Identification of multi-scale hierarchical brain functional networks using deep matrix factorization

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1 Identfcaton of mult-scale herarchcal bran functonal networks usng deep matrx factorzaton Hongmng L, Xaofeng Zhu, Yong Fan Center for Bomedcal Image Computng and Analytcs, Department of Radology, Perelman School of Medcne, Unversty of Pennsylvana Abstract. We present a deep sem-nonnegatve matrx factorzaton method for dentfyng subject-specfc functonal networks (FNs) at multple spatal scales wth a herarchcal organzaton from restng state fmri data. Our method s bult upon a deep sem-nonnegatve matrx factorzaton framework to jontly detect the FNs at multple scales wth a herarchcal organzaton, enhanced by group sparsty regularzaton that helps dentfy subject-specfc FNs wthout loss of nter-subject comparablty. The proposed method has been valdated for predctng subject-specfc functonal actvatons based on functonal connectvty measures of the herarchcal mult-scale FNs of the same subjects. Expermental results have demonstrated that our method could obtan subject-specfc mult-scale herarchcal FNs and ther functonal connectvty measures across dfferent scales could better predct subject-specfc functonal actvatons than those obtaned by alternatve technques. Keywords: bran functonal networks, mult-scale, herarchcal, subjectspecfc, deep matrx factorzaton 1 Introducton The human bran can be represented as a multscale herarchcal network [1, 2]. However, exstng functonal bran network analyss studes of restng-state functonal magnetc resonance magng (rsfmri) data typcally defne network nodes at a specfc scale, based on regons of nterest (ROIs) obtaned from anatomcal atlases or functonal bran parcellatons [3-5]. Recent work has demonstrated mportant subjectspecfc varaton n the functonal neuroanatomy of large-scale bran networks [6], emphaszng the need for tools whch can flexbly adapt to ndvduals varaton whle smultaneously mantanng correspondence for group-level analyses. To capture subject-specfc FC wthout loss of nter-subject comparablty, datadrven bran decomposton methods have been wdely adopted to dentfy spatal ntrnsc functonal networks (FNs) and estmate functonal network connectvty. In order to obtan subject-specfc FNs from rsfmri data of ndvdual subjects whle facltatng groupwse nference, ndependent vector analyss (IVA) and groupnformaton guded ICA (GIGICA) methods have been proposed [7, 8]. More recently, several methods have been proposed to dscover FNs from rsfmri data wth nonndependence assumptons [9-11], and drectly work on ndvdual subject fmri data

2 and smultaneously enforce correspondence across FNs of dfferent subjects by assumng that loadngs of correspondng FNs of dfferent subjects follow Gaussan [9] or delta-gaussan [10] dstrbutons. Non-negatve matrx decomposton technques have been adopted to smultaneously compute subject-specfc FNs for a group of subjects regularzed by group sparsty n order to separate ant-correlated FNs properly so that ant-correlaton nformaton between them could be preserved [11, 12]. However, these methods are not equpped to characterze mult-scale herarchcal organzaton of the bran networks [2]. Although clusterng and module detecton algorthms could be adopted to detect the herarchcal organzaton of bran networks, ther performance s hnged on the network nodes/fns used [13, 14]. To address the aforementoned lmtatons of exstng technques, we develop a novel bran decomposton model based on a collaboratve sparse bran decomposton approach [11] and deep matrx factorzaton technques [15], amng to dentfy subject-specfc, mult-scale herarchal FNs from rsfmri data. Based on rsfmri data and task actvaton maps of unrelated subjects from the HCP dataset [16], we have quanttatvely evaluated our method for predctng subject-specfc functonal actvatons based on functonal connectvty measures of the herarchcal mult-scale FNs of the same subjects. Expermental results have demonstrated that the mult-scale herarchcal subject-specfc FNs dentfed by our method from rsfmri data could better predct the subject-specfc functonal actvatons evoked by dfferent tasks than those dentfed by alternatve technques. Fg. 1. Framework of the deep decomposton model for dentfyng mult-scale herarchcal subject-specfc functonal networks (a two-scale decomposton s llustrated). 2 Methods A deep matrx decomposton framework s proposed to dentfy FNs at multple scales as schematcally llustrated n Fg. 1. Partcularly, a deep sem-nonnegatve matrx factorzaton s adopted to jontly detect a herarchy of FNs from fne to coarse spatal scales n a data-drven way, and a group sparsty regularzaton term s adopted for FNs of dfferent subjects at each scale to ensure the subject-specfc FNs to share smlar spatal patterns. Besdes enforcng the groupwse correspondence of FNs across subjects, the group sparsty term also encourages FNs at the fnest scale to have sparse spatal patterns and FNs at coarse scales to comprse functonally correlated FNs at fner scales. Our decomposton model s further enhanced by a data localty

3 regularzaton term that makes the decomposton robust to magng nose and mproves spatal smoothness and functonal coherences of the subject specfc FNs. 2.1 Deep sem-nonnegatve matrx factorzaton for bran decomposton Gven rsfmri data X R T S of subject, consstng of S voxels and T tme ponts, we am to fnd K j nonnegatve FNs V K j = (V j:k,s ) R j S + and ther correspondng tme courses U j = (U j:t,k ) R T K j at j = 1,, h scales, so that X U 1 V 1 ; X U 2 V 2 V 1, V 2 = V 2 V 1 ; ; X U h V h V 1, V h = V h V 1. The FNs V j at 1 j h scales are constraned to have a herarchcal structure and to be non-negatve so that each FN does not contan any ant-correlated functonal unts. A deep sem-nonnegatve matrx factorzaton (DSNMF) framework smlar to [15] s adopted to dentfy the mult-scale herarchcal FNs by optmzng mn X U h V h V 1 2 {U j,v j } F, s. t. V j 0, V 1 = V 1, 1 j h. (1) The FNs at dfferent scales are represented by V h = V h V 1, 1 j h for subject, and the FNs at 2 consecutve scales are lnked herarchcally accordng to weghts determned durng the jont decomposton. The decomposton model does not constran tme courses U h to be non-negatve so that t can be appled to preprocessed fmri data wth negatve values. 2.2 DSNMF based collaboratve bran decomposton Gven a group of n subjects, each havng fmri data X R T S, = 1,, n, we dentfy subject-specfc, multscale herarchcal FNs by optmzng a jont model wth ntegrated data fttng and regularzaton terms as llustrated n Fg. 1: n mn X U h V h V 1 2 F + λ c,j R c,j + λ M R M, {U j,v j } =1 s. t. V j 0, V j:kj, = 1, 1 k j K j, 1 n, 1 j h, where λ c,j = α n T T and λ K M = β are parameters for balancng data fttng and j K 1 n M regularzaton terms, T s the number of tme ponts, K j s the number of FNs at scale j, n M s the number of spatally neghborng voxels at voxel level, α and β are 2 parameters, R c,j and R M are regularzaton terms. Partcularly, R c,j s an nter-subject group sparsty term that enforces FNs of dfferent subjects to have common spatal structures at the same scale j. The group sparsty regularzaton term s defned as S j K R c,j = j K ( (V j:k,s ) k=1 V j:k, = n s=1 =1 2 ) 1/2 j k=1 S j for each row of V j, 1 n, 1 j h, 1,..,n 2,1 h ( s=1 n =1(V j:k,s ) 2 ) 1/2 where S j = K j 1 when j > 1 and S 1 = S. The group sparsty regularzaton (term 1 n Fg.1) enforces correspondng FNs of dfferent subjects to have non-zero elements at the same spatal locatons. Moreover, t encourages FNs to have spatally localzed loadngs. It s worth notng that the group sparsty term does not force dfferent FNs to be non-overlappng, thus certan functonal unts may be ncluded n multple FNs j n =1 (2)

4 smultaneously at each scale. We also adopt a data localty regularzaton term (term 2 n Fg.1), R M, to encourage spatal smoothness and functonal coherence of the FNs usng a graph regularzaton technque [17] at the fnest spatal scale, whch s defned as R M = Tr(V 1 L M (V 1 ) ), where L M = D M W M s a Laplacan matrx for subject, W M s a parwse affnty matrx to measure spatal closeness or functonal smlarty between voxels, and D M s ts correspondng degree matrx, the affnty between each par of spatally connected voxels s calculated as (1 + corr(x,a, X,b ))/2, where corr(x,a, X,b ) s the Pearson correlaton coeffcent between ther rsfmri sgnals, and others are set to zero so that W M had a sparse structure. We optmze the jont model usng an alternatve update strategy. When U h and V k (k j) are fxed, V j s updated as V j = V j [U j X ] + V j +[U j U j ] V j V jvj +λ c,j V j G L21 j /(G L2 j ) 3 [U j X ] V j +[U j U j ] + V j V j Vj +λ c,j V j /(G j G j L2 ) f j > 1 and V 1 = V 1 [U 1 X ] + +[U 1 U 1 ] V 1 +λ M V 1 W M +λ c,1 V 1 G L21 1 /(G L2 1 ) 3 f j = 1 [U 1 X ] +[U 1 U 1 ] + V 1 +λ M V 1 D M +λ c,1 V 1 /(G 1 G L2 1 ) L21 S = repmat( ( n =1 (V j:k,s G j:k, G j:k, s=1 ) 2 ) 1/2, 1, S), S s=1 n =1 ) 2 ) 1/2, 1, S), L2 = repmat(( (V j:k,s G j:k,s = ( n =1 (V j:k,s ) 2 ) 1/2, V j = V j 1 V j 2 V 1, where denotes element-wse multplcaton, repmat(b, r, c) denotes matrx obtaned by replcatng r and c copes of vector b n the row and column dmensons, [a] + = abs(a)+a, [a] 2 = abs(a) a, M 2 j:k, denotes the k-th row of matrx M j, and M j:k,s denotes the (k, s)-th element n the matrx. V j s normalzed by the row-wse maxmum value along the row dmenson after each update teraton. When V k, 1 k j are fxed, U j s updated as U j = X j (V j V j 1 V 1 ), (4) where M denotes the Moore-Penrose pseudo-nverse of matrx M. To expedte the convergence of the optmzaton, a pre-tranng step at group level s adopted before the jont optmzaton. In partcular, we compute (U 1, V 1 ) sparsenmf(x) frst, where X denotes the temporal concatenated data of { X, = 1,.., n}, and (U 2, V 2) sparsenmf(u 1 ),, (U h, V h) sparsenmf(u h 1 ), respectvely. {V 1, V 2,, V h} are then used to ntalze {V j, = 1,, n, j = 1,, h} for the jont optmzaton. {V 1, V 2,, V h} contans a herarchcal structure wth overlappng as they are obtaned by decomposton n a greedy manner. Once the ntalzaton s done, all FNs at dfferent scales are optmzed jontly. We set the parameters α to 1, and β to 10 accordng to [11] n the present study. (3) 3 Expermental results We evaluated our method based on rsfmri data and task actvaton maps of 40 unrelated subjects obtaned from the Human Connectome Project (HCP) [16], amng to evaluate the performance of predctng task-evoked actvaton responses based on

5 functonal connectvty measures of FNs at multple scales. The FNs derved from rsfmri data have demonstrated promsng performance for predctng task-evoked bran actvatons [18]. The proposed decomposton model was appled to the mnmal-preprocessed, cortcal gray-coordnates based rsfmri data. The number of scales was set to 3, and the number of FNs at the frst scale was set to 90, whch was estmated by MELODIC automatcally [19], the numbers of FNs at the 2 nd and 3 rd scales were set to 50 and 25 respectvely, whch were decreased by half approxmately along the scales. We compared the proposed model wth two alternatve decomposton strateges: mult-scale decomposton performed ndependently at dfferent scales, and greedy agglomeratve herarchcal mult-scale decomposton, wth the same settng of scales and the same numbers of FNs. For the ndependent decomposton, the collaboratve sparse decomposton model was adopted to obtan subject-specfc FNs ndependent at 3 scales (wth ndependent ntalzaton). For the agglomeratve one, the ntalzaton obtaned by greedy decomposton as descrbed secton 2.2 was adopted for each scale, but the fnal decomposton at each scale was obtaned usng the collaboratve sparse model separately. (a) (b) Fg. 2. Mult-scale functonal networks dentfed by dfferent methods. (a) A herarchy of FNs obtaned by the proposed method correspondng to sensormotor networks. FNs at dfferent scales are denoted by boundng boxes n dfferent colors. (b) Sensormotor regons n separate FNs at the coarsest scale (the 3 rd ) dentfed ndependently of other scales. 3.1 Mult-scale bran functonal networks wth a herarchcal organzaton An example herarchy of FNs (mean of 40 subjects) correspondng to sensormotor functon obtaned by the proposed method s llustrated n Fg. 2 (a). The FN at the 3 rd scale (top left) comprsed the sensormotor networks and part of vsual networks, and was a weghted composton of FNs at the 2 nd scale whle the FNs at the 2 nd scale were composed of FNs at the 1 st scale. Ths example herarchy of FNs llustrated that FNs correspondng to sensormotor functon gradually merged from fne to coarse scales n the herarchy. However, no clear herarchcal organzaton was observed for

6 FNs ndependently dentfed at dfferent scales. Partcularly, as shown n Fg. 2 (b) sensormotor regons appeared n separate FNs at the coarsest scale (the 3 rd ), nstead of formng a sngle FN. We postulated that the ndependent decomposton at dfferent scales favored to better data-fttng and therefore was affected by data nose, whle the jont decomposton model was more robust to data nose, facltatng accurate dentfcaton of FNs wth coherent functons, such as the FN comprsng sensormotor regons shown n Fg. 2 (a). 3.2 Predcton of task evoked actvatons based on mult-scale FN connectvty As no ground truth s avalable for FNs derved from rsfmri data, we evaluated the mult-scale herarchcal FNs for predctng functonal actvatons evoked by dfferent tasks based on ther functonal connectvty measures wth an assumpton that better FNs could provde more dscrmnatve nformaton for predctng the bran actvatons. We also compared FNs obtaned usng dfferent strateges n terms of ther predcton performance. Fg. 3. Quanttatve comparson of predcton performance for functonal actvatons of dfferent tasks (top: motor task events ncludng left foot, left hand, and tongue; bottom: workng memory task events ncludng 2bk_body, 2bk_face, and 0bk_tool). s1 to s3, g1 to g3, and h1 to h3 denote models traned based on FNs at sngle scale dentfed ndependently, n a greedy agglomeratve manner, and herarchcally by our method, respectvely, whle m_s, m_g and m_h denote models traned based on correspondng mult-scale FNs. The models bult upon FNs obtaned by our method performed sgnfcantly better than others (Wlcoxon sgned rank test, p < 0.02). Partcularly, a whole bran voxelwse functonal connectvty (FC) map was obtaned for each FN by computng voxelwse Pearson correlaton coeffcent between the FN s tme course and every cortcal gray-coordnate s tme course of the rsfmri data. All the FC maps were then transferred to z-score maps usng Fsher Z transformaton. All the z-score values of FNs on each cortcal gray-coordnate were used as features to predct ts actvaton measures under dfferent tasks of the same subject. Smlar to [18], the whole cortcal surface was dvded nto 90 parcels accordng to FNs obtaned at the fnest scale, and one ordnary least square model was traned for each parcel and every task event. The predcton performance was evaluated usng a

7 leave-one-subject-out cross-valdaton, where one subject s actvaton was predcted by a model bult upon data of the remanng 39 subjects. The predcton was conducted usng sngle-scale FNs (90, 50, 25) or mult-scale FNs (165) obtaned usng dfferent strateges respectvely, and the predcton accuracy was evaluated as the Pearson correlaton coeffcents between the predcted and real actvaton maps of 47 task events from 7 tasks categores. The predcton performance of 6 randomly selected task events s llustrated n Fg. 3. For all task events and FNs dentfed by dfferent strateges, the predcton models bult upon mult-scale FNs outperformed all predcton models bult upon any sngle scale FNs alone, ndcatng that mult-scale FNs could provde complementary nformaton for the task actvaton predcton. The predcton models bult upon mult-scale herarchcal FNs obtaned by our method had sgnfcantly better performance than those buld upon mult-scale FNs obtaned by ether the ndependent decomposton or the greedy agglomeratve decomposton (Wlcoxon sgned rank test, p < 0.02), ndcatng that the jont optmzaton of mult-scale herarchcal FNs could beneft from each other and characterze the ntrnsc FNs better. 4 Conclusons In ths study, we have developed a deep decomposton model to dentfy mult-scale, herarchcal, subject-specfc FNs wth group level correspondence across dfferent subjects. Our method s bult upon deep sem-nonnegatve matrx factorzaton framework, enhanced by a group sparsty regularzaton and graph regularzaton for mantanng nter-subject correspondence and better functonal coherence. Expermental results based on rsfmri data and task actvaton maps of the same subjects have demonstrated that the mult-scale herarchcal subject-specfc FNs could capture nformatve ntrnsc functonal networks and mprove the predcton performance of task actvatons evoked by dfferent tasks, compared to FNs dentfed at dfferent scales ndependently or n a greedy agglomeratve way. In concluson, our method provdes an mproved soluton for characterzng subject-specfc, mult-scale herarchcal organzaton of the bran functonal networks. 5 Acknowledgements Ths work was supported n part by Natonal Insttutes of Health grants [CA223358, EB022573, DK114786, DA039215, and DA039002]. References 1. Doucet, G., et al., Bran actvty at rest: a multscale herarchcal functonal organzaton. J Neurophysol, (6): p Park, H.J. and K. Frston, Structural and functonal bran networks: from connectons to cognton. Scence, (6158): p

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