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.
The processing and routing of information in the brain are implemented through the interplay of two general constraints: 1) the specifics of physiological dynamics, controlling how connected neurons respond to each other locally, and 2) the entire wiring diagram of anatomical connections, channeling the possible exchange of information between neuronal ensembles according to the properties of its cables [1–6]. While large-scale anatomical connections are expected to remain relatively stable over time, neuronal processes may be expected to produce complex transient patterns related to different processing phases, either in a stationary or non-stationary manner [7]. Since the same anatomical network may be used to implement variable physiological dynamics at different times [5,8, 9], many degrees of freedom could be expected in the relationship between structure and function, that is, between the wiring diagram and the resulting patterns of activity [5,10–12]. In the context of neuroimaging, the wiring diagram of connections, or ‘structural connectivity’ (SC), has generally been estimated using DWI-based tractography [13–15]. In blood oxygen level dependent (BOLD) resting-state functional magnetic resonance imaging (rs-fMRI, where subjects passively lie in the magnet [16,17]), measures of ‘functional connectivity’ (FC) have emerged as a convenient proxy to quantify functional patterns of interaction between neural ensembles [16,18–21]. In this scientific field, standard Pearson correlation has become a gold standard [22–24]; more elaborate measures, though rarely applied so far, have also been proposed, such as mutual information as well as estimators of higher-order relationships [25–30]. If FC were sensitive to features of dynamics related to ongoing cognition, we could expect a relatively large variability and complexity of this measure, in particular as compared to SC. In the resting state, for instance, cognition is allowed to roam free range, and FC could thus be expected to be quite variable across arbitrary acquisition times, both within and between subjects. Beyond theoretical rationale, empirical evidence displays mixed results making FC hard to assess as a measure. Several studies support the hypothesis that FC can express a variety of patterns and complex dynamical behaviors, which can demonstrate empirical correlations with normal and pathological cognitive processes [31]. As a matter of fact, it has already been shown that FC could be sensitive to brain maturity [32], age [33–38], the global level of awareness [8,39], personality traits [40,41], the current mental state [31,42,43], recent experience [44], and varies over time for a given subject [8,39]. It has even been suggested that the between-subject differences could serve as individual fingerprints [45,46]. Likewise, the variability of FC patterns across time, as observed with sliding time windows shorter than a standard acquisition run (‘dynamic FC’) has been recently featured with relation to cognition [47]. Yet, the origins of such differences and variations, whether structural, neuronal, hemodynamic/metabolic, or methodological, and their relative weights in the generation of FC—and thus the potential of FC as a measure—remain to be clarified [38,48–51]. Besides, many other studies have suggested that FC reflects SC [22–24,52–54] and patterns of FC extracted from rs-fMRI have been repeatedly found to be reproducible both within and across subjects [55–57]. Even though robust FC patterns can be related to general task-based networks from activation studies [58–64], overall these results suggested that the sensitivity of both the BOLD signal and the different measures of FC to the variability and complexity of the underlying dynamics might be quite limited [65]. The fact that FC is related to SC (which is relatively reproducible across subjects and stable over time—at least as seen by DWI tractography [66]), suggests a FC that would itself reflect the wiring diagram expressed in SC. Evidence for FC patterns that are both reproducible across subjects and sensitive to various subject-specific factors suggests that FC as usually measured might be composed of an invariant core that accounts for a large part of the variance, and a residual variable part. FC might thus have existing but limited sensitivity to important features of brain dynamics, while being strongly dependent on more invariant factors of interaction such as SC, its communication channel capacity and global steady routing schemes. FC would therefore appear as a degenerate measure of the expected complexity of brain interactions, all the more so that it would reflect SC, which, as a measure of static anatomical wiring diagram, implies the highest degeneracy with respect to brain dynamics [11]. Likewise, it remains unclear how this could depend on the kind of measure used to quantify FC or the time window considered. Addressing these questions is important as they concern the potential of FC for capturing relevant aspects of the complexity of brain dynamics and our ability to infer meaningful neurocomputational properties from such measures. In the present study, we characterized and compared qualitatively and quantitatively the relative variability of patterns of SC and FC, across different measures of FC (correlation, mutual information, 3-way connectivity, dynamic FC). We assessed: 1) the relative sensitivity of FC to variable states and dynamics as putatively underlying different phases and modes of brain activity, and 2) the extent to which FC reflects SC. We used DWI-based estimates of SC and rs-fMRI based estimates of FC from 21 normal participants over a partition of the brain into 160 regions [54]. In addition to the analysis of empirical data, in order to systematically assess the robustness of FC to different types of dynamics, we performed simulations of FC using seven different mainstream computational models of neuronal activity and BOLD response, with a wide range of complexity [54,67,68]. We used a unified analysis framework based on singular value decomposition (SVD) of matrices of connectivity and bootstrap statistics in order to compare patterns of connectivity between and within subjects and assess their variability in a multivariate framework. For each connectivity pattern under investigation, this method extracted a reproducible linear subspace with limited dimension that had the key feature of explaining the most variance or, equivalently, best explaining the data in a least square sense for that dimension (see Fig 1 and Methods).
We then adopted an approach based on computational modeling as a heuristic tool to study the sensitivity of FC to the specificity of the dynamical behaviors controlling interactions between connected regions. We generated simulated FC based on an array of standard computational models, all taking SC as input but with broadly different dynamical behaviors and equations. We wondered whether the patterns of FC generated by the different models would be able to fit empirical FC in the same manner or would retain irreducible components proper to the specific properties of the models (see Methods). We then assessed whether pooled empirical and simulated FC across subjects would project onto the same small number of dimensions by performing a global bootstrap SVD analysis. We found a reproducible linear subspace with dimension 2 (63. 7% ± 1. 6% of variance explained for cFC and 56. 0% ± 0. 5% for mFC) (Fig 4a). In both cases, the first dimension was tightly related to intrahemispheric connections, while the second dimension largely reflected homotopic and, more largely, interhemispheric connections (Fig 4b and 4c). For both cFC and mFC, the first dimension appeared again tightly related to SC, but not the second one. Finally, it was mostly empirical FC (which, contrary to simulations, demonstrated clear patterns of interhemispheric connections) that loaded onto the second dimension (Fig 4d). We reasoned that customary errors of DWI-based SC estimation, concerning more particularly homotopic and interhemispheric connections, could account for the appearance of a second linear dimension when adding empirical FC to simulated FC. Homotopic and interhemispheric connections are known to exist but to be difficult to measure, remaining largely undetected by standard DWI acquisition and processing pipelines [54,69,70]. Such connections were mostly absent of the SC dataset that we used as an input to the generative models. First, we investigated residual cFC by computing the difference between empirical and simulated FC for each generative model and subject. A bootstrap SVD analysis yielded a reproducible linear subspace with dimension 1 (47. 9% ± 1. 7% of explained variance). This dimension saliently featured interhemispheric and, more specifically, homotopic connections (Fig 5a). Thus a substantial amount of the differences between empirical and simulated FC could be reduced to errors pertaining to homotopic connections. Second, we performed an SVD analysis in which simulated and empirical cFC were pooled together but on intrahemispheric connections only, as a way of excluding putative errors of prediction directly and explicitly related to interhemispheric connections. This analysis yielded a reproducible linear subspace with dimension 2. Nonetheless, compared to bootstrap SVD over all (intra- and interhemispheric) connections, the part of variance accounted for by the first dimension (corresponding to the dimension embedding the variability of generative models) increased from 51. 5% ± 1. 2% to 63. 1% ± 1. 0%, while the part of variance accounted for by the second dimension (which corresponds to the dimension hypothesized to bear the functional consequences of SC estimation errors) decreased from 12. 2% ± 0. 6% to 7. 0% ± 0. 4%. Even though we excluded interhemispheric data from the SVD analysis, this approach was not fully satisfactory, since simulated FC was generated to begin with by computational models taking interhemispheric information from SC into account. Thus we might expect that intrahemispheric simulated FC could also suffer from SC estimation errors concerning interhemispheric and homotopic connections. In order to further tackle this issue in a heuristic manner, we performed a new set of simulations and analyses restricted to SC and cFC averaged across subjects, as we could not afford an analysis on individual data at this stage given the heavy computational burden of the simulations [68]. This precluded further bootstrap analysis and thus made the results only descriptive, but we had a directed hypothesis and were only looking for confirming the impact of failing to detect homotopic connections on the observed discrepancies between empirical and simulated FC. We manipulated the average SC by artificially setting homotopic connections to an arbitrary value (0. 5). Two new batches of simulations across all seven computational models were run in order to generate simulated matrices of FC as predicted from the original average SC matrix on the one hand and, on the other hand, from the modified average SC matrix. We hypothesized that adding homotopic connections would increase the variance explained by the first linear dimension and decrease that of the second one in a standard SVD analysis. We observed that adding homotopic connections increased the part of variance explained by the first dimension from 67. 2% to 78. 4%, while it decreased the part of variance explained by the second dimension from 13. 3% to 7. 7% (Fig 5b). Altogether this series of results supports the hypothesis that, independently from unmodeled differences that would be related to the complexity of the dynamics being captured, an incorrect estimation of SC, which is itself related to current limitation in DWI, plays an important role in the observed discrepancies between FC predicted by the different generative models and empirical FC. They also further support the hypothesis of a strong relationships between FC as measured and SC. In all the analyses so far, we computed FC over a whole rs-fMRI session, that is, over time series of about 11 minutes. We hypothesized that in resting state data acquired at arbitrary time points across different subjects, brain computation and dynamics would be maximally independent and that, consequently, such sampling strategy would foster the expression of putative differences in patterns related to this free state, and minimize the impact of more invariant constraints. However, it is possible that computing FC over time series of several minutes tends to bias the sensitivity of the analysis toward the most stationary aspects of the processes, thus missing the relevant variability because of the wrong choice of time scale. We thus conducted a bootstrap SVD analysis, concentrating on empirical data and using cFC, but this time on concatenated matrices of cFC independently computed over shorter time windows. Bootstrap SVD extracted a reproducible linear subspace with dimension 1 for all window sizes (from 32. 9 s to 5 min 29 s) but the shortest (26. 32 s), where no reproducible linear subspace could be extracted. The part of variance explained by the first dimension decreased however from 52. 6% ± 1. 6% for 2 windows of size 5 min 29 s to 21. 5% ± 2. 0% for 20 windows of size 32. 9 s and 19. 8% ± 2. 4% for 25 windows of size 26. 32 s (Fig 6a). This is to be compared to the 59. 8% ± 1. 6% of total variance explained by the analysis with cFC computed over the whole session. Dynamic FC featured a reproducible linear dimension that was similar to that of cFC, with a residual variance that increased as the window size decreased (Fig 6b and 6c). One of the findings of these analyses is that FC from rs-fMRI, as measured with correlation, mutual information, or 3-way connectivity, contained a degenerate projection of the complexity and variability of the dynamics underlying brain computation, which reflects a robust and stable common core. This core of FC also appeared to be clearly related to SC, both considering empirical data and simulations from generative models in which the relation is explicit. We reasoned that the empirical relationship between SC and FC (as measures) suggests that their common features reflect the same underlying anatomical network. Given the customary issues of false negatives and false positives in standard DWI pipelines, measurements of FC from rs-fMRI might thus appear as a fair additional ground for (re) estimating SC in a more encompassing way. While developing and testing a valid and full-fledged procedure for SC estimation through inversion of FC is beyond the scope of this article, we here give a proof of concept in favor of the possibility of such a procedure. The procedure relies on two main assumptions: (i) There is a simple, one-to-one relationship between SC and FC which is relatively insensitive to dynamical regimes (and that loads on the first component of the SVD); and (ii) deviation from this relationship (at the level of this first component) is mostly a consequence of incorrect estimation of SC. According to (i), one could quantify the relationship between SC and FC from simulations and then apply its inverse to empirical FC. More specifically, from the analyses, we were led to the conclusion that SVD on the pooled data extracted a first component that is a reflection of the underlying steady functional-anatomical organization, and a second one that is related to errors in SC estimation. We reasoned that, should DWI provide correct SC, the second component of SVD on the pooled data would be associated to vanishing variance, while the fraction of variance explained by the first component would increase to 1, and the relationship between SC and FC would appear on this component. As a consequence, the empirical data would mostly project on the first component. We then also postulated the relative stability of the relationship between SC and FC regardless of the quality of estimation of SC by DWI. In other words, a better estimation of SC would reduce the fraction of variance explained by the second component and increase the fraction of variance explained by the first component but not drastically change the relationship already observed between SC and FC on the first component. Practically, we first quantified the relationship observed between the first reproducible linear dimension of empirical SC and that of the pooled data (Fig 7a), then applied the inverse of this transform to empirical FC (see Methods). Such an approach produced an estimated matrix of SC that was similar to, but richer than, the original DWI-based SC matrix, featuring more interhemispheric connections, especially homotopic ones (Fig 7b and 7c). Beyond this preliminary example, a Bayesian iterative scheme could be developed in the future, embedding a similar approach, and aiming at converging towards optimal estimates of SC.
From a theoretical standpoint, the existence of a low-dimensional approximation of FC from SVD implies that strong linear relationships control FC variability, whereas approximations of significantly higher dimension (or no approximation at all) would imply more complex relationships. Our findings suggest that, to the least, FC as measured by these different metrics from rs-fMRI is a strongly degenerate representation of the potential complexity of underlying interactions. This relative degeneracy of FC was not simply attributable to the BOLD signal, whose SVD did not feature any main linear directions, but was a direct effect of measuring FC. Previous studies emphasized that a broad range of computational models generating time courses with quite different spectra and oscillatory properties featured limited yet similar overall predictive powers of simulated FC [54,68]; such results also suggest that FC is degenerate with respect to dynamical regimes. It remains that part of the variance was not reducible to this main linear dimension to an extent that depended on the metrics used to compute FC: according to SVD analyses, mutual information tended to explain less variance along the first linear dimension than correlation, and could also feature a second reproducible linear dimension. Likewise, even though the main linear dimension was still reproducibly present when reducing the time window of analysis for the computation of FC down to about 30 seconds, it explained less and less variance. It is difficult to assess how much this additional unexplained variance reflect increasingly worse performance in estimating FC or meaningful changes that reflect brain computation [31,71]. Degrees of freedom between SC and FC could be expected, as adaptive brain computation implies dynamic interactions for the routing and integration of information. However, at the level of our measurement space, such degrees of freedom appeared reduced to a main linear dimension relating all observations. Subjects’ cognition may be free ranging in the resting state, but this did not manifest as irreducibly complex differences between independently generated patterns of FC. BOLD-related FC, in particular along 5–10 minute runs, appeared fundamentally dominated by SC, which itself appears dominated by one main linear dimension, explaining up to 86% of the variance across subjects. Altogether this emphasizes that anatomical connections of brain structures themselves must have a strong backbone that is invariant across subjects and that drives a relatively invariant net pattern of bidirectional information transfer. This pattern is reflected in FC and implies an overall bandwidth across communication channels that is a function of the quantity of fibers (empirically estimated by DWI-SC). By manipulating SC, [54] demonstrated the importance of the integrity of SC and of the accuracy of its measurement on the predictive power of generative models for FC. Here, beyond predictive power, we further showed that the main differences between the patterns of simulated and empirical FC largely loaded on connections that were known to be poorly estimated in DWI tractography and that were absent in our SC matrices. Therefore the main factor limiting the predictive power of generative models might turn out to largely reduce to problems of estimation of SC (when we artificially added homotopic connections in our SC data, the variance explained by the first linear dimension increased from 67% to 78%). Improving tractography is therefore expected to greatly improve the fit of simulated FC to empirical FC [69,70]. Now, in spite of their strong relationships, there are deeper systematic differences between SC and FC that need to be emphasized. FC as measured is the image of (linear or nonlinear) correlations among time-courses that are generated by an underlying dynamical process (either simulated or empirical). While being nominally channeled by SC, FC is ultimately induced by an underlying hidden weighting of SC, sometimes called ‘effective connectivity’ (EC) [72,73]. This weighting of connections operates as multiplicative gains (which can be positive or negative) in the local transfer functions between connected regions, which themselves drive the overall dynamical process. Obviously, estimates of the specific weighting of EC cannot be simply inferred from DWI-based measurements of SC. However, based on the relationships demonstrated herein, procedures of inversion of FC, as envisioned and preliminarily explored above, could yield fair estimates of EC itself, which would be of great interest for neurocomputational modeling and inference. Likewise, BOLD-related measures of FC could provide, after adequate inversion, the basis for novel multimodal approaches to tractography, supporting inference on existing fibers in the context of ambiguous diffusion signal based on EC or SC estimates from FC. In the introduction, we referred to two bodies of literature, one showing the relative reproducibility of FC between subjects, and the other one showing that FC could be influenced by a wealth of factors, such as brain maturity, age, the global level of awareness, personality traits, the current mental state, recent experience, and time. In the present study, we found that the information contained in FC was rather degenerate and in a large part reflective of SC. Together with the fact that SC (as seen by DWI tractography) is rather reproducible across subjects, this is in agreement with studies showing the reproducibility of FC. As to the relative influence of the two main (anatomical and dynamical) factors on FC, our study hints that SC strongly dominated the signal and loaded on the first reproducible SVD component. Dynamical functional patterns (and noise) might load on the remaining components, which are not associated with reproducible patterns. The effect of some factors which cannot easily be associated with a modification of structural connections but are most likely related to a change in dynamical regime (e. g. , the global level of awareness, the current mental state), are probably observed on the part that is not accounted for by the first reproducible SVD component. This distinction could be used to classify data accordingly, even though some factors might influence both structure and dynamics. A future study could apply the framework developed in the present manuscript to experimental settings where a variation of FC is observed and quantified concomitantly with a change in state (e. g. , sleep or loss of consciousness [8,74]), behavior, or cognition, and investigate how such change would affect the various singular values from the SVD and corresponding reproducible patterns of FC. In any case, an important point is that the range of variability available to FC is quite reduced compared to the same range at the level of neuronal processes or even at the level of the BOLD signal. Among remaining questions for future studies, one may wonder whether the degeneracy of FC would be as strong with input signals offering more sensitivity to more complex and potentially faster dynamics than the slow and indirect BOLD signal, and how much it relates to the spatial scale of observation. While we showed that increasing the parcellation scale from 160 to 461 to 825 regions essentially led to similar conclusions in terms of degeneracy of FC, a more drastic change in scale together with a change in imaging method might provide complementary insight into the origin and function of brain interactions [75]. The same methodological framework could be applied to electrophysiological measurements and computational models directly, though measuring the required empirical data for the analysis at the scale of the matrix of SC could be challenging. We may also wonder if the brain’s ability to re-route information flow after brain damage, a process that might underlie mechanisms of functional recovery and be involved in resilience to neurological disorders, could not be related to a loosening of the relationship between SC and FC [5].
All datasets previously mentioned were analyzed in a similar fashion using singular value decomposition (SVD). For all but Dataset #9, we applied bootstrap SVD. In order to infer SC from empirical FC, we first quantified the relationship observed between the average projection of empirical SC onto its first left-singular vector and the average projection of all FC onto its first left-singular vector. Empirical observations suggested to consider only values of SC ≥10−6 and to apply the following transformation: SC ′ = ln SC 0. 0001. We then approximated the relation between SC’ and FC through 4-order polynomial interpolation, FC ≈ P4 (SC′). Estimation of SC through inversion of the relationship between SC and FC was then carried out using the relationship SC ^ = 0. 001 exp P 4 - 1 (FC).
The human brain is characterized by both the way its neurons are connected (the anatomy) and the way they emit signals to interact (the dynamics). At the typical scale of measurements, by analogy with the road network, anatomy (the ‘roads’) is expected to remain relatively stable over time and reproducible from person to person, while neuronal dynamics propagating in the network (the ‘traffic’) can be expected to vary widely, both over time and between people. But, paradoxically, patterns of functional connectivity (FC), a convenient proxy to quantify interactions between pairs of brain regions in magnetic resonance imaging (MRI), have been found to be reproducible both within and between subjects across several studies, while other studies, on the contrary, have featured variable FC patterns, often in relation to cognitive processes. We investigate this unsettled issue quantitatively using multivariate statistics and generative models of brain activity. We show, across a range of estimators, that FC offers a degenerate representation of brain interactions and strongly depends on anatomy.
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Functional Connectivity’s Degenerate View of Brain Computation