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. 2016 Oct 13;12(10):e1005031.
doi: 10.1371/journal.pcbi.1005031. eCollection 2016 Oct.

Functional Connectivity's Degenerate View of Brain Computation

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Free PMC article

Functional Connectivity's Degenerate View of Brain Computation

Guillaume Marrelec et al. PLoS Comput Biol. .
Free PMC article

Abstract

Brain computation relies on effective interactions between ensembles of neurons. In neuroimaging, measures of functional connectivity (FC) aim at statistically quantifying such interactions, often to study normal or pathological cognition. Their capacity to reflect a meaningful variety of patterns as expected from neural computation in relation to cognitive processes remains debated. The relative weights of time-varying local neurophysiological dynamics versus static structural connectivity (SC) in the generation of FC as measured remains unsettled. Empirical evidence features mixed results: from little to significant FC variability and correlation with cognitive functions, within and between participants. We used a unified approach combining multivariate analysis, bootstrap and computational modeling to characterize the potential variety of patterns of FC and SC both qualitatively and quantitatively. Empirical data and simulations from generative models with different dynamical behaviors demonstrated, largely irrespective of FC metrics, that a linear subspace with dimension one or two could explain much of the variability across patterns of FC. On the contrary, the variability across BOLD time-courses could not be reduced to such a small subspace. FC appeared to strongly reflect SC and to be partly governed by a Gaussian process. The main differences between simulated and empirical data related to limitations of DWI-based SC estimation (and SC itself could then be estimated from FC). Above and beyond the limited dynamical range of the BOLD signal itself, measures of FC may offer a degenerate representation of brain interactions, with limited access to the underlying complexity. They feature an invariant common core, reflecting the channel capacity of the network as conditioned by SC, with a limited, though perhaps meaningful residual variability.

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Summary of analyses.
We performed bootstrap SVD of various measures of structural and functional connectivity. In each case, we represented the dimension of the reproducible linear subspace extracted by bootstrap SVD, the part of variance explained by each dimension (bootstrap average) as well as the (bootstrap) standard deviation associated to all reproducible dimensions.
Fig 2
Fig 2. Empirical SC and FC.
(a) Boxplot of the part of variance explained by the successive linear dimensions obtained by bootstrap SVD for SC (left), correlation based FC (cFC, middle), and mutual information based FC (mFC, right). (b) Average projection of data onto first dimension for SC (left), cFC (middle), and mFC (right). (c) Relationship between SC and FC according to these average projections for cFC (left) and mFC (right). (d) Relationship between empirical cFC and mFC. The dotted line is the relationship that holds in the case of a bivariate normal distribution, see Eq (1). (e) Boxplot of the fraction of variance explained by the successive dimensions obtained by bootstrap SVD of 3-way connectivity (CT3). (f) Relationship between empirical CT3 and CT3 computed under the assumption of a trivariate Gaussian distribution, see Eq (2). The dotted line stands for the identity, that is CT3 = Gaussian CT3. See §3 of S1 Text for a view of the reproducible part of empirical SC, cFC, and mFC projected on anatomical template.
Fig 3
Fig 3. Robustness of simulated FC to dynamics.
(a) Boxplot of the part of variance explained by the successive linear dimension obtained by bootstrap SVD for cFC (top) and mFC (bottom). (b) Average projection of data onto first dimension of cFC: representation (left) and relationship as a function of the average projection of empirical SC onto its first dimension (right). (c) Average projection of data onto first and second dimensions of mFC: representation and relationship as a function of the average projection of empirical SC onto its first dimension. (d) Relationship between simulated cFC and mFC for the different generative models used. The dotted lines stand for the relationship that hold for a bivariate Gaussian distribution, see Eq (1). (e) Boxplot of the part of variance explained by the successive linear dimensions obtained by bootstrap SVD for simulated CT3. (f) Relationship between empirical CT3 and CT3 computed under the assumption of a trivariate Gaussian distribution for the different generative models used, see Eq (2). The dotted line stands for the identity, that is CT3 = Gaussian CT3. See §3 of S1 Text for a view of the reproducible part of simulated cFC and mFC projected on anatomical template.
Fig 4
Fig 4. Joint SVD of simulated and empirical FC.
(a) Boxplot of the part of variance explained by the successive linear dimensions obtained by bootstrap SVD for cFC (left) and mFC (right). (b) Average projection of data onto first (left) and second (right) dimensions for cFC: representation (top) and characterization as a function of the average projection of empirical SC onto its first dimension (bottom). (c) Same information for mFC. (d) Loading of data on the first and second dimensions broken down by model and empirical data, for cFC (left) and mFC (right). See §3 of S1 Text for a view of the reproducible part of cFC and mFC projected on anatomical template for pooled data.
Fig 5
Fig 5. Joint SVD of simulated and empirical FC (continued).
(a) Bootstrap SVD for residual cFC, computed as the difference between empirical and simulated cFC: boxplot of the part of variance explained by the successive linear dimensions (top), average projection of data onto first dimension (bottom left) and relationship of this projection as a function of the average projection of empirical SC onto its first dimension (bottom right). (b) SVD of data from average subject, where empirical SC and FC are computed by averaging across subjects. Simulated FC is obtained by feeding the average SC to the different generative models, either in its original form or after artificially setting all homotopic connections to an arbitrary value. Top: part of variance explained by the successive linear dimensions; bottom left: average projection of manipulated data onto first dimension; bottom right: relationship of this projection with manipulated average SC.
Fig 6
Fig 6. Dynamic FC.
Bootstrap SVD extracted a reproducible linear space of dimension one for all window sizes but the shortest, where no reproducible linear space was found. (a) Part of variance explained by the first dimension as a function of the size of the time window used to compute dynamic FC. (b) First dimension of dynamic FC as a function of the first dimension of cFC. (c) First dimension of dynamic FC as a function of the first dimension of empirical SC.
Fig 7
Fig 7. Inverting empirical FC.
(a) Average projection of all (empirical and simulated) data onto first reproducible linear dimension as a function of the average projection of empirical SC onto its first reproducible linear dimension thresholded at SC ≥10−6 and transformed according to SC′ = ln(SC/0.0001) (dots). The red line stands for a 4-order polynomial interpolation approximation. (b) SC estimated by inverting the previous relationship applied to empirical cFC. (c) As a reference, average projection of empirical SC onto its first reproducible linear dimension.

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Grant support

AM is supported by Deutsche Forschungsgemeinschaft (DFG) grant SFB 936/Z3. DR is supported by the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement #616268 “F-TRACT”. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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