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Figure 1. Data preparation and matrix construction. (A): This cartoon describes the production of basal Xenobots from frog embryonic tissue. Caps are excised by hand and cultured for seven days without perturbation; the resulting autonomously moving epidermal cell structure is a “basal” (i. e. unmodified) Xenobot. (B): The pipeline for learning functional connectivity patterns from basal Xenobots. Videos of the bots are segmented into individual cells, from which average calcium traces are extracted. The covariance matrix defines the functional connectivity, which can be organized into meso-scale communities following clustering. (C): The same pipeline, but for human brain data recorded with an fMRI; in this case, the elements are cortical regions (corresponding to the Schaefer 200 parcellation [Citation67]) and the signal is BOLD rather than calcium, but the overall pipeline remains the same.
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Figure 2. Autocorrelation. Violin plots of the distribution of autocorrelation over all brain regions (for the fMRI scans) and all cell calcium signals (for the basal Xenobot recordings). All cells and all brain regions had statistically significant autocorrelation.
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Figure 4. Spatial embedding of functional connections. (A) The Pearson correlation coefficient between the strength of a functional edge between two brain elements, and the Euclidean distance between them. Note that all correlations are negative, across all systems, showing a consistent drop off of functional integration with distance. (B) The difference between the average distance between elements that both reside within a single module and the average distance between elements in different modules. All differences were positive, showing that elements grouped within a functional module tend to be closer together than elements assigned to different modules. Collectively, these results show that the functional statistics are embedded in space in plausible ways: cells that are closer together tend to be more integrated and are more likely to belong to the same meso-scale community.
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Figure 3. Sample communities. A set of sample functional covariance matrices, organized by the communities detected by the multi-resolution community detection algorithm [Citation73] (see materials and Methods 3.5). The on-diagonal squares (outlined in black) correspond to the communities. Three fMRI and three basal Xenobot matrices are shown. The full set of all clustered functional connectivity matrices, for basal Xenobots and fMRI scans Supplementary material.
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Figure 5. Variance in pairwise co-fluctuation magnitude. For both fMRI scans and basal Xenobot recordings, the variance of the RSS of co-fluctuation amplitude across all pairs of elements is greater than in the circular-shifted, autocorrelation-preserving nulls. These show that both fMRI scans and basal Xenobot calcium imaging data show transient, periods of both increased integration and increased segregation, indicating a kind of dynamical “richness”.
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Figure 6. Higher-order information: total correlation and dual total correlation. (A) The proportion of subsystems of size 3, 4, and 5 that had significant total correlation for both basal Xenobots and fMRI scans. Both systems showed significant integration for multiple samples. (B) Both fMRI scans and basal Xenobots showed significantly greater total correlation [Citation79] than their autocorrelation-preserving nulls. (C) The proportion of systems of size 3, 4, and 5 that had significant dual total correlation [Citation80] for both system types. (D) Both fMRI scans and basal Xenobot recordings showed greater dual total correlation than their respective nulls.
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Figure 7. Positively- and negatively signed O-information. (A) The proportion of subsystems of size 3, 4, and 5 that had significant positive O-information (i.e. were significantly redundancy-dominated) for both basal Xenobots and fMRI scans. Both systems showed significant higher-order redundancy for multiple samples. (B) For positively signed samples, both fMRI scans and basal Xenobots showed significantly greater O-information than their autocorrelation-preserving nulls. (C) The proportion of systems of size 3, 4, and 5 that had significant negative O-information (i.e. were significantly synergy-dominated) for both system types. As with redundancy, both systems displayed significant synergy. (D) for negatively signed samples, both fMRI scans and basal Xenobot recordings showed more strongly negative O-information than their respective nulls.
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Figure 8. Tononi-Sporns-Edelman complexity. Both fMRI scans and basal Xenobot recordings showed higher TSE complexity [Citation82] than their respective autocorrelation-preserving null models. These results show that the higher-order structure in brains and bots spans the whole system, involving multiple orders of interaction. The TSE complexity is understood as being maximized by a balanced combination of integration and segregation: it is low both when the whole system is synchronized, and also when all elements are independent. It is maximized when higher-order integration co-exists with lower-order independence.
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Figure 9. Bivariate transfer entropy. (A) This plot shows the proportion of possible pairs (i.e. Of all possible 𝑋𝑖,𝑋𝑗∈𝑋) that showed significant transfer entropy. Both the fMRI scans and basal Xenobot recordings showed some multi-cellular directed information flows, although basal Xenobots were considerably sparser. (B) For both fMRI scans and basal Xenobot recordings, the empirical, significant transfer entropies were significantly higher than their respective circular shifted nulls.
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Figure 10. Integrated information (Φ𝑅). Both the basal Xenobots and the fMRI scans have significantly higher global integrated information than their respective nulls do. This shows that both systems display at least some dynamical properties in the “whole” that are not trivially reducible to lower-order “parts”.
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