Functional connectivity change as shared signal dynamics

Abstract

Background: An increasing number of neuroscientific studies gain insights by focusing on differences in functional connectivity-between groups, individuals, temporal windows, or task conditions. We found using simulations that additional insights into such differences can be gained by forgoing variance normalization, a procedure used by most functional connectivity measures. Simulations indicated that these functional connectivity measures are sensitive to increases in independent fluctuations (unshared signal) in time series, consistently reducing functional connectivity estimates (e.g., correlations) even though such changes are unrelated to corresponding fluctuations (shared signal) between those time series. This is inconsistent with the common notion of functional connectivity as the amount of inter-region interaction.

New method: Simulations revealed that a version of correlation without variance normalization - covariance - was able to isolate differences in shared signal, increasing interpretability of observed functional connectivity change. Simulations also revealed cases problematic for non-normalized methods, leading to a "covariance conjunction" method combining the benefits of both normalized and non-normalized approaches.

Results: We found that covariance and covariance conjunction methods can detect functional connectivity changes across a variety of tasks and rest in both clinical and non-clinical functional MRI datasets.

Comparison with existing method(s): We verified using a variety of tasks and rest in both clinical and non-clinical functional MRI datasets that it matters in practice whether correlation, covariance, or covariance conjunction methods are used.

Conclusions: These results demonstrate the practical and theoretical utility of isolating changes in shared signal, improving the ability to interpret observed functional connectivity change.

Keywords: Functional MRI; Functional connectivity; Resting-state functional connectivity; Schizophrenia; Task functional connectivity.

Figure 1. Differences between correlations and covariances for estimating functional connectivity differences

A) Diagrams and equations illustrating simulated communication changes between brain regions (or neurons) X and Y. Left, only the portion of the time series shared across both regions is amplified relative to the unshared and noise portions. Center, only the unshared portion is amplified. Right, both the shared and unshared portions are amplified. B) A single subject’s simulated data are shown for illustration across the three conditions. Results of the group simulation are shown in the upper left of each panel. The correlation (corr_diff) and covariance (cov_diff) results are in agreement when only shared signal is increased, but not for the other two cases.

Figure 2. Differences between correlations and covariances for estimating functional connectivity differences, due to interaction with a third region

Interaction between region Y and Z can stand in for “unshared signal” (in Figure 1) when testing for functional connectivity differences for regions Y and X. A) Diagrams and equations illustrating simulated communication changes between brain regions (or neurons) X, Y, and Z. Left, only the portion of the time series shared between regions X and Y is amplified in region Y. Center, only the portion shared between regions Z and Y is amplified in region Y. Right, the XY shared and ZY shared portions are both amplified in region Y. B) A single subject’s simulated data are shown for illustration across the three conditions. Results of the group simulation are shown in the upper left of each panel. The correlation (corr_diff) and covariance (cov_diff) results are in agreement when only XY shared signal is increased, but not for the other two cases. This suggests correlation-like measures are sensitive to a wider variety of interactions irrelevant to the interactions between the two regions being tested.

Figure 3

A flowchart illustrating a “covariance conjunction” approach to interpreting functional connectivity differences. A similar line of reasoning would also work for most functional connectivity measures (not just correlation; e.g., PPI). Note that simply using covariance would result in a simpler line of reasoning: a significant covariance difference signifies a shared variance difference. However, as noted, a potential confound related to a change in overall variance could invalidate a result significant for covariance only. We suggest that the most conservative approach involves conducting both covariance and correlation analyses, assigning the most confidence to results that are consistent across both approaches (the upper-most route in the flowchart).

Figure 4. All possible shared and unshared variance change combinations

A) The full parameter space is shown for changes in shared and unshared variance relative to a central point (in white). The correlation and covariance values were calculated using the simple mathematical formulation described in the Results section (not the simulations, though note the similarity to results with the realistic neural simulations presented in Figure 5). The boxed numbers refer to the combinations listed in part B. B) Group simulation results (using the same methods as Figure 1) are shown across all possible manipulations of shared variance and unshared variance (p<0.05). Figure 1 illustrates cases 2, 4, and 1. Note that correlation and covariance give different answers in 4 out of the 8 cases, and that covariance matches shared variance changes in all cases. The suggested covariance conjunction approach results are highlighted in green. A variety of other common functional connectivity measures are also included to illustrate how general these results are. Code for these simulations can be found at: https://github.com/ColeLab/simplesims/ + increase, 0 no change, - decrease, cov=covariance, scov=spectral covariance, corr=Pearson correlation, coh=coherence, MI=mutual information, reg=regression, PPI=psycho-physiological interaction, spcorr=Spearman correlation.

Figure 5. Neurobiologically realistic simulations reveal the relationship between network changes and functional connectivity measures

A) Shared and unshared neural signals were systematically manipulated across 66 simulated brain regions. The two-dimensional parameter space illustrates the effects of these manipulations for correlations (corr, squares, far right color bar) and covariances (cov, circles in each square, the adjacent color bar), averaged across all connections for parsimony (i.e., global connectivity across the entire set of simulated regions). The color scales indicate increases (red) and decreases (blue) relative to the central point in the parameter space (white, marked with gray border). The approximate portion of the parameter space in which both correlation and covariance gave the same results (i.e., the conjunction) is highlighted by green triangles in the upper left and lower right corners. Note that these large-scale neural network dynamics are nearly isomorphic to the pure mathematical solution (see Figure 4A), supporting the theoretical formulation. B) The same simulations for each variable in one dimension, indicating that simulations of a neurobiologically realistic network are consistent with the simpler simulations in Figure 1. Note that correlation here (as throughout this article) is the Fisher’s Z-transformed Pearson correlation, which can exceed 1.

Figure 6. Validating covariance by estimating functional connectivity relative to zero

A) A set of 264 previously identified regions were used because of an associated partition consistent with known neural systems (e.g., visual, auditory, default-mode). B) Standard resting-state functional connectivity estimation with fMRI was carried out with 118 subjects using Pearson correlation. Group t-tests versus 0 are reported for each connection, placing correlation results on the same scale as covariances. Labels are indicated on the right for the putative systems that the regions group into based on functional connectivity (Power et al. 2011). C) The analysis was repeated using covariance, resulting in a virtually identical whole-brain functional connectivity pattern (r²=0.98, p<0.00001 between the correlation and covariance t-value matrices). Results were similar for raw correlation and covariance matrices, and without global signal regression. These results validate covariance as a functional connectivity measure, while the following results focusing on functional connectivity change demonstrate distinctions between the measures.

Figure 7. Correlation versus covariance across major brain systems and diverse cognitive domains

A) T-tests compared all 34716 connections for an example task (the Emotion task) versus rest (p<0.05, corrected for multiple comparisons), separately using correlation and covariance. There was a rough similarity in the pattern of results, but also noticeable differences. Generally, there were many differences across methods that would alter interpretation of functional connectivity effects. Further, the above simulations suggest any observed difference with covariance has a clearer interpretation (i.e., results are unlikely to be driven by unshared signal changes). B) The percentage of connections significantly changed (each task versus rest) was computed when using covariance and correlation, then subtracted. Results from all seven tasks are shown. C) The percentage of the time that covariance and correlation gave different answers. For each task, the total number of differences in results (e.g., a connection that was significantly increased with covariance but significantly decreased with correlation) divided by the total number of significant results across both covariance and correlation approaches.

Figure 8. The covariance conjunction approach

A) The statistically significant correlation (p<0.05, FDR corrected) and statistically significant covariance (p<0.05, FDR corrected) results from Figure 7A were combined via conjunction to implement the “covariance conjunction” approach. Conjunctions were calculated separately for increases and decreases from 0. B) The percentage of different results between covariance conjunction and correlation are shown. C) The percentage of different results between covariance conjunction and covariance are shown.

Figure 9. Detected disruptions across functional networks in schizophrenia differ between covariance and correlation

A) Here we show altered covariance structure between large-scale associative networks in schizophrenia (SCZ), similar to recent findings (Baker et al. 2014) [t(143)=2.37, p=0.019, Cohen’s d=0.4]. B) We recently discovered elevated variance across the entire brain in chronic SCZ, which was particularly evident for associative networks (Yang et al. 2014). C) Based on this elevated non-shared variance, it follows that the difference in correlations between SCZ and healthy control subjects (HCS) across the two networks will be attenuated and no longer reveal a significant clinical effect [t(143)=1.48, p=0.14, Cohen’s d=0.25]. The equation on the bottom is presented for illustrative purposes, to highlight the importance of carefully decomposing the final correlation into variance and covariance components (Figure 3). FPCN, fronto-parietal control network; DMN, defaultmode network.