Hierarchical Complexity of the Macro-Scale Neonatal Brain

Abstract The human adult structural connectome has a rich nodal hierarchy, with highly diverse connectivity patterns aligned to the diverse range of functional specializations in the brain. The emergence of this hierarchical complexity in human development is unknown. Here, we substantiate the hierarchical tiers and hierarchical complexity of brain networks in the newborn period, assess correspondences with hierarchical complexity in adulthood, and investigate the effect of preterm birth, a leading cause of atypical brain development and later neurocognitive impairment, on hierarchical complexity. We report that neonatal and adult structural connectomes are both composed of distinct hierarchical tiers and that hierarchical complexity is greater in term born neonates than in preterms. This is due to diversity of connectivity patterns of regions within the intermediate tiers, which consist of regions that underlie sensorimotor processing and its integration with cognitive information. For neonates and adults, the highest tier (hub regions) is ordered, rather than complex, with more homogeneous connectivity patterns in structural hubs. This suggests that the brain develops first a more rigid structure in hub regions allowing for the development of greater and more diverse functional specialization in lower level regions, while connectivity underpinning this diversity is dysmature in infants born preterm.


S1. Designation of tiers
The group-aggregated degree distributions (Figure 3 in main text) were clearly not unimodal distributions, exhibiting several noticeable distinct peaks suggesting they were composed of several components.
The modelling also provides the proportions of each of these distributions, w 1 , w 2 , w 3 , w 4 which determines contribution of a given distribution to the mixture. This gives the probability density function (pdf) of where each g is the pdf of a Gaussian distribution with the mean and variance in the conditional argument, is the collection of all parameters. For full details see e.g. Reynolds (2009). Each modelling iteration was tested for fit against the real data using the Bayesian Information Criterion (BIC), which is lowest for models achieving a trade-off between high accuracy and low complexity, aiding to avoid selection of models which overfit the data. From a set of models, the best model is the one achieving  Figure S1: Average Bayesian Information Criterion values (relative to minimum across component number) for Gaussian mixture models applied to a range of density thresholds of the structural connectomes. The colour axis is cut off at 100 (bright yellow) to make comparisons easier between the more likely models. For each density, the minimum (dark blue) is the best model. smallest BIC. Fig S1 shows results for connectomes thresholded at different densities, from 0.2 up to 0.4 in steps of 0.0005. Those shown are BIC scores minus the minimum found over all models (2,3,4,5, and 6 component models) at that density, so that the minimum model at a given density has a value of 0. Results are averaged over 1000 iterations for each model. It is clear that the adult group was best represented by a 4 component model over almost all densities, while neonates were best represented mostly by 4 components, but also by 3 components at larger densities. To define tiers within adult and neonate groups, we used the 4-component GMM achieving minimum BIC score (out of the 1000 iterations) for each group. Note, to get a consistent tier definition for term and preterm born neonates, these groups were aggregated. From these models, we automatically set the threshold between tiers at the point where the probability density functions of two adjacent components crossed each other, i.e. the point where the next component began to have more likelihood of being the source of the node at that point, see Fig. 3 in the main text for details. Therefore, for neonates, Tier 1 was defined from the maximum degree down to a degree of 37, Tier 2 from 36 down to 29, Tier 3 from 28 down to 19 and Tier 4 from 18 down to 0. For adults, Tier 1 was defined from the maximum degree to a degree of 46, Tier 2 from 45 down to 29, Tier 3 from 28 down to 13 and Tier 1 from 12 down to 1.
For neonates it was notable that the GMM model was unable to pick up on the two smaller distribution components observable at the right-end of the distribution, instead treating them as a single distribution with a very large standard deviation. In our initial analysis, we kept these two observable components separate to study this further.
Next, for the best fit 4-component models, we inspected the consistency of the makeup of the ROIs in each component between groups. For each ROI and each group we counted how many times an ROI was assigned to each tier and divided these numbers by the group size. This provided the proportion of times an ROI was in each tier for that group. We then computed Pearson's correlation coefficients of these proportions over all ROIs between the groups, Table S1. Correlations between neonate groups were extremely high, with all tiers being above 0.95, indicating a strong alignment between the hierarchical structure of term and preterm neonate connectomes, see Fig S3. Correlations between neonate groups and adults were much more variable.
While Tier 1 and Tier 4 showed moderately strong correlations, Tiers 2 and 3 had only medium correlations.
However, on combining Tiers 2 and 3 together, the correlations significantly increased, Table S1, right. To make the analysis of more relevance for the adult and neonate groups, we thus designated an adapted tier system. We defined i) Tier A as Tier 1, ii) Tier B as the combination of Tiers 2 and 3, and iii) Tier C as Tier 4, in both neonates and adults. This is also reflected in the relative tier proportions between adults and term-born neonates using both the original 4-tier system, Fig S4, Figure S6 shows the spatial distribution of the tiers and table S2 the average number of nodes by tier within groups.   to ROI i, and the second vector x i as the row of A corresponding to ROI i's symmetric counterpart, the hemispheric symmetry of a single ROI is defined as where δ is the Kronecker delta function, being 1 if the two inputs are equal and 0 otherwise, x i (j) is the jth element of vector x i and n is the maximum number of possible connections. We normalized our results with respect to the expected number of matching connections between two sets of connections selected independently at random. Let V be the set of all possible connections and define S i as the set of connections of ROI i with size |S i | = k i and S i as the set of connections of ROI i with size |S i | = k i . Then Hence the normalized measure of hemispheric symmetry we use is For each tier of each participant in each group, we computed the average of this value.

S4. Common tier connections
Since hemispheric symmetry was not seen to contribute to hierarchical complexity, we asked instead if highly common and uncommon neighbours of Tiers explained the trends in hierarchical complexity. An ROI was defined as commonly connected to a given tier if it shared links with at least 80% of that tier's ROIs. An ROI was defined as uncommonly connected to a given tier if it shared links with at most 20%, but not none, of that tier's ROIs. The common and uncommon neighbours for each tier are shown per ROI and summed as distributions over participants in Fig S8. Common connections for Tier A and uncommon connections for Tiers B and C were also computed for 100 configuration models per participant and averaged. The results are shown in Fig S9 with Wilcoxon rank sum tests and effect sizes above each plot.