Functional brain networks: random, "small world" or deterministic?

Abstract

Lately the problem of connectivity in brain networks is being approached frequently by graph theoretical analysis. In several publications based on bivariate estimators of relations between EEG channels authors reported random or "small world" structure of networks. The results of these works often have no relation to other evidence based on imaging, inverse solutions methods, physiological and anatomical data. Herein we try to find reasons for this discrepancy. We point out that EEG signals are very much interdependent, thus bivariate measures applied to them may produce many spurious connections. In fact, they may outnumber the true connections. Giving all connections equal weights, as it is usual in the framework of graph theoretical analysis, further enhances these spurious links. In effect, close to random and disorganized patterns of connections emerge. On the other hand, multivariate connectivity estimators, which are free of the artificial links, show specific, well determined patterns, which are in a very good agreement with other evidence. The modular structure of brain networks may be identified by multivariate estimators based on Granger causality and formalism of assortative mixing. In this way, the strength of coupling may be evaluated quantitatively. During working memory task, by means of multivariate Directed Transfer Function, it was demonstrated that the modules characterized by strong internal bonds exchange the information by weaker connections.

Figure 1. Simulation description and results.

Top: simulation scheme, signal from channel 1 is propagating to the other channels with different delays. Graphs—left: ordinary coherences; right: multivariate DTFs as functions of frequency. Propagation by DTF is shown from the channel marked above to the channels marked at the left of the picture (on the diagonal power spectra). At the bottom obtained corresponding connectivity schemes. Incorrect flows shown by broken lines.

Figure 2. Connectivity patterns obtained for awake state, eyes closed by: DTF (upper left), bivariate coherences (upper right), SL (lower right), pattern obtained from SL by giving all connections equal weight (threshold = 0.2).

Figure 3. Connectivity patterns obtained for a working memory task by DTF (upper left), bivariate coherences (upper right), SL (lower right), pattern obtained from SL by giving all connections equal weight (threshold = 0.2).

Figure 4. Node degrees for DTF and SL coded by the dimensions of the circles.

From left to the right: DTF outflows, DTF inflows, SL node degree.

Figure 5. The results of assortative mixing for four modules in the whole (4–60 Hz) frequency band.

At the left for DTF, at the right for SL. On the diagonal coupling within the modules, off-diagonal between the modules. For DTF the causal coupling from the module marked below the column of boxes to the module marked at the left. In each box the strength of coupling is illustrated in color scale: dark red-the strongest, dark blue-the weakest.