Important information on the structure of complex systems can be obtained by measuring to what extent the individual components exchange information among each other. The linear Granger approach, to detect cause-effect relationships between time series, has emerged in recent years as a leading statistical technique to accomplish this task. Here we generalize Granger causality to the nonlinear case using the theory of reproducing kernel Hilbert spaces. Our method performs linear Granger causality in the feature space of suitable kernel functions, assuming arbitrary degree of nonlinearity. We develop a new strategy to cope with the problem of overfitting, based on the geometry of reproducing kernel Hilbert spaces. Applications to coupled chaotic maps and physiological data sets are presented.

We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel-Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: ͑i͒ the nonlinearity of the regression model can be controlled by choosing the kernel function and ͑ii͒ the problem of false causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no a priori assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of the bivariate Granger causality is a better choice than methods using L1 minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to wellknown transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

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 propose a formal expansion of the transfer entropy to put in evidence irreducible sets of variables which provide information for the future state of each assigned target. Multiplets characterized by a large contribution to the expansion are associated to the informational circuits present in the system, with an informational character which can be associated to the sign of the contribution. For the sake of computational complexity, we adopt the assumption of Gaussianity and use the corresponding exact formula for the conditional mutual information. We report the application of the proposed methodology on two electroencephalography (EEG) data sets.

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.

When evaluating causal influence from one time series to another in a multivariate data set it is necessary to take into account the conditioning effect of the other variables. In the presence of many variables and possibly of a reduced number of samples, full conditioning can lead to computational and numerical problems. In this paper, we address the problem of partial conditioning to a limited subset of variables, in the framework of information theory. The proposed approach is tested on simulated data sets and on an example of intracranial EEG recording from an epileptic subject. We show that, in many instances, conditioning on a small number of variables, chosen as the most informative ones for the driver node, leads to results very close to those obtained with a fully multivariate analysis and even better in the presence of a small number of samples. This is particularly relevant when the pattern of causalities is sparse.

There were clear differences in ongoing EEG under visual stimulation, which emerged between the two forms of migraine, probably subtended by increased cortical activation in migraine with aura, and compensatory phenomena of reduced connectivity and functional networks segregation, occurring in patients not experiencing aura symptoms. Further investigation may confirm whether the clinical manifestation of aura symptoms is subtended by a peculiar neuronal connectivity pattern.

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...