Hierarchical organization of functional connectivity in the mouse brain: a complex network approach

This paper represents a contribution to the study of the brain functional connectivity from the perspective of complex networks theory. More specifically, we apply graph theoretical analyses to provide evidence of the modular structure of the mouse brain and to shed light on its hierarchical organization. We propose a novel percolation analysis and we apply our approach to the analysis of a resting-state functional MRI data set from 41 mice. This approach reveals a robust hierarchical structure of modules persistent across different subjects. Importantly, we test this approach against a statistical benchmark (or null model) which constrains only the distributions of empirical correlations. Our results unambiguously show that the hierarchical character of the mouse brain modular structure is not trivially encoded into this lower-order constraint. Finally, we investigate the modular structure of the mouse brain by computing the Minimal Spanning Forest, a technique that identifies subnetworks characterized by the strongest internal correlations. This approach represents a faster alternative to other community detection methods and provides a means to rank modules on the basis of the strength of their internal edges.

. The neuroanatomical ROI considered for our analysis, mapped into a mouse brain..

From time series to correlation matrices
Our data consist of 41 sets of 54 fMRI BOLD-signals each, collected as the time series shown in fig. 2.
The information carried by each mouse-specific set of time series has been condensed into a correlation matrix, whose generic entry C i j is the Pearson coefficient between time series X i and X j , defined as where and T is the total temporal length of the series.
In order to create an average adjacency matrix describing brain functional connectivity at the population level, subject-wise matrices were first Fisher-transformed, i.e.
then averaged across subjects (i.e. the generic entry of the average Fisher-transformed matrix is the arithmetic mean of the corresponding individual entries, z 1 i j , z 2 i j . . . z 54 i j ) and then back-transformed: Analysing single individuals The same analysis described in the main text to study the average correlation matrix has been also applied to the subject-wise matrices, in order to highlight the discrepancies between the individual connectivity structures and the average ones. In what follows, we sum up the results of the analysis of the whole sample of 54 mice, by explicitly showing the plots of four mice only (see figs. ??-9).
As mentioned in the main test, the empirical cumulative distributions of correlations are always well reproduced by the cumulative density functions of normal distributions whose mean and standard deviation are estimated from the data via a likelihood-maximization procedure.
As for the average mouse brain, our variation of the percolation analysis reveals the presence of plateaus, as well as the existence of a nested structure of highly correlated areas. Analogously, constraining the distribution of correlations allows one to recover the presence of multiple percolation thresholds, even if the brain-specific step-wise plots are not reproduced.

Testing our approach on synthetic networks
In order to test the effectiveness of our method in detecting hierarchical modular structures, we have repeated the same analysis described in the main text to study two synthetic networks.
The first case we considered is a correlation matrix with modular, but not hierarchical, structure. As fig. 7 shows, we considered a network with three well-identifiable modules, weakly inter-connected and lacking an internal hierarchy of correlation coefficients. The dendrogram correctly identifies the presence of three clusters (the different heights are induced by the different values of the internal correlations) with no internal structure; on the other hand, our percolation analysis detects the presence of (three) connected components for lower values of the threshold r (explicitly shown in the bottom panel of fig.  8), while providing an explicit representation of the detachment of single areas in correspondence of higher values. Such a  Result of the analysis of an individual brain: (left panels) experimental correlation matrices C i j , whose rows and columns have been ordered according to the dissimilarity measure D i j = 1 −C i j , ∀ i, j; (middle panels) empirical cumulative distributions of the correlations -blue trends -and of normal distributions whose parameters have been estimated through the likelihood-maximization procedure -red trends; (right panels) comparison between the result of our modified percolation analysis run on the observed brains -red trends -on the synthetic brains -brown trends -generated by preserving the observed distributions of correlations and averaged on the ensemble -green trends.
behavior signals that a deeper structural organization is, indeed, missing. Notice how such trend differs dramatically from the trend characterizing our average mouse brain, where different regimes are not clearly distinguishable.
The behavior of our second case-study is completely different. In this case, in fact, we have considered a ring of cliques (i.e. not a correlation matrix), each one composed by five nodes, whose internal links are characterized by different values of pseudo-correlation coefficients (see fig. 9). As our percolation analysis reveals, the network progressively breaks down in components characterized by higher internal correlations than with the remaining sub-sets of nodes. As larger correlations are removed, the cliques are correctly recovered as independent modules; further rising the threshold value allows the (internal) modules constituting the cliques to emerge.  Result of the analysis of an individual brain: (left panels) experimental correlation matrices C i j , whose rows and columns have been ordered according to the dissimilarity measure D i j = 1 −C i j , ∀ i, j; (middle panels) empirical cumulative distributions of the correlations -blue trends -and of normal distributions whose parameters have been estimated through the likelihood-maximization procedure -red trends; (right panels) comparison between the result of our modified percolation analysis run on the observed brains -red trends -on the synthetic brains -brown trends -generated by preserving the observed distributions of correlations and averaged on the ensemble -green trends.  Figure 5. Result of the analysis of an individual brain: (left panels) experimental correlation matrices C i j , whose rows and columns have been ordered according to the dissimilarity measure D i j = 1 −C i j , ∀ i, j; (middle panels) empirical cumulative distributions of the correlations -blue trends -and of normal distributions whose parameters have been estimated through the likelihood-maximization procedure -red trends; (right panels) comparison between the result of our modified percolation analysis run on the observed brains -red trends -on the synthetic brains -brown trends -generated by preserving the observed distributions of correlations and averaged on the ensemble -green trends.  Figure 6. Result of the analysis of an individual brain: (left panels) experimental correlation matrices C i j , whose rows and columns have been ordered according to the dissimilarity measure D i j = 1 −C i j , ∀ i, j; (middle panels) empirical cumulative distributions of the correlations -blue trends -and of normal distributions whose parameters have been estimated through the likelihood-maximization procedure -red trends; (right panels) comparison between the result of our modified percolation analysis run on the observed brains -red trends -on the synthetic brains -brown trends -generated by preserving the observed distributions of correlations and averaged on the ensemble -green trends. 1  3  18  5  11  15  2  14  12  17  7  4  6  10  16  19  9  20  13  8  48  51  44  42  41  43  50  46  53  52  45  54  47  49  36  21  26  39  27  35  37  23  33  29  34  28  40  38  24  31  25  22 1  2  4  5  3  6  9  10  7  8  11  13  15  14  12  16  17  19  18  20  52  51  54  53  47  48  46  49  50  42  43  41  44  45  22  21  24  23  25  26  29  27  28  30  35  33  34  31  32  36  37  39  40  38 1 1 Figure 9. Dendrogram and adjacency matrix of a synthetic network characterized by a ring of cliques structure, whose nodes are linked by pseudo-correlation coefficients, ranging between 0 and 1.  Figure 11. Explicit representation of the modules recovered by our percolation analysis on the ring of cliques. Our modified percolation detects the presence of modules within modules (for r 0.15), characterized by higher internal correlations than with the remaining sub-sets of nodes. As larger correlations are removed, the single cliques are correctly recovered (r 0.5); further rising the threshold value (r 0.86) allows the internal modules constituting the cliques to emerge.