Non redundant functional brain connectivity in schizophrenia

Abstract

Schizophrenia is considered a disorder of abnormal brain connectivity. Although whole brain maps of averaged bivariate voxel correlations have been successfully applied to study connectivity abnormalities in schizophrenia these maps do not adequately explore the multivariate nature of brain connectivity. Here we adapt a novel method for high-dimensional regression (supervised principal component regression) to estimate brain maps of multivariate non redundant connectivity (NRC) from resting functional Magnetic Resonance Imaging (fMRI) data of 116 patients with schizophrenia and 122 matched controls. Disorder related differences in NRC involved caudate hyper-connectivity and hypo-connectivity of several cortical areas such as the dorsal cingulate, the cuneus and the right postcentral cortex. These abnormalities were coupled with abnormalities in the amplitude of signal fluctuations and, to a minor extent, with differences in the dimensionality of connectivity patterns as quantified by the number of supervised principal components. Second level seed correlation analyses linked the observed abnormalities to an additional set of brain regions relevant to schizophrenia such as the thalamus and the temporal cortex. The non redundant connectivity maps proposed here are a new tool that will complement the information provided by other already available voxel based whole brain connectivity measures.

Appendix 1 Development of a computationally efficient formula to calculate NRC values

Equation 2 of the main text involves a strong reduction in the dimensionality of the independent variable set from p to m variables (with m < N) making the calculation of a multiple correlation feasible. Still, the computational burden of fitting Eq. 2 for each one of the grey matter voxels is very high. Luckily, as shown below Eq. 2 can be estimated without the need to obtain the PCs or their correlations with Y at each single voxel.

We start with the definition of

$$ PC={L}^T\;X $$

(A1)

where L ^T is the matrix of m orthogonal and normalized eigenvectors related to the m components (given in rows). Each one of the components PC _i has a different variance given by its respective eigenvalue Λ _i (and Λ is the diagonal covariance matrix of PC).

Now, due to the scale invariance of correlations, we can consider an alternative formula for Eq. 2 based on normalized components $NPC=\left\{P{C}_1/\sqrt{\varLambda_1},P{C}_2/\sqrt{\varLambda_2},\dots, P{C}_m/\sqrt{\varLambda_m}\right\}$

$$ {R}^2=\mathrm{c}\mathrm{o}\mathrm{r}\;{\left(Y,NPC\right)}^T\;\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,NPC\right) $$

(A2)

and since Y and NPC have unit variance

$$ \mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,PC\right)=\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,NPC\right)=\mathrm{c}\mathrm{o}\mathrm{v}\;\left(Y,NPC\right)=\sqrt{\varLambda^{-1}}\;\mathrm{c}\mathrm{o}\mathrm{v}\;\left(Y,PC\right) $$

which, by considering Eq. A1, becomes $\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,PC\right)=\sqrt{\varLambda^{-1}}\;\mathrm{c}\mathrm{o}\mathrm{v}\;\left(Y,{L}^TX\right)=\sqrt{\varLambda^{-1}}\;{L}^T\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,X\right)$.

Finally, Eq. 2 (main text) may be reduced to

$$ {R}^2={\left[{L}^T\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,X\right)\right]}^T{\varLambda}^{-1}{L}^T\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,X\right) $$

And since Λ ^− 1 is diagonal, it may be further simplified to a sum

$$ {R}^2={\displaystyle \sum_{i=1}^m\frac{1}{\varLambda_{ii}}{\left[{L}_i^T\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,X\right)\right]}^T{L}_i^T\mathrm{c}\mathrm{o}\mathrm{r}\;\left(Y,X\right)} $$

(A3)

in which every term is the squared correlation between each component and Y.

In summary, since both eigenvectors (L) and eigenvalues (Λ) are obtained from the diagonalization of correlation matrices, once the matrix containing correlations between each pair of voxels is initially calculated there is no need to operate with X anymore.

Finally, computational time can be further reduced by implementing sub-sampling strategies, which are acceptable considering the high levels of spatial autocorrelation in fMRI images. In this study a sub-sample of randomly chosen voxels was used for the computation of NRC at each voxel.