We implement the Ising model on a structural connectivity matrix describing the brain at two different resolutions. Tuning the model temperature to its critical value, i.e. at the susceptibility peak, we find a maximal amount of total information transfer between the spin variables. At this point the amount of information that can be redistributed by some nodes reaches a limit and the net dynamics exhibits signature of the law of diminishing marginal returns, a fundamental principle connected to saturated levels of production. Our results extend the recent analysis of dynamical oscillators models on the connectome structure, taking into account lagged and directional influences, focusing only on the nodes that are more prone to became bottlenecks of information. The ratio between the outgoing and the incoming information at each node is related to the the sum of the weights to that node and to the average time between consecutive time flips of spins. The results for the connectome of 66 nodes and for that of 998 nodes are similar, thus suggesting that these properties are scale-independent. Finally, we also find that the brain dynamics at criticality is organized maximally to a rich-club w.r.t. the network of information flows.

We investigate phase synchronization in EEG recordings from migraine patients. We use the analytic signal technique, based on the Hilbert transform, and find that migraine brains are characterized by enhanced alpha band phase synchronization in the presence of visual stimuli. Our findings show that migraine patients have an overactive regulatory mechanism that renders them more sensitive to external stimuli.

We discuss the use of multivariate Granger causality in presence of redundant variables: the application of the standard analysis, in this case, leads to under estimation of causalities. Using the un-normalized version of the causality index, we quantitatively develop the notions of redundancy and synergy in the frame of causality and propose two approaches to group redundant variables: ͑i͒ for a given target, the remaining variables are grouped so as to maximize the total causality and ͑ii͒ the whole set of variables is partitioned to maximize the sum of the causalities between subsets. We show the application to a real neurological experiment, aiming to a deeper understanding of the physiological basis of abnormal neuronal oscillations in the migraine brain. The outcome by our approach reveals the change in the informational pattern due to repetitive transcranial magnetic stimulations. Wiener ͓1͔ and Granger ͓2͔ formalized the notion that if the prediction of one time series could be improved by incorporating the knowledge of past values of a second one, then the latter is said to have a causal influence on the former. Initially developed for econometric applications, Granger causality has gained popularity also among physicists ͑see, e.g., ͓3-7͔͒. A kernel method for Granger causality, introduced in ͓8͔, deals with the nonlinear case by embedding data onto an Hilbert space, and searching for linear relations in that space. Geweke ͓9͔ has generalized Granger causality to a multivariate fashion in order to identify conditional Granger causality; as described in ͓10͔, multivariate causality may be used to infer the structure of dynamical networks ͓11͔ from data.Granger causality is connected to the information flow between variables ͓12͔. Another important notion in information theory is the redundancy in a group of variables, formalized in ͓13͔ as a generalization of the mutual information. A formalism to recognize redundant and synergetic variables in neuronal ensembles has been proposed in ͓14͔ and generalized in ͓15͔; the information theoretic treatments of groups of correlated degrees of freedom can reveal their functional roles in complex systems.The purpose of this work is to show that the presence of redundant variables influences the performance by multivariate Granger causality and to propose a novel approach to exploit redundancy so as to identify functional patterns in data. In the following, we provide a quantitative definition to recognize redundancy and synergy in the frame of causality and show that the maximization of the total causality is connected to the detection of groups of redundant variables.Let us consider n time series ͕x ␣ ͑t͖͒ ␣=1,. . .,n ͓16͔; the state vectors are denotedm being the window length ͑the choice of m can be done using the standard cross-validation scheme͒. Let ⑀͑x ␣ ͉ X͒ be the mean squared error prediction of x ␣ on the basis of all the vectors X ͑corresponding to linear regression or non linear regression by the kernel approach described in ͓8͔͒. The multivariate Gra...