthe brain affecting brain functional connectivity. Structural MRI gives information about the extent\nof white-matter lesions, but provides no information on the impact on functional brain connectiv-\nity. Given the widespread demyelinization in the brain (i.e., affecting the brain\u2019s anatomical and\nfunctional \u2019wiring\u2019) MS represents a disease state which is particular suited for relational modeling.\nHere, relational modeling is able to provide a global view of the communication in the functional\nnetwork between the extracted functional units. Furthermore, the method facilitates the examina-\ntion of all brain networks simultaneously in a completely data driven manner. An illustration of the\nproposed analysis is given in \ufb01gure 1.\n\n2\n\n\f2 Methods\n\nData: 42 clinically stable patients with relapsing-remitting (RR) and secondary progressive multiple\nsclerosis (27 RR; 22 females; mean age: 43.5 years; range 25-64 years) and 30 healthy individuals\n(15 females; mean age: 42.6 years; range 22-69 years) participated in this cross-sectional study.\nPatients were neurologically examined and assigned a score according to the EDSS which ranged\nfrom 0 to 7 (median EDSS: 4.25; mean disease duration: 14.3 years; range 3-43 years). rs-fMRI\nwas performed with the subjects being at rest and having their eyes closed (3 Tesla Magnetom Trio,\nSiemens, Erlangen, Germany). We used a gradient echo T2*-weighted echo planar imaging se-\nquence with whole-brain coverage (repetition time: 2490 ms; 3 mm isotropic voxels). The rs-fMRI\nsession lasted 20 min (482 brain volumes). During the scan session the cardiac and respiratory cy-\ncles were monitored using a pulse oximeter and a pneumatic belt.\nPreprocessing: After exclusion of 2 pre-saturation volumes each remaining volume was realigned\nto the mean volume using a rigid body transformation. The realigned images were then normalized\nto the MNI template. In order to remove nuisance effects related to residual movement or physio-\nlogical effects a linear \ufb01lter comprised of 24 motion related and a total of 60 physiological effects\n(cardiac, respiratory and respiration volume over time) was constructed [14]. After \ufb01ltering, the\nvoxel were masked [23] and divided into 5039 voxel groups consisting of 2 \u00d7 2 \u00d7 2 voxels for the\nestimation of pairwise MI.\n\n2.1 Mutual Information Graphs\n\nI(i, j) = (cid:80)\n\nThe mutual information between voxel groups i and j is given by\n\nuv Pij(u, v) log Pij (u,v)\n\nPi(u)Pj (u). Thus, the mutual information hinges on the estimation of\nthe joint density Pij(u, v). Several approaches exists for the estimation of mutual information [25]\nranging from parametric to non-parametric methods such as nearest neighbor density estimators [7]\nand histogram methods. The accuracy of both approaches relies on the number of observations\npresent. We used the histogram approach. We used equiprobable rather than equidistant bins [25]\nbased on 10 percentiles derived from the individual distribution of each voxel group, i.e. Pi(u) =\n10. Pij(u, v) counts the number of co-occurrences of observations from voxels in voxel\nPj(v) = 1\ngroup i that are at bin u while the corresponding voxels from group j are at bin v at time t. As such,\nwe had a total of 8\u00b7 480 = 3840 samples to populate the 100 bins in the joint histogram. To generate\nthe mutual information graphs for each subject a total of 72\u00b75039\u00b7(5039\u22121)/2 \u2248 1 billion pairwise\nMI were evaluated. We thresholded each graph keeping the top 100, 000 pairwise MI as links in the\ngraph. As such, each graph had size 5039\u00d7 5039 with a total of 200,000 directed links (i.e. 100, 000\nundirected link) which resulted in each graph having link density\n5039\u00b7(5039\u22121)/2 = 0.0079 while\nthe total number of links was 72 \u00b7 100, 000 = 7.2 million links (when counting links only in the one\ndirection).\n\n100,000\n\n2.2\n\nIn\ufb01nite Relational Modeling (IRM)\n\nThe importance of modeling brain connectivity and interactions is widely recognized in the litera-\nture on fMRI [13, 28, 20]. Approaches such as dynamic causal modeling [13], structural equation\nmodels [20] and dynamic Bayes nets [28] are normally limited to analysis of a few interactions be-\ntween known brain regions or prede\ufb01ned regions of interest. The bene\ufb01ts of the current relational\nmodeling approach are that regions are de\ufb01ned in a completely data driven manner while the method\nestablishes interaction at a low computational complexity admitting the analysis of large scale brain\nnetworks. Functional connectivity graphs have previously been considered in [6] for the discrim-\nination of schizophrenia. In [24] resting state networks were de\ufb01ned based on normalized graph\ncuts in order to derive functional units. While normalized cuts are well suited for the separation of\nvoxels into groups of disconnected components the method lacks the ability to consider coherent\ninteraction between groups. In [17] the stochastic block model also denoted the relational model\n(RM) was proposed for the identi\ufb01cation of coherent groups of nodes in complex networks. Here,\neach node i belongs to a class zir where ir denote the ith row of a clustering assignment matrix Z,\nand the probability, \u03c0ij, of a link between node i and j is determined by the class assignments zir\njr. Here, \u03c1k(cid:96) \u2208 [0, 1] denotes the probability of generating a link between\nand zjr as \u03c0ij = zir \u03c1z(cid:62)\na node in class k and a node in class (cid:96). Using the Dirichlet process (DP), [16, 27] propose a non-\nparametric generalization of the model with a potentially in\ufb01nite number of classes, i.e. the in\ufb01nite\n\n3\n\n\frelational model (IRM). Inference in IRM jointly determines the number of latent classes as well as\nclass assignments and class link probabilities. To our knowledge this is the \ufb01rst attempt to explore\nthe IRM model for fMRI data.\nFollowing [16] we have the following generative model for the in\ufb01nite relational model\n\nZ|\u03b1 \u223c DP(\u03b1)\n\n\u03c1(n)(a, b)|\u03b2+(a, b), \u03b2\u2212(a, b) \u223c Beta(\u03b2+(a, b), \u03b2\u2212(a, b))\n\nA(n)(i, j)|Z, \u03c1(n) \u223c Bernoulli(zir \u03c1(n)z(cid:62)\n\n)\n\njr\n\nAs such an entitys tendency to participate in relations is determined solely by its cluster assignment\nin Z. Since the prior on the elements of \u03c1 is conjugate the resulting integral\n\nP (A(n)|Z, \u03b2+, \u03b2\u2212) = (cid:82) P (A(n)|\u03c1(n), Z)P (\u03c1(n)|\u03b2+, \u03b2\u2212)d\u03c1(n) has an analytical solution such\n\nthat\n\nP (A(n)|Z, \u03b2+, \u03b2\u2212) =\n\nBeta(M (n)\n\n+ (a, b) + \u03b2+(a, b), M (n)\u2212 (a, b) + \u03b2\u2212(a, b))\n\n,\n\n(cid:89)\n+ (a, b) = (1 \u2212 1\nM (n)\nM (n)\u2212 (a, b) = (1 \u2212 1\n\na\u2265b\n\nBeta(\u03b2+(a, b), \u03b2\u2212(a, b))\n\n2 \u03b4a,b)z(cid:62)\n2 \u03b4a,b)z(cid:62)\n\na (A(n) + A(n)(cid:62)\na (ee(cid:62) \u2212 I)zb \u2212 M (n)\n\n)zb\n\n+ (a, b)\n\nM (n)\n+ (a, b) is the number of links between functional units a and b whereas M (n)\u2212 (a, b) is the\nnumber of non-links between functional unit a and b when disregarding links between a node and\nitself. e is a vector of length J with ones in all entries where J is the number of voxel groups.\nWe will assume that the graphs are independent over subjects such that\n\nP (A(1), . . . , A(N )|Z, \u03b2+, \u03b2\u2212) =\n\nBeta(M (n)\n\n+ (a, b) + \u03b2+(a, b), M (n)\u2212 (a, b) + \u03b2\u2212(a, b))\n\nBeta(\u03b2+(a, b), \u03b2\u2212(a, b))\n\n(cid:89)\n\n(cid:89)\n\na\u2265b\n\nn\n\nAs a result, the posterior likelihood is given by\n\n\uf8eb\uf8ed(cid:89)\n\nn\n\n(cid:89)\n\na\u2265b\n\nP (Z|A(1), . . . , A(N ), \u03b2+, \u03b2\u2212, \u03b1) \u221d\n\nP (A(n)|Z, \u03b2+, \u03b2\u2212)\n\nP (Z|\u03b1) =\n\nBeta(M (n)\n\n+ (a, b) + \u03b2+(a, b), M (n)\u2212 (a, b) + \u03b2\u2212(a, b))\n\nBeta(\u03b2+(a, b), \u03b2\u2212(a, b))\n\nn\n\n\u03b1D \u0393(\u03b1)\n\n\u0393(J + \u03b1)\n\n\u0393(na)\n\n.\n\nWhere D is the number of expressed functional units and na the number of voxel groups assigned\nto functional unit a. The expected value of \u03c1(n) is given by\n(cid:104)\u03c1(n)(a, b)(cid:105) =\n\n+ (a,b)+\u03b2+(a,b)\n\nM (n)\n\n.\n\nM (n)\n\n+ (a,b)+M (n)\u2212 (a,b)+\u03b2+(a,b)+\u03b2\u2212(a,b)\n\nMCMC Sampling the IRM model: As proposed in [16] we use a Gibbs sampling scheme in\ncombination with split-merge sampling [15] for the clustering assignment matrix Z. We used the\nsplit-merge sampling procedure proposed in [15] with three restricted Gibbs sampling sweeps. We\ninitialized the restricted Gibbs sampler by the sequential allocation procedure proposed in [8]. For\nthe MCMC sampling, the posterior likelihood for a node assignment given the assignment of the\nremaining nodes is needed both for the Gibbs sampler as well as for calculating the split-merge\nacceptance ratios [15].\n\n.\n\n(cid:33)\n\n(cid:33)\n(cid:89)\n\na\n\n(cid:32)(cid:89)\n\n(cid:32)\n\n\uf8f6\uf8f8 \u00b7\n\n\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 ma\n(cid:81)\n\u03b1(cid:81)\n\nn\n\n(cid:81)\n(cid:81)\n\nb\n\nn\n\n4\n\nP (zia = 1|Z\\zir , A(1), ..., A(N )) \u221d\n\nwhere ma =(cid:80)\n\nj(cid:54)=i zj,a is the size of the ath functional unit disregarding the assignment of the ith\nnode. We note that this posterior likelihood can be ef\ufb01ciently calculated only considering the parts\nof the computation of M (n)\n+ (a, b) and M (n)\u2212 (a, b) as well as evaluation of the Beta function that are\naffected by the considered assignment change.\n\nBeta(M\n\n(n)\n+ (a,b)+\u03b2+(a,b),M\n\n(n)\u2212 (a,b)+\u03b2\u2212(a,b))\n\nb\nBeta(M\n\nBeta(\u03b2+(a,b),\u03b2\u2212(a,b))\n\n(n)\n+ (a,b)+\u03b2+(a,b),M\n\n(n)\u2212 (a,b)+\u03b2\u2212(a,b))\n\nBeta(\u03b2+(a,b),\u03b2\u2212(a,b))\n\nif ma > 0\n\notherwise .\n\n\fScoring the functional units in terms of stability: By sampling we obtain a large amount of\npotential solutions, however, for visualization and interpretation it is dif\ufb01cult to average across all\nsamples as this requires that the extracted groups in different samples and runs can be related to\neach other. For visualization we instead selected the single best extracted sample r\u2217 (i.e., the MAP\nestimate) across 10 separate randomly initialized runs each of 500 iterations.\nTo facilitate interpretation we displayed the top 20 extracted functional units most reproducible\nacross the separate runs. To identify these functional units we analyzed how often nodes co-occurred\nin the same cluster across the extracted samples from the other random starts r according to C =\n\n(cid:80)\nr(cid:54)=r\u2217 (Z(r)Z(r)(cid:62) \u2212 I) using the following score \u03b7c\nCz(r\u2217)\n\nsc = 1\n\n\u03b7c =\n\n2 z(r\u2217)(cid:62)\n\n,\n\nc\n\nc\n\nsc\nstot\nc\n\nc = z(r\u2217)(cid:62)\nstot\n\nc\n\nCe \u2212 sc.\n\n,\n\nsc counts the number of times the voxels in group c co-occurred with other voxels in the group\nwhereas stot gives the total number of times voxels in group c co-occurred with other voxels in the\ngraph. As such 0 \u2264 \u03b7c \u2264 1 where 1 indicates that all voxels in the cth group were in the same\ncluster across all samples whereas 0 indicates that the voxels never co-occurred in any of the other\nsamples.\n\n3 Results and Discussion\nFollowing [11] we calculated the average shortest path length (cid:104)L(cid:105), average clustering coef\ufb01cient\n(cid:104)C(cid:105), degree distribution \u03b3 and largest connected component (i.e., giant component) G for each\nsubject speci\ufb01c graph as well as the MI threshold value tc used to de\ufb01ne the top 100, 000 links. In\ntable 1 it can be seen that the derived graphs are far from Erd\u00a8os-R\u00b4enyi random graphs. Both the\nclustering coef\ufb01cient, degree distribution parameter \u03b3 and giant component G differ signi\ufb01cantly\nfrom the random graphs. However, there are no signi\ufb01cant differences between the Normal and MS\ngroup indicating that these global features do not appear to be affected by the disease.\nFor each run, we initialized the IRM model with D = 50 randomly generated functional units.\nWe set the prior \u03b2+(a, b) =\nfavoring a\npriori higher within functional unit link density relative to between link density. We set \u03b1 = log J\n(where J is the number of voxel groups). In the model estimation we treated 2.5% of the links\nand an equivalent number of non-links as missing at random in the graphs. When treating entries\nas missing at random these can be ignored maintaining counts only over the observed values [16].\nThe estimated models are very stable as they on average extracted D = 72.6 \u00b1 0.6 functional units.\nIn \ufb01gure 2 the area under curve (AUC) scores of the receiver operator characteristic for predicting\nlinks are given for each subject where the prediction of links was based on averaging over the \ufb01nal\n100 samples. While these AUC scores are above random for all subjects we see a high degree of\nvariability across the subjects in terms of the model\u2019s ability to account for links and non-links in\nthe graphs. We found no signi\ufb01cant difference between the Normal and MS group in terms of the\n\nand \u03b2\u2212(a, b) =\n\n(cid:26) 5\n\n(cid:26) 1\n\notherwise\n\notherwise\n\na = b\n\na = b\n\n1\n\n5\n\nTable 1: Median threshold values tc, average shortest path (cid:104)L(cid:105), average clustering coef\ufb01cient (cid:104)C(cid:105),\ndegree distribution exponent \u03b3 (i.e. p(k) \u221d k\u2212\u03b3) and giant component G (i.e. largest connected\ncomponent in the graphs relative to the complete graph) for the normal and multiple-sclerosis group\nas well as a non-parametric test of difference in median between the two groups. The random graph\nis an Erd\u00a8os-R\u00b4enyi random graph with same density as the constructed graphs.\n\nNormal\nMS\nRandom\nP-value(Normal vs. MS)\nP-value(Normal and MS vs. Random)\n\ntc\n\n0.0164\n0.0163\n\n-\n\n0.9964\n\n-\n\n(cid:104)L(cid:105)\n2.77\n2.70\n2.73\n0.4509\n0.6764\n\n(cid:104)C(cid:105)\n0.1116\n0.0898\n0.0079\n0.9954\np \u2264 0.001\n\n\u03b3\n1.40\n1.36\n0.88\n0.7448\np \u2264 0.001\n\nG\n\n0.8587\n0.8810\n\n1\n\n0.7928\np \u2264 0.001\n\n5\n\n\fFigure 2: AUC score across the 10 different runs for each subject in the Normal group (top) and\nMS group (bottom). At the top right the distribution of the AUC scores is given for the two groups\n(Normal: blue, MS: red). No signi\ufb01cant difference between the median value of the two distributions\nare found (p \u2248 0.34).\n\nmodel\u2019s ability to account for the network dynamics. Thus, there seem to be no difference in terms\nof how well the IRM model is able to account for structure in the networks of MS and Normal\nsubjects. Finally, we see that the link prediction is surprisingly stable for each subject across runs as\nwell as links and non-links treated as missing. This indicate that there is a high degree of variability\nin the graphs extracted from resting state fMRI between the subjects relative to the variability within\neach subject.\nConsidering the inference a stochastic optimization procedure we have visualized the sample with\nhighest likelihood (i.e. the MAP estimate) over the runs in \ufb01gure 3. We display the top 20 most\nreproducible extracted voxel groups (i.e., functional units) across the 10 runs. Fifteen of the 20\nfunctional units are easily identi\ufb01ed as functionally relevant networks. These selected functional\nunits are similar to the networks previously identi\ufb01ed on resting-state fMRI data using ICA [9].\nThe sensori-motor network is represented by the functional units 2, 3, 13 and 20; the posterior part\nof the default-mode network [19] by functional units 6, 14, 16, 19; a fronto-parietal network by the\nfunctional units 7,10 and 12; the visual system represented by the functional units 5, 11, 15, 18. Note\nthe striking similarity to the sensori-motor ICA1, posterior part of the default network ICA2 and\nfronto-parietal network ICA3 and visual component ICA4. Contrary to ICA the current approach is\nable to also model interactions between components and a consistent pattern is revealed where the\nfunctional units with the highest within connectivity also show the strongest between connectivity.\nFurthermore the functional units appear to have symmetric connectivity pro\ufb01les e.g. functional unit\n2 is strongly connected to functional unit 3 (sensori-motor system), and these both strongly connect\nto the same other functional units, in this case 6 and 16 (default-mode network). Functional units\n1, 4, 8, 9, 17 we attribute to vascular noise and these units appear to be less connected with the\nremaining functional units.\nIn panel C of \ufb01gure 3 we tested the difference between medians in the connectivity of the extracted\nfunctional units. Given are connections that are signi\ufb01cant at p \u2264 0.05. Healthy individuals show\nstronger connectivity among selected functional units relative to patients. The functional units in-\nvolved are distributed throughout the brain and comprise the visual system (functional unit 5 and 11),\nthe sensori-motor network (functional unit 2), and the fronto-parietal network (functional unit 10).\nThis is expected since MS affects the brain globally by white-matter changes disseminated through-\nout the brain [12]. Patients with MS show stronger connectivity relative to healthy individuals\nbetween selected parts of the sensori-motor (functional unit 13) and fronto-parietal network (func-\ntional units 7 and 12). An interpretation of this \ufb01nding could be that the communication increases\nbetween the fronto-parietal and the sensori-motor network either as a maladaptive consequence of\nthe disease or as part of a bene\ufb01cial compensatory mechanism to maintain motor function.\n\n6\n\n\fFigure 3: Panel A: Visualization of the MAP model over the 10 restarts. Given are the functional\nunits indicated in red while circles indicate median within unit link density and lines median between\nfunctional unit link density. Gray scale and line width code the link density between and within the\nfunctional units using a logarithmic scale. Panel B: Selected resting state components extracted\nfrom a group independent component analysis (ICA) are given. After temporal concatenation over\nsubjects the Infomax ICA algorithm [2] was used to identify 20 spatially independent components.\nSubsequently the individual component time series was used in a regression model to obtain subject\nspeci\ufb01c component maps [5]. The displayed ICA maps are based on one sample t-tests corrected\nfor multiple comparisons p \u2264 0.05 using Gaussian random \ufb01elds theory. Panel C: AUC score for\nrelations between the extracted groups thresholded at a signi\ufb01cance level of \u03b1 = 5% based on a two\nsided rank-sum test. Blue indicates that the link density is larger for Normal than MS, yellow that\nMS is larger than Normal. (A high resolution version of the \ufb01gure can be found in the supplementary\nmaterial).\n\n7\n\n\fTable 2: Leave one out classi\ufb01cation performance based on support vector machine (SVM) with\na linear kernel, linear discriminant analysis (LDA) and K-nearest neighbor (KNN). Signi\ufb01cance\nlevel estimated by comparing to classi\ufb01cation performance for the corresponding classi\ufb01ers with\nrandomly permuted class labels, bold indicates signi\ufb01cant classi\ufb01cation at a p \u2264 0.05.\n\nRaw data\n\n51.39\n59.72\n38.89\n\nPCA\n55.56\n51.39\n58.33\n\nSVM\nLDA\nKNN\n\nICA\n\n63.89 (p \u2264 0.04)\n63.89 (p \u2264 0.05)\n\n56.94\n\nDegree\n59.72\n51.39\n51.39\n\nIRM\n\n72.22(p \u2264 0.002)\n75.00(p \u2264 0.001)\n66.67(p \u2264 0.01)\n\nDiscriminating Normal subjects from MS: We evaluated the classi\ufb01cation performance of the\nsubject speci\ufb01c group link densities \u03c1(n) based on leave one out cross-validation. We considered\nthree standard classi\ufb01ers, soft margin support vector machine (SVM) with linear kernel (C = 1),\nlinear discriminant analysis (LDA) based on the pooled variance estimate (features projected by prin-\ncipal component analysis to a 20 dimensional feature space prior to analysis), as well as K-nearest\nneighbor (KNN), K = 3. We compared the classi\ufb01er performances to classifying the normalized\nraw subject speci\ufb01c voxel\u00d7 time series, i.e. the matrix given by subject\u00d7 voxel\u2212 time as well as\nthe data projected to the most dominant 20 dimensional subspace denoted (PCA). For comparison\nwe also included a group ICA [5] analysis as well as the performance using node degree (Degree)\nas features which has previously been very successful for classi\ufb01cation of schizophrenia [6]. For\nthe IRM model we used the Bayesian average over predictions which was dominated by the MAP\nestimate given in \ufb01gure 3. For all the classi\ufb01cation analyses we normalized each feature. In table\n2 is given the classi\ufb01cation results. Group ICA as well as the proposed IRM model signi\ufb01cantly\nclassify above random. The IRM model has a higher classi\ufb01cation rate and is signi\ufb01cant across all\nthe classi\ufb01ers.\nFinally, we note that contrary to analysis based on temporal correlation such as the ICA and PCA\napproaches used for the classi\ufb01cation the bene\ufb01t of mutual information is that it can take higher\norder dependencies into account that are not necessarily re\ufb02ected by correlation. As such, a brain\nregion driven by the variance of another brain region can be captured by mutual information whereas\nthis is not necessarily captured by correlation.\n\n4 Conclusion\n\nThe functional units extracted using the IRM model correspond well to previously described RSNs\n[19, 9]. Whereas conventional models for assessing functional connectivity in rs-fMRI data often\naim to divide the brain into segregated networks the IRM explicitly models relations between func-\ntional units enabling visualization and analysis of interactions. Using classi\ufb01cation models to predict\nthe subject disease state revealed that the IRM model had a higher prediction rate than discrimina-\ntion based on the components extracted from a conventional group ICA approach [5]. IRM readily\nextends to directed graphs and networks derived from task related functional activation. As such\nwe believe the proposed method constitutes a promising framework for the analysis of functionally\nderived brain networks in general.\n\nReferences\n\n[1] S. Achard, R. Salvador, B. Whitcher, J. Suckling, and E. Bullmore. A resilient, low-frequency, small-\nworld human brain functional network with highly connected association cortical hubs. 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Advances in Neural Information Processing Systems, 18:1593\u20131600, 2006.\n\n9\n\n\f", "award": [], "sourceid": 1259, "authors": [{"given_name": "Morten", "family_name": "M\u00f8rup", "institution": null}, {"given_name": "Kristoffer", "family_name": "Madsen", "institution": null}, {"given_name": "Anne-marie", "family_name": "Dogonowski", "institution": null}, {"given_name": "Hartwig", "family_name": "Siebner", "institution": null}, {"given_name": "Lars", "family_name": "Hansen", "institution": null}]}