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128 nips-2010-Infinite Relational Modeling of Functional Connectivity in Resting State fMRI


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Author: Morten Mørup, Kristoffer Madsen, Anne-marie Dogonowski, Hartwig Siebner, Lars K. Hansen

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. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 dk 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. [sent-9, score-1.113]

2 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. [sent-10, score-0.834]

3 While these models can identify coherently behaving groups in terms of correlation they give little insight into how these groups interact. [sent-11, score-0.282]

4 In this paper we take a different view on the analysis of functional resting state networks. [sent-12, score-0.694]

5 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. [sent-13, score-2.211]

6 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. [sent-14, score-2.151]

7 1 Introduction Neuronal elements of the brain constitute an intriguing complex network [4]. [sent-15, score-0.307]

8 Functional magnetic resonance imaging (fMRI) can be applied to study the functional connectivity of the neural elements which form this complex network at a whole brain level. [sent-16, score-1.113]

9 It has been suggested that fluctuations in the blood oxygenation level-dependent (BOLD) signal during rest reflecting the neuronal baseline activity of the brain correspond to functionally relevant networks [9, 3, 19]. [sent-17, score-0.429]

10 All pairwise mutual information (MI) are calculated between the 2x2x2 group of voxels for each subjects resting state fMRI activity. [sent-20, score-0.815]

11 The graph of pairwise mutual information is thresholded such that the top 100,000 un-directed links are kept. [sent-21, score-0.353]

12 The graphs are analyzed by the infinite relational model (IRM) assuming the functional units Z are the same for all subjects but their interactions ρ(n) are individual. [sent-22, score-0.974]

13 these models identify coherently behaving groups in terms of correlation they give limited insight into how these groups interact. [sent-24, score-0.282]

14 Furthermore, while correlation is optimal for extracting second order statistics it easily fails in establishing higher order interactions between regions of the brain [22, 7]. [sent-25, score-0.372]

15 In this paper we take a different view on the analysis of functional resting state networks. [sent-26, score-0.694]

16 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. [sent-27, score-2.088]

17 Consequently, what define functional units are the way in which they interact with the remaining parts of the network. [sent-28, score-0.63]

18 We will consider functional connectivity between regions as measured by mutual information. [sent-29, score-0.726]

19 Mutual information (MI) is well rooted in information theory and given enough data MI can detect functional relations between regions regardless of the order of the interaction [22, 7]. [sent-30, score-0.516]

20 Thereby, resting state fMRI can be represented as a mutual information graph of pairwise relations between voxels constituting a complex network. [sent-31, score-0.721]

21 Here common procedures have been to extract various summary statistics of the networks and compare them to those of random networks and these analyses have demonstrated that fMRI derived graphs behave far from random [11, 1, 4]. [sent-33, score-0.19]

22 In this paper we propose to use relational modeling [17, 16, 27] in order to quantify functional coherent groups of resting state networks. [sent-34, score-1.027]

23 In particular, we investigate how this line of modeling can be used to discriminate patients with multiple sclerosis from healthy individuals. [sent-35, score-0.235]

24 Focal axonal demyelinization and secondary axonal degeneration results in variable delays or even in disruption of signal transmission along cortico-cortical and cortico-subcortical connections [21, 26]. [sent-37, score-0.182]

25 In addition to the characteristic macroscopic white-matter lesions seen on structural magnetic resonance imaging (MRI), pathology- and advanced MRI-studies have shown demyelinated lesions in cortical gray-matter as well as in white-matter that appear normal on structural MRI [18, 12]. [sent-38, score-0.548]

26 These findings show that demyelination is disseminated throughout the brain affecting brain functional connectivity. [sent-39, score-0.967]

27 Structural MRI gives information about the extent of white-matter lesions, but provides no information on the impact on functional brain connectivity. [sent-40, score-0.682]

28 , affecting the brain’s anatomical and functional ’wiring’) MS represents a disease state which is particular suited for relational modeling. [sent-43, score-0.736]

29 Here, relational modeling is able to provide a global view of the communication in the functional network between the extracted functional units. [sent-44, score-1.162]

30 Furthermore, the method facilitates the examination of all brain networks simultaneously in a completely data driven manner. [sent-45, score-0.31]

31 2 2 Methods Data: 42 clinically stable patients with relapsing-remitting (RR) and secondary progressive multiple sclerosis (27 RR; 22 females; mean age: 43. [sent-47, score-0.171]

32 The rs-fMRI session lasted 20 min (482 brain volumes). [sent-55, score-0.248]

33 After filtering, the voxel were masked [23] and divided into 5039 voxel groups consisting of 2 × 2 × 2 voxels for the estimation of pairwise MI. [sent-60, score-0.711]

34 1 Mutual Information Graphs The mutual information between voxel groups i and j is given by Pij (u,v) I(i, j) = uv Pij (u, v) log Pi (u)Pj (u) . [sent-62, score-0.404]

35 We used equiprobable rather than equidistant bins [25] based on 10 percentiles derived from the individual distribution of each voxel group, i. [sent-67, score-0.173]

36 Pij (u, v) counts the number of co-occurrences of observations from voxels in voxel group i that are at bin u while the corresponding voxels from group j are at bin v at time t. [sent-70, score-0.789]

37 To generate the mutual information graphs for each subject a total of 72·5039·(5039−1)/2 ≈ 1 billion pairwise MI were evaluated. [sent-72, score-0.266]

38 We thresholded each graph keeping the top 100, 000 pairwise MI as links in the graph. [sent-73, score-0.23]

39 100, 000 100,000 undirected link) which resulted in each graph having link density 5039·(5039−1)/2 = 0. [sent-76, score-0.166]

40 2 million links (when counting links only in the one direction). [sent-78, score-0.23]

41 2 Infinite Relational Modeling (IRM) The importance of modeling brain connectivity and interactions is widely recognized in the literature on fMRI [13, 28, 20]. [sent-80, score-0.417]

42 Approaches such as dynamic causal modeling [13], structural equation models [20] and dynamic Bayes nets [28] are normally limited to analysis of a few interactions between known brain regions or predefined regions of interest. [sent-81, score-0.424]

43 The benefits of the current relational modeling approach are that regions are defined in a completely data driven manner while the method establishes interaction at a low computational complexity admitting the analysis of large scale brain networks. [sent-82, score-0.439]

44 Functional connectivity graphs have previously been considered in [6] for the discrimination of schizophrenia. [sent-83, score-0.189]

45 In [24] resting state networks were defined based on normalized graph cuts in order to derive functional units. [sent-84, score-0.801]

46 While normalized cuts are well suited for the separation of voxels into groups of disconnected components the method lacks the ability to consider coherent interaction between groups. [sent-85, score-0.408]

47 In [17] the stochastic block model also denoted the relational model (RM) was proposed for the identification of coherent groups of nodes in complex networks. [sent-86, score-0.333]

48 Here, each node i belongs to a class z ir where ir denote the ith row of a clustering assignment matrix Z, and the probability, πij , of a link between node i and j is determined by the class assignments z ir and z jr as πij = z ir ρz jr . [sent-87, score-0.677]

49 Here, ρk ∈ [0, 1] denotes the probability of generating a link between a node in class k and a node in class . [sent-88, score-0.164]

50 e is a vector of length J with ones in all entries where J is the number of voxel groups. [sent-96, score-0.173]

51 a Where D is the number of expressed functional units and na the number of voxel groups assigned to functional unit a. [sent-105, score-1.395]

52 where ma = j=i zj,a is the size of the ath functional unit disregarding the assignment of the ith node. [sent-115, score-0.58]

53 To facilitate interpretation we displayed the top 20 extracted functional units most reproducible across the separate runs. [sent-121, score-0.788]

54 sc counts the number of times the voxels in group c co-occurred with other voxels in the group whereas stot gives the total number of times voxels in group c co-occurred with other voxels in the graph. [sent-123, score-1.257]

55 As such 0 ≤ ηc ≤ 1 where 1 indicates that all voxels in the cth group were in the same cluster across all samples whereas 0 indicates that the voxels never co-occurred in any of the other samples. [sent-124, score-0.565]

56 , giant component) G for each subject specific graph as well as the MI threshold value tc used to define the top 100, 000 links. [sent-127, score-0.185]

57 Both the o e clustering coefficient, degree distribution parameter γ and giant component G differ significantly from the random graphs. [sent-129, score-0.172]

58 For each run, we initialized the IRM model with D = 50 randomly generated functional units. [sent-131, score-0.434]

59 5 a=b 1 a=b We set the prior β + (a, b) = and β − (a, b) = favoring a 1 otherwise 5 otherwise priori higher within functional unit link density relative to between link density. [sent-132, score-0.693]

60 We set α = log J (where J is the number of voxel groups). [sent-133, score-0.173]

61 In figure 2 the area under curve (AUC) scores of the receiver operator characteristic for predicting links are given for each subject where the prediction of links was based on averaging over the final 100 samples. [sent-140, score-0.27]

62 While these AUC scores are above random for all subjects we see a high degree of variability across the subjects in terms of the model’s ability to account for links and non-links in the graphs. [sent-141, score-0.371]

63 We found no significant difference between the Normal and MS group in terms of the Table 1: Median threshold values tc , average shortest path L , average clustering coefficient C , degree distribution exponent γ (i. [sent-142, score-0.209]

64 largest connected component in the graphs relative to the complete graph) for the normal and multiple-sclerosis group as well as a non-parametric test of difference in median between the two groups. [sent-146, score-0.312]

65 001 Figure 2: AUC score across the 10 different runs for each subject in the Normal group (top) and MS group (bottom). [sent-172, score-0.283]

66 Finally, we see that the link prediction is surprisingly stable for each subject across runs as well as links and non-links treated as missing. [sent-178, score-0.31]

67 This indicate that there is a high degree of variability in the graphs extracted from resting state fMRI between the subjects relative to the variability within each subject. [sent-179, score-0.548]

68 We display the top 20 most reproducible extracted voxel groups (i. [sent-183, score-0.402]

69 Fifteen of the 20 functional units are easily identified as functionally relevant networks. [sent-186, score-0.673]

70 These selected functional units are similar to the networks previously identified on resting-state fMRI data using ICA [9]. [sent-187, score-0.692]

71 Contrary to ICA the current approach is able to also model interactions between components and a consistent pattern is revealed where the functional units with the highest within connectivity also show the strongest between connectivity. [sent-190, score-0.799]

72 Furthermore the functional units appear to have symmetric connectivity profiles e. [sent-191, score-0.753]

73 functional unit 2 is strongly connected to functional unit 3 (sensori-motor system), and these both strongly connect to the same other functional units, in this case 6 and 16 (default-mode network). [sent-193, score-1.402]

74 Functional units 1, 4, 8, 9, 17 we attribute to vascular noise and these units appear to be less connected with the remaining functional units. [sent-194, score-0.826]

75 In panel C of figure 3 we tested the difference between medians in the connectivity of the extracted functional units. [sent-195, score-0.681]

76 Healthy individuals show stronger connectivity among selected functional units relative to patients. [sent-198, score-0.787]

77 The functional units involved are distributed throughout the brain and comprise the visual system (functional unit 5 and 11), the sensori-motor network (functional unit 2), and the fronto-parietal network (functional unit 10). [sent-199, score-1.146]

78 This is expected since MS affects the brain globally by white-matter changes disseminated throughout the brain [12]. [sent-200, score-0.533]

79 Patients with MS show stronger connectivity relative to healthy individuals between selected parts of the sensori-motor (functional unit 13) and fronto-parietal network (functional units 7 and 12). [sent-201, score-0.526]

80 Given are the functional units indicated in red while circles indicate median within unit link density and lines median between functional unit link density. [sent-204, score-1.481]

81 Gray scale and line width code the link density between and within the functional units using a logarithmic scale. [sent-205, score-0.751]

82 Panel B: Selected resting state components extracted from a group independent component analysis (ICA) are given. [sent-206, score-0.479]

83 Panel C: AUC score for relations between the extracted groups thresholded at a significance level of α = 5% based on a two sided rank-sum test. [sent-211, score-0.267]

84 01) Discriminating Normal subjects from MS: We evaluated the classification performance of the subject specific group link densities ρ(n) based on leave one out cross-validation. [sent-237, score-0.303]

85 We compared the classifier performances to classifying the normalized raw subject specific voxel × time series, i. [sent-239, score-0.213]

86 the matrix given by subject × voxel − time as well as the data projected to the most dominant 20 dimensional subspace denoted (PCA). [sent-241, score-0.213]

87 For comparison we also included a group ICA [5] analysis as well as the performance using node degree (Degree) as features which has previously been very successful for classification of schizophrenia [6]. [sent-242, score-0.171]

88 As such, a brain region driven by the variance of another brain region can be captured by mutual information whereas this is not necessarily captured by correlation. [sent-249, score-0.619]

89 4 Conclusion The functional units extracted using the IRM model correspond well to previously described RSNs [19, 9]. [sent-250, score-0.72]

90 Whereas conventional models for assessing functional connectivity in rs-fMRI data often aim to divide the brain into segregated networks the IRM explicitly models relations between functional units enabling visualization and analysis of interactions. [sent-251, score-1.571]

91 Using classification models to predict the subject disease state revealed that the IRM model had a higher prediction rate than discrimination based on the components extracted from a conventional group ICA approach [5]. [sent-252, score-0.332]

92 IRM readily extends to directed graphs and networks derived from task related functional activation. [sent-253, score-0.562]

93 As such we believe the proposed method constitutes a promising framework for the analysis of functionally derived brain networks in general. [sent-254, score-0.353]

94 A resilient, low-frequency, smallworld human brain functional network with highly connected association cortical hubs. [sent-261, score-0.741]

95 Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. [sent-278, score-0.581]

96 Complex brain networks: graph theoretical analysis of structural and functional systems. [sent-283, score-0.765]

97 A method for making group inferences from functional MRI data using independent component analysis. [sent-294, score-0.563]

98 Exploring functional connectivities of the human brain using multivariate information analysis. [sent-316, score-0.682]

99 Blood oxygenation level dependent contrast resting state networks are relevant to functional activity in the neocortical sensorimotor system. [sent-347, score-0.798]

100 MRI evidence for multiple sclerosis as a diffuse disease of the central nervous system. [sent-365, score-0.192]


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Our experimental results show that the optimized ROIs achieved our optimization objectives and demonstrated promising results. 2 Mat eria l s a nd Metho ds 2.1 Data acquisition and preprocessing Twenty-five university students were recruited to participate in this study. Each participant performed an fMRI modified version of the OSPAN task (3 block types: OSPAN, Arithmetic, and Baseline) while fMRI data was acquired. DTI scans were also acquired for each participant. FMRI and DTI scans were acquired on a 3T GE Signa scanner. Acquisition parameters were as follows : fMRI: 64x64 matrix, 4mm slice thickness, 220mm FOV, 30 slices, TR=1.5s, TE=25ms, ASSET=2; DTI: 128x128 matrix, 2mm slice thickness, 256mm FOV, 60 slices, TR=15100ms, TE= variable, ASSET=2, 3 B0 images, 30 optimized gradient directions, b-value=1000). Each participant’s fMRI data was analyzed using FSL. Individual activation map Fig.2. working memory reflecting the OSPAN (OSPAN > Baseline) contrast was used. In ROIs mapped on a total, we identified the 16 highest activated ROIs, including left WM/GM surface and right insula, left and right medial frontal gyrus, left and right precentral gyrus, left and right paracingulate gyrus, left and right dorsolateral prefrontal cortex, left and right inferior parietal lobule, left occipital pole, right frontal pole, right lateral occipital gyrus, and left and right precuneus. Fig.2 shows the 16 ROIs mapped onto a WM(white matter)/GM(gray matter) cortical surface. For some individuals, there may be missing ROIs on their activation maps. Under such condition, we adapted the group activation map as a guide to find these ROIs using linear registration. DTI pre-processing consisted of skull removal, motion correction, and eddy current correction. After the pre-processing, fiber tracking was performed using MEDINRIA (FA threshold: 0.2; minimum fiber length: 20). Fibers were extended along their tangent directions to reach into the gray matter when necessary. Brain tissue segmentation was conducted on DTI data by the method in [16] and the cortical surface was reconstructed from the tissue maps using the marching cubes algorithm. The cortical surface was parcellated into anatomical regions using the HAMMER tool [17]. DTI space was used as the standard space from which to generate the GM (gray matter) segmentation and from which to report the ROI locations on the cortical surface. Since the fMRI and DTI sequences are both EPI (echo planar imaging) sequences, their distortions tend to be similar and the misalignment between DTI and fMRI images is much less than that between T1 and fMRI images [18]. Co-registration between DTI and fMRI data was performed using FSL FLIRT [12]. The activated ROIs and tracked fibers were then mapped onto the cortical surface for joint modeling. 2.2 Joint modeling of anatomical, structural and functional profiles Despite the high degree of variability across subjects, there are several aspects of regularity on which we base the proposed solution. Firstly, across subjects, the functional ROIs should have similar anatomical locations, e.g., similar locations in the atlas space. Secondly, these ROIs should have similar structural connectivity profiles across subjects. In other words, fibers penetrating the same functional ROIs should have at least similar target regions across subjects. Lastly, individual networks identified by task-based paradigms, like the working memory network we adapted as a test bed in this paper, should have similar functional connectivity pattern across subjects. The neuroscience bases of the above premises include: 1) structural and functional brain connectivity are closely related [19], and cortical gyrification and axongenesis processes are closely coupled [20]; Hence, it is reasonable to put these three types of information in a joint modeling framework. 2) Extensive studies have already demonstrated the existence of a common structural and functional architecture of the human brain [21, 22], and it makes sense to assume that the working memory network has similar structural and functional connectivity patterns across individuals. Based on these premises, we proposed to optimize the locations of individual functional ROIs by jointly modeling anatomic profiles, structural connectivity patterns, and functional connectivity patterns, as illustrated in Fig 3. The Fig.3. ROIs optimization scheme. goal was to minimize the group-wise variance (or maximize group-wise consistency) of these jointly modeled profiles. Mathematically, we modeled the group-wise variance as energy E as follows. A ROI from fMRI analysis was mapped onto the surface, and is represented by a center vertex and its neighborhood. Suppose đ?‘… đ?‘–đ?‘— is the ROI region j on the cortical surface of subject i identified in Section 2.1; we find a corresponding surface ROI region đ?‘† đ?‘–đ?‘— so that the energy E (contains energy from n subjects, each with m ROIs) is minimized: đ??¸ = đ??¸ đ?‘Ž (đ?œ† đ??¸ đ?‘? −đ?‘€ đ??¸ đ?‘? đ?œŽ đ??¸đ?‘? + (1 − đ?œ†) đ??¸ đ?‘“ −đ?‘€ đ??¸ đ?‘“ đ?œŽđ??¸đ?‘“ ) (1) where Ea is the anatomical constraint; Ec is the structural connectivity constraint, M Ec and ď ł E are the mean and standard deviation of Ec in the searching space; E f is the functional c connectivity constraint, M E f and ď ł E f are the mean and standard deviation of E f respectively; and ď Ź is a weighting parameter between 0 and 1. If not specified, and m is the number of ROIs in this paper. The details of these energy terms are provided in the following sections. 2.2.1 n is the number of subjects, Anatomical constraint energy Anatomical constraint energy Ea is defined to ensure that the optimized ROIs have similar anatomical locations in the atlas space (Fig.4 shows an example of ROIs of 15 randomly selected subjects in the atlas space). We model the locations for all ROIs in the atlas space using a Gaussian model (mean: đ?‘€ đ?‘‹ đ?‘— ,and standard deviation: ď ł X j for ROI j ). The model parameters were estimated using the initial locations obtained from Section 2.1. Let X ij be the center coordinate of region Sij in the atlas space, then Ea is expressed as đ??¸đ?‘Ž = { 1 đ?‘’ đ?‘‘đ?‘šđ?‘Žđ?‘Ľâˆ’1 Fig.4. ROI distributions in Atlas space. (đ?‘‘đ?‘šđ?‘Žđ?‘Ľâ‰¤1) (đ?‘‘đ?‘šđ?‘Žđ?‘Ľ>1) (2) ‖ , 1 ≤ đ?‘– ≤ đ?‘›; 1 ≤ đ?‘— ≤ đ?‘š. } (3) where đ?‘‘đ?‘šđ?‘Žđ?‘Ľ = đ?‘€đ?‘Žđ?‘Ľ { ‖ đ?‘‹ đ?‘–đ?‘— −đ?‘€ đ?‘‹ đ?‘— 3đ?œŽ đ?‘‹ đ?‘— Under the above definition, if any X ij is within the range of 3s X from the distribution model j center M X , the anatomical constraint energy will always be one; if not, there will be an j exponential increase of the energy which punishes the possible involvement of outliers. In other words, this energy factor will ensure the optimized ROIs will not significantly deviate away from the original ROIs. 2.2.2 Structural connectivity constraint energy Structural connectivity constraint energy Ec is defined to ensure the group has similar structural connectivity profiles for each functional ROI, since similar functional regions should have the similar structural connectivity patterns [19], n m Ec  ďƒĽďƒĽ (Cij  M C j )Covc 1 (Ci j  M C j )T (4) i 1 j 1 where Cij is the connectivity pattern vector for ROI j of subject i , M C j is the group mean 1 for ROI j , and Covc is the inverse of the covariance matrix. The connectivity pattern vector Cij is a fiber target region distribution histogram. To obtain this histogram, we first parcellate all the cortical surfaces into nine regions ( as shown in Fig.5a, four lobes for each hemisphere, and the subcortical region) using the HAMMER algorithm [17]. A finer parcellation is available but not used due to the relatively lower parcellation accuracy, which might render the histogram too sensitive to the parcellation result. Then, we extract fibers penetrating region Sij , and calculate the distribution of the fibers’ target cortical regions. Fig.5 illustrates the ideas. Fig.5. Structural connectivity pattern descriptor. (a): Cortical surface parcellation using HAMMER [17]; (b): Joint visualization of the cortical surface, two ROIs (blue and green spheres), and fibers penetrating the ROIs (in red and yellow, respectively); (c): Corresponding target region distribution histogram of ROIs in Fig.5b. There are nine bins corresponding to the nine cortical regions. Each bin contains the number of fibers that penetrate the ROI and are connected to the corresponding cortical region. Fiber numbers are normalized across subjects. 2.2.3 Functional connectivity constraint energy Functional connectivity constraint energy E f is defined to ensure each individual has similar functional connectivity patterns for the working memory system, assuming the human brain has similar functional architecture across individuals [21]. đ?‘› đ??¸ đ?‘“ = ∑ đ?‘–=1‖đ??šđ?‘– − đ?‘€ đ??š ‖ (5) Here, Fi is the functional connectivity matrix for subject i , and M F is the group mean of the dataset. The connectivity between each pair of ROIs is defined using the Pearson correlation. The matrix distance used here is the Frobenius norm. 2.3 Energy minimization solution The minimization of the energy defined in Section 2.2 is known as a combinatorial optimization problem. Traditional optimization methods may not fit this problem, since there are two noticeable characteristics in this application. First, we do not know how the energy changes with the varying locations of ROIs. Therefore, techniques like Newton’s method cannot be used. Second, the structure of search space is not smooth, which may lead to multiple local minima during optimization. To address this problem, we adopt the simulated annealing (SA) algorithm [23] for the energy minimization. The idea of the SA algorithm is based on random walk through the space for lower energies. In these random walks, the probability of taking a step is determined by the Boltzmann distribution, - (E - E )/ ( KT ) p = e i+ 1 i (6) if Ei 1  Ei , and p  1 when Ei 1 ď‚Ł Ei . Here, đ??¸ đ?‘– and đ??¸ đ?‘–+1 are the system energies at solution configuration đ?‘– and đ?‘– + 1 respectively; đ??ž is the Boltzmann constant; and đ?‘‡ is the system temperature. In other words, a step will be taken when a lower energy is found. A step will also be taken with probability p if a higher energy is found. This helps avoid the local minima in the search space. 3 R esult s Compared to structural and functional connectivity patterns, anatomical profiles are more easily affected by variability across individuals. Therefore, the anatomical constraint energy is designed to provide constraint only to ROIs that are obviously far away from reasonableness. The reasonable range was statistically modeled by the localizations of ROIs warped into the atlas space in Section 2.2.1. Our focus in this paper is the structural and functional profiles. 3.1 Optimization using anatomical and structural connectivity profile s In this section, we use only anatomical and structural connectivity profiles to optimize the locations of ROIs. The goal is to check whether the structural constraint energy Ec works as expected. Fig.6 shows the fibers penetrating the right precuneus for eight subjects before (top panel) and after optimization (bottom panel). The ROI is highlighted in a red sphere for each subject. As we can see from the figure (please refer to the highlighted yellow arrows), after optimization, the third and sixth subjects have significantly improved consistency with the rest of the group than before optimization, which proves the validity of the energy function Eq.(4). Fig.6. Comparison of structural profiles before and after optimization. Each column shows the corresponding before-optimization (top) and after-optimization (bottom) fibers of one subject. The ROI (right precuneus) is presented by the red sphere. 3.2 Optimization using anatomical and functional connectivity profiles In this section, we optimize the locations of ROIs using anatomical and functional profiles, aiming to validate the definition of functional connectivity constraint energy E f . If this energy constraint worked well, the functional connectivity variance of the working memory system across subjects would decrease. Fig.7 shows the comparison of the standard derivation for functional connectivity before (left) and after (right) optimization. As we can see, the variance is significantly reduced after optimization. This demonstrated the effectiveness of the defined functional connectivity constraint energy. Fig.7. Comparison of the standard derivation for functional connectivity before and after the optimization. Lower values mean more consistent connectivity pattern cross subjects. 3.3 Consistency between optimization of functional profiles and structural profiles Fig.8. Optimization consistency between functional and structural profiles. Top: Functional profile energy drop along with structural profile optimization; Bottom: Structural profile energy drop along with functional profile optimization. Each experiment was repeated 15 times with random initial ROI locations that met the anatomical constraint. The relationship between structure and function has been extensively studied [24], and it is widely believed that they are closely related. In this section, we study the relationship between functional profiles and structural profiles by looking at how the energy for one of them changes while the energy of the other decreases. The optimization processes in Section 3.1 and 3.2 were repeated 15 times respectively with random initial ROI locations that met the anatomical constraint. As shown in Fig.8, in general, the functional profile energies and structural profile energies are closely related in such a way that the functional profile energies tend to decrease along with the structural profile optimization process, while the structural profile energies also tend to decrease as the functional profile is optimized. This positively correlated decrease of functional profile energy and structural profile energy not only proves the close relationship between functional and structural profiles, but also demonstrates the consistency between functional and structural optimization, laying down the foundation of the joint optimiza tion, whose results are detailed in the following section. 3.4 Optimization connectivity profiles using anatomical, structural and functional In this section, we used all the constraints in Eq. (1) to optimize the individual locations of all ROIs in the working memory system. Ten runs of the optimization were performed using random initial ROI locations that met the anatomical constraint. Weighting parameter ď Ź equaled 0.5 for all these runs. Starting and ending temperatures for the simulated annealing algorithm are 8 and 0.05; Boltzmann constant K  1 . As we can see from Fig.9, most runs started to converge at step 24, and the convergence energy is quite close for all runs. This indicates that the simulated annealing algorithm provides a valid solution to our problem. By visual inspection, most of the ROIs move to more reasonable and consistent locations after the joint optimization. As an example, Fig.10 depicts the location movements of the ROI in Fig. 6 for eight subjects. As we can see, the ROIs for these subjects share a similar anatomical landmark, which appears to be the tip of the upper bank of the parieto-occipital sulcus. If the initial ROI was not at this landmark, it moved to the landmark after the optimization, which was the case for subjects 1, 4 and 7. The structural profiles of these ROIs are very similar to Fig.6. The results in Fig. 10 indicate the significant improvement of ROI locations achieved by the joint optimization procedure. Fig.9. Convergence performance of the simulated annealing . Each run has 28 temperature conditions. Fig.10. The movement of right precuneus before (in red sphere) and after (in green sphere) optimization for eight subjects. The

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Abstract: Functional segregation and integration are fundamental characteristics of the human brain. Studying the connectivity among segregated regions and the dynamics of integrated brain networks has drawn increasing interest. A very controversial, yet fundamental issue in these studies is how to determine the best functional brain regions or ROIs (regions of interests) for individuals. Essentially, the computed connectivity patterns and dynamics of brain networks are very sensitive to the locations, sizes, and shapes of the ROIs. This paper presents a novel methodology to optimize the locations of an individual's ROIs in the working memory system. Our strategy is to formulate the individual ROI optimization as a group variance minimization problem, in which group-wise functional and structural connectivity patterns, and anatomic profiles are defined as optimization constraints. The optimization problem is solved via the simulated annealing approach. 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Without accurate and reliable individualized ROIs, the validity of brain connectivity analysis, and computational modeling of dynamics and interactions among brain networks , would be questionable. In response to this fundamental issue, this paper presents a novel computational methodology to optimize the locations of an individual's ROIs initialized from task-based fMRI. We use the ROIs identified in a block-based working memory paradigm as a test bed application to develop and evaluate our methodology. The optimization of ROI locations was formulated as an energy minimization problem, with the goal of jointly maximizing the group-wise consistency of functional and structural connectivity patterns and anatomic profiles. The optimization problem is solved via the well-established simulated annealing approach. Our experimental results show that the optimized ROIs achieved our optimization objectives and demonstrated promising results. 2 Mat eria l s a nd Metho ds 2.1 Data acquisition and preprocessing Twenty-five university students were recruited to participate in this study. Each participant performed an fMRI modified version of the OSPAN task (3 block types: OSPAN, Arithmetic, and Baseline) while fMRI data was acquired. DTI scans were also acquired for each participant. FMRI and DTI scans were acquired on a 3T GE Signa scanner. Acquisition parameters were as follows : fMRI: 64x64 matrix, 4mm slice thickness, 220mm FOV, 30 slices, TR=1.5s, TE=25ms, ASSET=2; DTI: 128x128 matrix, 2mm slice thickness, 256mm FOV, 60 slices, TR=15100ms, TE= variable, ASSET=2, 3 B0 images, 30 optimized gradient directions, b-value=1000). Each participant’s fMRI data was analyzed using FSL. Individual activation map Fig.2. working memory reflecting the OSPAN (OSPAN > Baseline) contrast was used. In ROIs mapped on a total, we identified the 16 highest activated ROIs, including left WM/GM surface and right insula, left and right medial frontal gyrus, left and right precentral gyrus, left and right paracingulate gyrus, left and right dorsolateral prefrontal cortex, left and right inferior parietal lobule, left occipital pole, right frontal pole, right lateral occipital gyrus, and left and right precuneus. Fig.2 shows the 16 ROIs mapped onto a WM(white matter)/GM(gray matter) cortical surface. For some individuals, there may be missing ROIs on their activation maps. Under such condition, we adapted the group activation map as a guide to find these ROIs using linear registration. DTI pre-processing consisted of skull removal, motion correction, and eddy current correction. After the pre-processing, fiber tracking was performed using MEDINRIA (FA threshold: 0.2; minimum fiber length: 20). Fibers were extended along their tangent directions to reach into the gray matter when necessary. Brain tissue segmentation was conducted on DTI data by the method in [16] and the cortical surface was reconstructed from the tissue maps using the marching cubes algorithm. The cortical surface was parcellated into anatomical regions using the HAMMER tool [17]. DTI space was used as the standard space from which to generate the GM (gray matter) segmentation and from which to report the ROI locations on the cortical surface. Since the fMRI and DTI sequences are both EPI (echo planar imaging) sequences, their distortions tend to be similar and the misalignment between DTI and fMRI images is much less than that between T1 and fMRI images [18]. Co-registration between DTI and fMRI data was performed using FSL FLIRT [12]. The activated ROIs and tracked fibers were then mapped onto the cortical surface for joint modeling. 2.2 Joint modeling of anatomical, structural and functional profiles Despite the high degree of variability across subjects, there are several aspects of regularity on which we base the proposed solution. Firstly, across subjects, the functional ROIs should have similar anatomical locations, e.g., similar locations in the atlas space. Secondly, these ROIs should have similar structural connectivity profiles across subjects. In other words, fibers penetrating the same functional ROIs should have at least similar target regions across subjects. Lastly, individual networks identified by task-based paradigms, like the working memory network we adapted as a test bed in this paper, should have similar functional connectivity pattern across subjects. The neuroscience bases of the above premises include: 1) structural and functional brain connectivity are closely related [19], and cortical gyrification and axongenesis processes are closely coupled [20]; Hence, it is reasonable to put these three types of information in a joint modeling framework. 2) Extensive studies have already demonstrated the existence of a common structural and functional architecture of the human brain [21, 22], and it makes sense to assume that the working memory network has similar structural and functional connectivity patterns across individuals. Based on these premises, we proposed to optimize the locations of individual functional ROIs by jointly modeling anatomic profiles, structural connectivity patterns, and functional connectivity patterns, as illustrated in Fig 3. The Fig.3. ROIs optimization scheme. goal was to minimize the group-wise variance (or maximize group-wise consistency) of these jointly modeled profiles. Mathematically, we modeled the group-wise variance as energy E as follows. A ROI from fMRI analysis was mapped onto the surface, and is represented by a center vertex and its neighborhood. Suppose đ?‘… đ?‘–đ?‘— is the ROI region j on the cortical surface of subject i identified in Section 2.1; we find a corresponding surface ROI region đ?‘† đ?‘–đ?‘— so that the energy E (contains energy from n subjects, each with m ROIs) is minimized: đ??¸ = đ??¸ đ?‘Ž (đ?œ† đ??¸ đ?‘? −đ?‘€ đ??¸ đ?‘? đ?œŽ đ??¸đ?‘? + (1 − đ?œ†) đ??¸ đ?‘“ −đ?‘€ đ??¸ đ?‘“ đ?œŽđ??¸đ?‘“ ) (1) where Ea is the anatomical constraint; Ec is the structural connectivity constraint, M Ec and ď ł E are the mean and standard deviation of Ec in the searching space; E f is the functional c connectivity constraint, M E f and ď ł E f are the mean and standard deviation of E f respectively; and ď Ź is a weighting parameter between 0 and 1. If not specified, and m is the number of ROIs in this paper. The details of these energy terms are provided in the following sections. 2.2.1 n is the number of subjects, Anatomical constraint energy Anatomical constraint energy Ea is defined to ensure that the optimized ROIs have similar anatomical locations in the atlas space (Fig.4 shows an example of ROIs of 15 randomly selected subjects in the atlas space). We model the locations for all ROIs in the atlas space using a Gaussian model (mean: đ?‘€ đ?‘‹ đ?‘— ,and standard deviation: ď ł X j for ROI j ). The model parameters were estimated using the initial locations obtained from Section 2.1. Let X ij be the center coordinate of region Sij in the atlas space, then Ea is expressed as đ??¸đ?‘Ž = { 1 đ?‘’ đ?‘‘đ?‘šđ?‘Žđ?‘Ľâˆ’1 Fig.4. ROI distributions in Atlas space. (đ?‘‘đ?‘šđ?‘Žđ?‘Ľâ‰¤1) (đ?‘‘đ?‘šđ?‘Žđ?‘Ľ>1) (2) ‖ , 1 ≤ đ?‘– ≤ đ?‘›; 1 ≤ đ?‘— ≤ đ?‘š. } (3) where đ?‘‘đ?‘šđ?‘Žđ?‘Ľ = đ?‘€đ?‘Žđ?‘Ľ { ‖ đ?‘‹ đ?‘–đ?‘— −đ?‘€ đ?‘‹ đ?‘— 3đ?œŽ đ?‘‹ đ?‘— Under the above definition, if any X ij is within the range of 3s X from the distribution model j center M X , the anatomical constraint energy will always be one; if not, there will be an j exponential increase of the energy which punishes the possible involvement of outliers. In other words, this energy factor will ensure the optimized ROIs will not significantly deviate away from the original ROIs. 2.2.2 Structural connectivity constraint energy Structural connectivity constraint energy Ec is defined to ensure the group has similar structural connectivity profiles for each functional ROI, since similar functional regions should have the similar structural connectivity patterns [19], n m Ec  ďƒĽďƒĽ (Cij  M C j )Covc 1 (Ci j  M C j )T (4) i 1 j 1 where Cij is the connectivity pattern vector for ROI j of subject i , M C j is the group mean 1 for ROI j , and Covc is the inverse of the covariance matrix. The connectivity pattern vector Cij is a fiber target region distribution histogram. To obtain this histogram, we first parcellate all the cortical surfaces into nine regions ( as shown in Fig.5a, four lobes for each hemisphere, and the subcortical region) using the HAMMER algorithm [17]. A finer parcellation is available but not used due to the relatively lower parcellation accuracy, which might render the histogram too sensitive to the parcellation result. Then, we extract fibers penetrating region Sij , and calculate the distribution of the fibers’ target cortical regions. Fig.5 illustrates the ideas. Fig.5. Structural connectivity pattern descriptor. (a): Cortical surface parcellation using HAMMER [17]; (b): Joint visualization of the cortical surface, two ROIs (blue and green spheres), and fibers penetrating the ROIs (in red and yellow, respectively); (c): Corresponding target region distribution histogram of ROIs in Fig.5b. There are nine bins corresponding to the nine cortical regions. Each bin contains the number of fibers that penetrate the ROI and are connected to the corresponding cortical region. Fiber numbers are normalized across subjects. 2.2.3 Functional connectivity constraint energy Functional connectivity constraint energy E f is defined to ensure each individual has similar functional connectivity patterns for the working memory system, assuming the human brain has similar functional architecture across individuals [21]. đ?‘› đ??¸ đ?‘“ = ∑ đ?‘–=1‖đ??šđ?‘– − đ?‘€ đ??š ‖ (5) Here, Fi is the functional connectivity matrix for subject i , and M F is the group mean of the dataset. The connectivity between each pair of ROIs is defined using the Pearson correlation. The matrix distance used here is the Frobenius norm. 2.3 Energy minimization solution The minimization of the energy defined in Section 2.2 is known as a combinatorial optimization problem. Traditional optimization methods may not fit this problem, since there are two noticeable characteristics in this application. First, we do not know how the energy changes with the varying locations of ROIs. Therefore, techniques like Newton’s method cannot be used. Second, the structure of search space is not smooth, which may lead to multiple local minima during optimization. To address this problem, we adopt the simulated annealing (SA) algorithm [23] for the energy minimization. The idea of the SA algorithm is based on random walk through the space for lower energies. In these random walks, the probability of taking a step is determined by the Boltzmann distribution, - (E - E )/ ( KT ) p = e i+ 1 i (6) if Ei 1  Ei , and p  1 when Ei 1 ď‚Ł Ei . Here, đ??¸ đ?‘– and đ??¸ đ?‘–+1 are the system energies at solution configuration đ?‘– and đ?‘– + 1 respectively; đ??ž is the Boltzmann constant; and đ?‘‡ is the system temperature. In other words, a step will be taken when a lower energy is found. A step will also be taken with probability p if a higher energy is found. This helps avoid the local minima in the search space. 3 R esult s Compared to structural and functional connectivity patterns, anatomical profiles are more easily affected by variability across individuals. Therefore, the anatomical constraint energy is designed to provide constraint only to ROIs that are obviously far away from reasonableness. The reasonable range was statistically modeled by the localizations of ROIs warped into the atlas space in Section 2.2.1. Our focus in this paper is the structural and functional profiles. 3.1 Optimization using anatomical and structural connectivity profile s In this section, we use only anatomical and structural connectivity profiles to optimize the locations of ROIs. The goal is to check whether the structural constraint energy Ec works as expected. Fig.6 shows the fibers penetrating the right precuneus for eight subjects before (top panel) and after optimization (bottom panel). The ROI is highlighted in a red sphere for each subject. As we can see from the figure (please refer to the highlighted yellow arrows), after optimization, the third and sixth subjects have significantly improved consistency with the rest of the group than before optimization, which proves the validity of the energy function Eq.(4). Fig.6. Comparison of structural profiles before and after optimization. Each column shows the corresponding before-optimization (top) and after-optimization (bottom) fibers of one subject. The ROI (right precuneus) is presented by the red sphere. 3.2 Optimization using anatomical and functional connectivity profiles In this section, we optimize the locations of ROIs using anatomical and functional profiles, aiming to validate the definition of functional connectivity constraint energy E f . If this energy constraint worked well, the functional connectivity variance of the working memory system across subjects would decrease. Fig.7 shows the comparison of the standard derivation for functional connectivity before (left) and after (right) optimization. As we can see, the variance is significantly reduced after optimization. This demonstrated the effectiveness of the defined functional connectivity constraint energy. Fig.7. Comparison of the standard derivation for functional connectivity before and after the optimization. Lower values mean more consistent connectivity pattern cross subjects. 3.3 Consistency between optimization of functional profiles and structural profiles Fig.8. Optimization consistency between functional and structural profiles. Top: Functional profile energy drop along with structural profile optimization; Bottom: Structural profile energy drop along with functional profile optimization. Each experiment was repeated 15 times with random initial ROI locations that met the anatomical constraint. The relationship between structure and function has been extensively studied [24], and it is widely believed that they are closely related. In this section, we study the relationship between functional profiles and structural profiles by looking at how the energy for one of them changes while the energy of the other decreases. The optimization processes in Section 3.1 and 3.2 were repeated 15 times respectively with random initial ROI locations that met the anatomical constraint. As shown in Fig.8, in general, the functional profile energies and structural profile energies are closely related in such a way that the functional profile energies tend to decrease along with the structural profile optimization process, while the structural profile energies also tend to decrease as the functional profile is optimized. This positively correlated decrease of functional profile energy and structural profile energy not only proves the close relationship between functional and structural profiles, but also demonstrates the consistency between functional and structural optimization, laying down the foundation of the joint optimiza tion, whose results are detailed in the following section. 3.4 Optimization connectivity profiles using anatomical, structural and functional In this section, we used all the constraints in Eq. (1) to optimize the individual locations of all ROIs in the working memory system. Ten runs of the optimization were performed using random initial ROI locations that met the anatomical constraint. Weighting parameter ď Ź equaled 0.5 for all these runs. Starting and ending temperatures for the simulated annealing algorithm are 8 and 0.05; Boltzmann constant K  1 . As we can see from Fig.9, most runs started to converge at step 24, and the convergence energy is quite close for all runs. This indicates that the simulated annealing algorithm provides a valid solution to our problem. By visual inspection, most of the ROIs move to more reasonable and consistent locations after the joint optimization. As an example, Fig.10 depicts the location movements of the ROI in Fig. 6 for eight subjects. As we can see, the ROIs for these subjects share a similar anatomical landmark, which appears to be the tip of the upper bank of the parieto-occipital sulcus. If the initial ROI was not at this landmark, it moved to the landmark after the optimization, which was the case for subjects 1, 4 and 7. The structural profiles of these ROIs are very similar to Fig.6. The results in Fig. 10 indicate the significant improvement of ROI locations achieved by the joint optimization procedure. Fig.9. Convergence performance of the simulated annealing . Each run has 28 temperature conditions. Fig.10. The movement of right precuneus before (in red sphere) and after (in green sphere) optimization for eight subjects. The

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