We analyzed a dataset discussed in the literature [6
], consisting of subjects undertaking a simple self-paced finger-tapping task, for the purpose of exploring the potential of our method. The task is composed of three variants, which were designed to be relatively similar, in the hope that the method could discriminate subtle differences in the patterns of brain activation. The variation between the tasks is based on the cue utilized to signal the start and stop of the finger-tapping. In the first task, a small featureless visual cue presented in the center of the screen; in the second task the cue is a short auditory tone; in the third one a visual cue similar to the first one is presented, the only difference being its larger size.
After standard functional pre-processing (see Methods), and delayed covariance analysis as explained above, the resulting networks were studied using the tools of statistical network theory. The first observation is that most links are undirected, comprising on average of 50% to 60% of total links; this is compatible with the notion that most neural interactions result from fast, local and presumably symmetric connections, whose subtle dynamics are for the most part beyond the present reach of functional MRI. However, directed links account for a significant number of the observed correlations between voxels, suggesting that our approach can indeed be fruitful in terms of capturing ongoing dynamics.
Although directed links have less statistical power than undirected ones, their degree distribution still shows a power-law behavior. This is exemplified in Fig. , where the histogram of source (blue) and sink (red) degrees, i.e. out- and in-directed links, averaged over all subjects and tasks, is represented. Observe that the distribution is very similar for both classes of links, and approaches a power-law with an exponent of 3/2, which is even more evident when all links are considered for each node (black). This result is in line with our previous findings, as it reinforces the notion of information hubs dominating the flow of information in the brain. The effect of removing some of the undirected edges is to reduce the scaling from an exponent near 2 as estimated with our previous approach. Figure depicts the result of analyzing the small-world topology of the functional networks. The panel represents the clustering vs. average minimal path for all the studied cases (open circles), the equivalent random networks (red crosses), and the equivalent regular lattices (blue x's). Observe that the clustering of the functional networks is several orders of magnitude larger than that of the equivalent random networks, and comparable to that of the equivalent regular lattices. The average minimal path of the functional networks, while larger than that of the random networks, is still smaller than that of the equivalent regular lattices. In comparison with our previous study, the main topological features are preserved: for the same threshold (0.7), there is an increase from 12.9 to 16.3 in the average path due to the removal of undirected links explained by a common source; interestingly, the clustering coefficient increases from 0.12 to 0.18, presumably due to triangulations that could not be detected before. Similarly, a comparison with our previous finding regarding the assortative nature of functional brain networks shows that indeed this property does not change when directed links are included. Figure represents the total degree of each node vs. the average degree of the first neighbors, i.e. nodes reachable through in-, out- and undirected edges. Again, we find a correlation between these two degrees, in contrast with all other biological networks investigated to date, which implies a lack of hierarchical organization in the networks: a hierarchical network would display a negative correlation, such that nodes with high degree (or hub nodes) tend to be connected with low degree nodes (or peripheral nodes). A more detailed study, analyzing differences between in- and out-hubs, will be reported in future publications.
Figure 1 (A) Degree distribution of links. The black trace corresponds to the degree for all the links (directed and undirected), the blue one to the degree of sources (out-directed links) and the red trace to sinks (in-directed links), averaged over subjects. (more ...)
Interestingly, the networks hold enough information about the dynamics of brain states so that even a global measure of their properties can discriminate between tasks. Figure shows the result of comparing different topological measures of the three resulting networks (small visual cue, auditory cue, large visual cue tasks) for 6 different subjects. The first observation in this regard is that the total number of nodes identified by the method (Panel 2A) is not a good indicator of the identity of the task, or of the subject for that matter; this is to be expected, as the number of nodes depends dramatically on the chosen threshold. The average degree, or connectivity (Panel 2B), does not seem to provide much information either; this is clearly a very local measure of the structure of the network, for the most part unrelated to the flow of information. We observed a tendency towards discriminating between the tasks when we computed the average minimal path of the networks (not shown), defined as the minimal number of links needed to reach a node from another one, averaged over all nodes. This tendency, though, is not consistent across subjects, as the auditory cue task is in some cases large and small in others. However, we were able to improve this measurement by considering the normalized mean path, as the mean path of the network over that of the equivalent Erdös random network, displayed in Panel 2C (see Methods). Observe that for 5 out of the 6 subjects, the auditory task is consistently larger (i.e. has a bigger normalized mean path) that the equivalent visual cue tasks. Non-parametric statistical analysis on the rank of the normalized mean path (a
) yields a p-value of 0.017. This indicates that the subtle differences in activation elicited by the tasks have a measurable effect in the overall structure of correlations and flow of information of the networks. This tendency was probably amplified by the fact that only the giant
component of the network was considered for the purpose of measuring the average minimal path. The giant component is defined as the largest connected sub-network of a graph; we observed that in all cases this component was at least one order of magnitude larger in size that the runner-ups. In other words, we targeted a core
of correlated activity and disregarded much smaller sub-networks that could otherwise be relevant. It is also worth mentioning that the directional nature of the links enters explicitly the computation of the mean path, as a link A
means that there is no direct access from B
Figure 2 Discrimination of tasks based on global topological properties. The first column of data points corresponds to the small visual cue task, the second to the auditory cue, and the third to the large visual cue in all panels. The upper row corresponds to (more ...)
A remarkable regularity displayed by these networks is the tendency for nodes to be mostly "sources" (i.e. heavy out-hubs) or "sinks" (heavy in-hubs). That is, nodes with a large number of out-links tend to have relatively few in-edges, and vice versa, although, interestingly, this is not a strictly enforced rule. Moreover, in-hubs tend to have relatively few undirected links, whereas out-hubs tend to be also undirected hubs. This seems to be counter-intuitive at face value, as one may naïvely think that the hubs are balanced; however, they need not be so, as one would expect in, for instance, tracffic hubs. In other words, there are no conserved quantities at the hub level to be balanced. These results are summarized in Fig. . The tendency for nodes to be either in- or out-hubs is particularly clear in Panel D, where the maximum between the in-degree and the out-degree is plotted against the absolute value of their difference, showing a strong correlation.
Figure 3 Panel A: relationship between the in-degree and the undirected-degree for each node; observe that there is a negative correlation, i.e. in-hubs tend to have very few undirected links and vice versa (insets correspond to covariance between the plotted (more ...)