Estimating brain’s functional graph from the structural graph’s Laplacian Abdelnour F. a , Dayan M.a , Devinsky O.b , Thesen T.b and Raj A.a a Weill
Cornell Medical College, New York City, USA; Medical Center, New York City, USA
b NYU
ABSTRACT The interplay between the brain’s function and structure has been of immense interest to the neuroscience and connectomics communities. In this work we develop a simple linear model relating the structural network and the functional network. We propose that the two networks are related by the structural network’s Laplacian up to a shift. The model is simple to implement and gives accurate prediction of function’s eigenvalues at the subject level and its eigenvectors at group level. Keywords: brain network, Laplacian, functional network, structural network
1. INTRODUCTION The interplay between the brain’s function and structure has been of immense interest to the neuroscience and connectomics communities. Elucidation of the large scale function-structure relationship can potentially open doors to novel and useful approaches for mapping and treating brain diseases. Structural connectivity (SC) consists of mapping the direct anatomical connections between cortical and subcortical regions of the brain obtained from DTI.1–3 Functional connectivity (FC) is usually defined as the temporal correlation of neurophysiological time series obtained via such modalities as fMRI or EEG,4 and FC may be task-oriented or resting state. Several works have independently indicated strong correlation between functional and structural networks, where the former appears to be restricted by the latter.5–8 In this work we revisit the graph diffusion model introduced in9 and recast it in terms of the eigenvalues and eigenvectors of the structural network’s normalized Laplacian, which we show can closely predict the functional network. Specifically, in the proposed model the estimated eigenvectors of the functional adjacency matrix are simply those of the Laplacian, while the estimated functional eigenvalues are the exponential of the Laplacian’s eigenvalues up to a shift. The latter in fact matches well the relationship between the true FC and Laplacian eigenvalues. Additionally, we show that the estimated FC is essentially captured by only a subset of the Laplacian eigenvectors spanning low graph frequency components. The proposed model yields improved estimates of function from structure as compared with our previous work on estimating function via graph diffusion methods.9 Fig 1(a) gives the mean structural adjacency matrix over a group of healthy subjects. Fig 1(b) gives the mean functional adjacency matrix taken over the same group. Fig 1(c) gives the estimated FC network obtained via the graph diffusion model proposed in,9 and lastly Fig 1(d) gives the estimated functional network using L’s eigen components.
2. THEORY In a brain network each node represents a GM region located on either the neocortex or in deep brain subcortical areas. We define a network G = (V, E) with a set of N nodes given by V = {vi | i ∈ 1, . . . , N } and a set of edges given by an ordered node pair E = {(i, j) | i ∈ V, j ∈ V}.10 Between any two nodes i and j there might exist a fibre pathway whose connectivity weight ci,j ∈ [0, ∞) can be measured from dMRI tractography, and this defines a connectivity matrix C = {ci,j |(i, j) ∈ E}. Although some individual neurons are known to be directional, dMRI does not allow measurement of directionality. Major fiber bundles resolvable by dMRI, Further author information: (Send correspondence to Abdelnour F.) Abdelnour F.: E-mail:
[email protected]. Wavelets and Sparsity XVI, edited by Manos Papadakis, Vivek K. Goyal, Dimitri Van De Ville, Proc. of SPIE Vol. 9597, 95970N · © 2015 SPIE · CCC code: 0277-786X/15/$18 doi: 10.1117/12.2203566 Proc. of SPIE Vol. 9597 95970N-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/14/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx
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Figure 1. (a) Mean structural connectivity matrix, (b) Functional connectivity matrix.
especially cortico-cortical pathways are generally bidirectional, having roughly equal number of connections in either direction.11 We define the connectivity strength or the weighted degree of a node i in this graph as the sum of all connection weights: X ci,j . (1) δi = j|(i,j)∈E
The Laplacian of the structural network C is given by L = I − ∆−1/2 C∆−1/2 where ∆ is a diagonal matrix with ∆ii = δi . Then eigenvectors and eigenvalues of L are given respectively by Ul and Λl . Similarly, let Cf denote the functional network’s connectivity matrix, with Cf = Uf Λf U′f . We model the functional eigenvalues Λf and their Laplacian counterparts Λl as related by a simple exponential relationship: b f = ae−αΛl + bI Λ (2)
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b f U′ , b f = Ul Λ C l
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a α b
Full 11.53 ± 1.43 4.03 ± 0.51 −0.76 ± 0.06
Left 6.26 ± 0.71 2.69 ± 0.47 −0.84 ± 0.11
Right 6.38 ± 0.73 2.76 ± 0.39 −0.82 ± 0.08
Table 1. Mean and standard deviation values of parameters {a, α, b} for the cases of full brain network, left hemisphere, and right hemisphere.
3. RESULTS We show that the SC and FC networks reveal properties predicted by the proposed model, namely the exponential relationship between the eigenvalues λf and λl , and the tight relationship between the corresponding eigenvectors. The Kendall tau functional networks Cf and the normalized symmetric Laplacian L of the structural network matrices were obtained for all subjects. Next, the Laplacian (L) and functional (Cf ) eigenvalues (Λl ,Λf ) and eigenvectors (Ul ,Uf ) were obtained. For each subject an exponential curve fit of the {λf } and {λl } eigenvalues of the form ae−αx + b was computed. To evaluate the accuracy of the FC estimate, we compute the Pearson correlation between the estimated and the true FC matrices, excluding the matrices’ diagonal elements.
3.1 Predicting FC from Λl & Ul In both the graph diffusion and Laplacian eigen decomposition models we would expect an exponential relationship between the functional eigenvalues λf and the Laplacian ones, λl . Additionally, the matrix U = U′f Ul should be nearly identity. Due to across the subjects variability we consider instead the product T = U′l Λf Ul . We evaluate the mean value of T over all subjects. If the matrix U indeed approximates identity, then we would expect T to be a near diagonal matrix. FC matrix is predicted from the Laplacian eigenvectors Ul and eigenvalues Λl , following equation (4), where parameters {a, α, b} are estimated from curve fitting of λf vs λl stacked over all subjects for the cases of full network, right, and left hemispheres. Fig 2(a) gives the true λf vs λl eigenvalues plot for all subjects. FC eigenvalues Λf closely approximate an exponential curve of the form ae−αx + b, with α = 4.08. Fig 2(b) gives the mean Laplacian eigenvalues λl over all subjects as a diagonal matrix. A semi-log scatter plot of all subjects’ FC eigenvalues λf vs the Laplacian eigenvalues λl is depicted in Fig 2(c). As expected, the plot reveals an approximately linear relationship between log(λf ) and λl for all subjects. The eigenvectors’ function-structure relationship is observed at group level, where we compute the matrix T = U′ Λf U for all subjects and compute mean T. The resulting matrix is given in Fig 2(d) where the matrix is nearly diagonal, similar to the true mean eigenvalues of Fig 2(b), suggesting that at least at group level eigenvector matrices Ul and Uf are nearly equal. We predict the FC eigenvalues λf given a subject’s Laplacian matrix L. First the coefficients {a, α, b} are estimated by stacking all subjects’ FC and Laplacian eigenvalues then performing a curve fit. For the case of the full network we obtain {a, α, b} = {11.46, 4.01, −0.76}, while the right and left hemispheres give respectively {a, α, b} = {6.34, 2.76, −0.81} and {a, α, b} = {6.21, 2.68, −0.82}. Fig 3 gives the scatter plots of the estimated FC eigenvalues using global values of the parameters {a, α, b} for the cases of the full network and the right hemisphere. For both cases the prediction is less accurate for the larger values of true λf . Table 3.1 provides the individual subjects’ mean and standard deviation of parameters {a, α, b} for the full network, left hemisphere, and right hemisphere. For the purpose of predicting FC eigenvalues, the global coefficients {a, α, b} are estimated by stacking λf and λs of all subjects then performing a curve fitting. The obtained parameters are then used for predicting the functional eigenvalues from the Laplacian eigenvalues.
4. CONCLUSION In this work we show that the graph diffusion property of brain networks relating structural and functional connectivities is preserved in TLE-MTS and TLE-no patients. This suggests that the brain shows plasticity by adjusting to epilepsy-related insults and preserving a graph diffusion nature of the functional and structural networks. This suggests that the graph diffusion relationship between function and structure may be optimal for the brain operation in some sense. For future work, we plan to incorporate networks’ nodes and edges delays for a more realistic modelling using EEG data.
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Mean FC eigenvalues, full network
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Figure 2. (a) Curve fitting when all subjects’ Laplacian and FC eigenvalues are stacked. The curve is exponential with α = 4.08. (b) Mean eigenvalues over the entire group. (c) Scatter semi-log plot of subjects’ Laplacian and FC eigenvalues, largely a linear plot. (d) Matrix obtained from mean U′l Cf Ul over all subjects. Resulting matrix is nearly diagonal.
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Figure 3. Scatter plots of the estimated functional eigenvalues vs true FC eigenvalues: full network (a), left hemisphere (b), and right hemisphere (c). For all cases the eigenvalues are accurately captured using the graph diffusion model.
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ACKNOWLEDGMENTS AR and FA were supported by the NIH grant R01 NS075425.
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