A joint subspace mapping between structural and functional brain connectomes

Abstract

Understanding the connection between the brain's structural connectivity and its functional connectivity is of immense interest in computational neuroscience. Although some studies have suggested that whole brain functional connectivity is shaped by the underlying structure, the rule by which anatomy constraints brain dynamics remains an open question. In this work, we introduce a computational framework that identifies a joint subspace of eigenmodes for both functional and structural connectomes. We found that a small number of those eigenmodes are sufficient to reconstruct functional connectivity from the structural connectome, thus serving as low-dimensional basis function set. We then develop an algorithm that can estimate the functional eigen spectrum in this joint space from the structural eigen spectrum. By concurrently estimating the joint eigenmodes and the functional eigen spectrum, we can reconstruct a given subject's functional connectivity from their structural connectome. We perform elaborate experiments and demonstrate that the proposed algorithm for estimating functional connectivity from the structural connectome using joint space eigenmodes gives competitive performance as compared to the existing benchmark methods with better interpretability.

Keywords: Brain connectivity; Eigen decomposition; Functional connectome; Laplacian; Structural connectome.

Fig. 1.

A pair of structural and functional connectomes (from public dataset Griffa et al. (2019) subject # 1) and their respective eigen spectra. The line plot in (b) shows eigen spectra of Laplacian of the structural connectome shown in (a). Similarly, the line plot in (d) displays eigen spectra of the functional connectome in (c). In second row, we show the top four eigenmodes (eigenvectors corresponding to least eigenvalues) of the structural Laplacian. Similarly, in the third row we show the top eigenmodes of functional connectome. The fourth row show the joint eigen spectra for the structural Laplacian and functional connectome. The bottom row shows the top four dominant (with respect to $\mathrm{\Psi }$) joint eigenmodes.

Fig. 2.

Joint eigen spectra for different representative subjects from dataset Griffa et al. (2019). For each subject, left plot is joint eigen spectrum ($\mathrm{\Phi }$) of structural Laplacian and right plot is joint eigen spectrum $\left(\mathrm{\Psi }\right)$ of functional connectome. Across all subjects, it can be noted that the first four joint modes lie in the lower-half of both the structural and functional eigen spectra. However, there is no consistent ordering in the dominant eigenmodes with reference to ascending ordering of joint eigen spectra $\left(\mathrm{\Phi }\right)$ of structural Laplacian.

Fig. 3.

Example of subspace relationship in structure-function joint eigen-spectrum. (a): $\mathrm{\Psi }$ obtained via joint diagonalization. Top $\mathrm{K}=20$ values are marked in red. (b): Plot of Pearson R value as a function of $K$. For each $K$, we estimate functional connectome as in (). The red dotted $\mathrm{x}$-line indicates the instance when estimated functional connectome almost matches with the ground-truth one. In the top example, it is $K=20$ where the Pearson $\mathrm{R}$ is 0.996. Similarly, for $K=30$, the Pearson $\mathrm{R}$ is 0.998 in case of $(219\times 219)$ atlas in the bottom example. The respective estimated functional connectomes are displayed in the last column. It is evident that both the estimated connectomes are visually indistinguishable to the true functional connectomes shown in third column. These examples demonstrate that functional connectome is contained within structural connectome. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4.

Similarity of joint eigenmodes with structural and functional eigenmodes. Pearson R for different combination of (first four) eigenmodes for subject #1 in dataset Griffa et al. (2019): (a) eigenmodes of structural Laplacian of subject #1 vs its joint eigenmodes, (b) eigenmodes of functional connectome of subject #1 vs its joint eigenmodes, (c) eigenmodes of structural Laplacian of subject #1 vs the group-level joint eigenmodes, and (d) eigenmodes of functional connectome of subject #1 vs the group-level joint eigenmodes. A higher value indicates high correlation. With reference to self joint eigenmode of this subject, we observe highest correlation between $\left({\mathit{v}}_4,{\mathit{a}}_4\right)$ pair in (b). For the group-level joint eigenmodes, the highest correlation is found between $\left({\mathit{v}}_2,{\mathit{a}}_2\right)$. In general however, the joint eigenmodes are more similar to the functional eigenmodes than the structural eigenmodes at both the individual and the group level.

Fig. 5.

Group-level joint eigenmodes: first four modes. These modes are obtained using Algorithm 1 on all 68 subjects in the public dataset Griffa et al. (2019).

Fig. 6.

Similarity between 7 canonical networks and group-level (first four) joint eigenmodes of 68 subjects in dataset Griffa et al. (2019). The pair (${\mathit{a}}_2$, default) has the highest similarity in terms of Pearson R metric. The pair ($a_1$, dorsal-attention) has the highest similarity in terms of geodesic metric..

Fig. 7.

Comparison of structure function mapping for a representative subject (#7). The Pearson R and geodesic distance values for the estimated FC (with reference to the ground-truth one) are reported in the sub-captions. A higher R value indicates superior prediction. On the other hand, a lower value of geodesic distance implies better quality. Among all the methods, our proposed method JESM achieves best results in terms of both visual quality and numerical metrics.

Fig. 8.

Performance comparisons of structure-function mapping on 68 healthy subjects from Griffa et al. (2019). Metrics of performance are: (a) Pearson R, (b) Geodesic distance, (c) Structural similarity index measure (SSIM), and (d) Mean square error (MSE).

Fig. 9.

Residual and variance measures on the predicted functional connectivity values for the experimental setting in Fig. 8. Notice that the residual histograms are fairly symmetric distributions with zero mean at (a) intra- vs inter hemispheric and (b) short vs long cases. The mean and skewness of the histograms in (a) are (0,−0.172) and (0,−0.167). This indicates similar residual nature between the intra- and inter-hemispheric regions. The mean and skewness of the histograms of short and long connections in (b) are (0, 0.174) and (0, 0.172). Therefore, a highly similar residual pattern is also found between short vs long connections. The variance of FC estimates across subjects at intra- and inter-hemispheric regions shown in (c) have the same mean and median values as (0.012, 0.011). Similarly, the variance at both short and long connections shown in (d) have mean and median values (0.013, 0.012). Therefore, very similar pattern of variances are found at both (c) intra- vs inter hemispheric and (d) short vs long cases. Moreover, JESM preserves the difference across subjects equally well at both intra- vs inter-hemispheric regions as well as short vs long connections in the brain..

Fig. 10.

Predicting FC on Schizophrenia dataset Vohryzek et al. (2020): group-level joint modes from healthy subjects. We train the model on connectome pairs of 27 control subjects and then estimate the FC of Schizophrenia patients using our proposed method JESM. The geodesic distance between (b, c) pair is 30.07 and (e, f) pair is 31.67.

Fig. 11.

Statistics of the functional connectivity estimation on Schizophrenia patients data Vohryzek et al. (2020): group-level joint modes from healthy subjects.

Fig. 12.

Variance (point-wise) across 68 subjects at each ROI pair. Left: from input functional connectomes. Right: from predicted FCs using our method JESM.

Fig. 13.

Group-level joint eigenmodes of schizophrenia subjects: first four modes. These modes are obtained using Algorithm 1 on all 27 subjects obtained from the public dataset Vohryzek et al. (2020).

Fig. 14.

Visual comparison of structure function mapping on Schizophrenia Subject #2. In this experiment, we perform 3-fold cross-validation on 27 schizophrenia subjects. In particular, data from 18 subjects are used to train the group-level joint modes..

Fig. 15.

Statistics of the functional connectivity estimation on Schizophrenia dataset Vohryzek et al. (2020) with 27 patients. The group-level joint modes are learned from schizophrenia subjects via 3-fold cross validation.