The arrow‐of‐time in neuroimaging time series identifies causal triggers of brain function

Abstract Moving from association to causal analysis of neuroimaging data is crucial to advance our understanding of brain function. The arrow‐of‐time (AoT), that is, the known asymmetric nature of the passage of time, is the bedrock of causal structures shaping physical phenomena. However, almost all current time series metrics do not exploit this asymmetry, probably due to the difficulty to account for it in modeling frameworks. Here, we introduce an AoT‐sensitive metric that captures the intensity of causal effects in multivariate time series, and apply it to high‐resolution functional neuroimaging data. We find that causal effects underlying brain function are more distinctively localized in space and time than functional activity or connectivity, thereby allowing us to trace neural pathways recruited in different conditions. Overall, we provide a mapping of the causal brain that challenges the association paradigm of brain function.

, for all brain regions (left to right), and spatial correlation between the AoT patterns obtained using two successive numbers of samples. The estimates reached using n * s = 8000 samples (the value selected for our main analyses) are highlighted by a dashed horizontal line (left heatmaps) or a red rectangle (right plots). WM: working memory. Fig. 3 shows the AoT patterns extracted from full task recordings using n * s = 8000 samples, 18 when convergence is already achieved as demonstrated above. Below, drawing from past work 1 , 19 we first briefly summarize the main components of each task. We then discuss the largest AoT 20 contributors in terms of how they fit each paradigm's demands. Since we consider full paradigms, 21 for which a given area may transit between acting as a causal source or sink over the course of 22 time, we do not take sign into account in what follows. 23 The working memory task was an N -back task in which images of faces, tools, places and body 24 parts were presented to the subjects. Half of the blocks consisted in a 0-back task, and half in a 25 2-back task.

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In terms of AoT strength, the most influential areas were largely confined to the occipital 27 cortex. There were also two anterior frontopolar regions from the right hemisphere (R341, R343), 28 known to be important in working memory tasks for the manipulation of integrated information 2 .

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In the relational task, for relational blocks, the subjects were simultaneously shown two pairs 30 of objects, with each object a combination of a shape and a texture. They had to determine which 31 dimension differed between the top objects, and whether the bottom objects also differed along 32 that same dimension. In matching blocks, they were instead shown two objects at the top of the 33 screen, one at the bottom, and a word (either "shape" or "texture") in the middle. contributes to relational integration during reasoning 5 .

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In the emotion task, in emotional blocks, participants were shown one face at the top of the 43 screen, and two at the bottom. They had to determine which of these two matches the top one. The faces had either an angry or a fearful expression. In the shape blocks, they instead had to 45 determine which of two bottom shapes matched the top one.
part of the visual system.

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In the social task, participants were shown movie clips of geometrical shapes that either inter-49 acted in a certain way (social blocks), or moved randomly (random blocks). They had to decide 50 whether the shapes were socially interacting or moving randomly, with the possibility to state that 51 they were unsure.

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As in the above cases, the strongest contributors were visual regions. This task was the one 53 with the second broadest set of influential visual areas. Similarly to the relational task, R282 and 54 R286 were detected, which makes sense as the social movie clips also involved multiple objects to 55 track. In addition, an area in the left angular gyrus (R72) previously linked to action awareness 56 representation 6 was pinpointed, as well as the left supramarginal gyrus (R95), which enables to 57 retain an abstract representation of serial order information 7 , and the right inferior parietal lobule 58 (R332), implicated in the discrimination of direction changes 8 .

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In the language task, participants were stimulated auditorily instead of visually. In story blocks,

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In the gambling task, subjects were asked to guess whether the number (between 1 and 9) on a 69 mystery card would be lower or larger than 5. In the win blocks, the outcome would be decided so Only a restricted set of visual areas were influential in this task.

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In summary, these observations collectively strengthen our main results ( counterparts (compare to Fig. 2 from the main results), while for the language task, there was no 81 major change, and for the working memory task, there was a switch to negative values.

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The pinpointed regional pattern without baseline epochs remained overall similar for the motor 83 Figure 4: For each task, estimation of τ when up to 10000 samples are used (top to bottom in the heatmaps), for all brain regions (left to right); convergence of mean AoT across regions (τ ) as more samples are considered (with error bars reflecting standard error of the mean); and spatial correlation between the AoT patterns obtained using two successive numbers of samples. The estimates reached using n * s = 8000 samples (the value selected for main analyses) are highlighted by a dashed horizontal line (top heatmaps) or a red rectangle (bottom plots). WM: working memory. 8 and emotion tasks, to the exception of some areas which switched sign (positive to negative τ in 84 the former case, and negative to positive τ in the latter case). This may be because their role as 85 causal source or sink fluctuates as a function of epoch type.

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A marked transition to negative-valued τ was seen in the working memory and social task 87 cases, particularly for visual areas. The working memory task was specifically designed to probe    from the IPS were two of the remaining three, also squaring well with previous static observations. 127 Similarly to the working memory task, R178 was also pinpointed as a highly dynamic area. fitting with their prefrontal location in the brain. In addition, another found area was R161

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(parietal default mode network), whose dynamic tracking of emotional stimuli was previously 137 shown to be disrupted in anxiety and depression 16 .

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From Table 2, it can be seen that AoT estimates remain extremely similar regardless of the 179 extent of scrubbing applied to the data. This is strong evidence that head motion does not impact 180 our results.

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In Table 3

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would initially perturb the fMRI signals (e.g., stronger heart rate fluctuations until one becomes 185 at ease in the scanner).

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In addition to the above, we also verified that the results were not affected by the use of a 187 coarser (R 2 = 219 regions) or finer-grained (R 3 = 819 regions) atlas. As can be seen from Fig. 13, 188 the extracted AoT patterns in the resting state and motor task cases remained similar regardless 189 of atlas granularity.
190 Figure 13: Regional AoT patterns obtained for the resting state and motor paradigms, using n * s = 8000 samples, when resorting to an atlas with 219, 419 or 819 regions of interest. ROI: region of interest.

Differences in effects captured by our approach, Granger causality and LiNGAM
To explore the similarities and differences between the results of LiNGAM, Granger causality 192 and our approach, we downscaled the dimensionality of our data from R = 419 regions to 15 193 networks (the 7 Yeo networks 21 for each hemisphere, plus subcortical regions). We estimated pa-194 rameters for each method using 56000 samples, a high enough number to ensure accurate outcomes, 195 and performed bootstrapping over 50 folds that included different subsets of subjects each time.

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For LiNGAM, we extracted the output set of causal coefficients following estimation and pruning 197 (B ∈ R R×R ), using a dedicated toolbox 22 . For Granger causality, we extracted the set of coefficients 198 obtained upon fitting a first-order autoregressive model to the forward time courses (having set 199 to zero the coefficients denoting the influence of a network onto itself), which we will refer to as 200Ã f ∈ R R×R . Finally, for our method, we computed τ ∈ R R×1 following Eqs.   Scatter plots depicting the relationships between in-degree and out-degree vectors for the in Fig. 15. There was a strong negative correlation between the in-degree and out-degree vectors 214 for the LiNGAM case (R = −0.62, p = 0.015) and to a milder extent, albeit not significantly, 215 for the multivariate autoregressive model case (R = −0.35, p = 0.21). Thus, as seen from both 216 methodologies, networks that tend to causally regulate others will not be so strongly modulated 217 themselves, and vice versa.

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There were also moderate, but non-significant similarities between the LiNGAM and multivari-219 ate autoregressive approaches: across methods, correlation for in-degree and out-degree vectors 220 was R = 0.49, p = 0.066 and R = 0.28, p = 0.31, respectively. When comparing LiNGAM-221 extracted features with the outputs from our approach, correlation also did not reach significance 222 (R = −0.07, p = 0.81 for in-degree and R = 0.19, p = 0.49 for out-degree, respectively), and the 223 same was seen when comparing Granger causality features to τ (R = −0.37, p = 0.18 for in-degree 224 and R = −0.01, p = 0.97 for out-degree, respectively).

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All in all, LiNGAM, Granger causality and our AoT-sensitive metric thus capture different 226 facets of fMRI activity, an expected finding given the differences between the three approaches.