Sebastiano Stramaglia

Kernel method for nonlinear granger causality.

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.

Nonlinear connectivity by Granger causality

The communication among neuronal populations, reflected by transient synchronous activity, is the mechanism underlying the information processing in the brain. Although it is widely assumed that the interactions among those populations (i.e. functional connectivity) are highly nonlinear, the amount of nonlinear information transmission and its functional roles are not clear. The state of the art to understand the communication between brain systems are dynamic causal modeling (DCM) and Granger causality. While DCM models nonlinear couplings, Granger causality, which constitutes a major tool to reveal effective connectivity, and is widely used to analyze EEG/MEG data as well as fMRI signals, is usually applied in its linear version. In order to capture nonlinear interactions between even short and noisy time series, a few approaches have been proposed. We review them and focus on a recently proposed flexible approach has been recently proposed, consisting in the kernel version of Granger causality. We show the application of the proposed approach on EEG signals and fMRI data.

Kernel-Granger causality and the analysis of dynamical networks.

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: (i) the nonlinearity of the regression model can be controlled by choosing the kernel function and (ii) 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 nodes 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 well-known transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

Altered processing of sensory stimuli in patients with migraine

Migraine is a cyclic disorder, in which functional and morphological brain changes fluctuate over time, culminating periodically in an attack. In the migrainous brain, temporal processing of external stimuli and sequential recruitment of neuronal networks are often dysfunctional. These changes reflect complex CNS dysfunction patterns. Assessment of multimodal evoked potentials and nociceptive reflex responses can reveal altered patterns of the brains electrophysiological activity, thereby aiding our understanding of the pathophysiology of migraine. In this Review, we summarize the most important findings on temporal processing of evoked and reflex responses in migraine. Considering these data, we propose that thalamocortical dysrhythmia may be responsible for the altered synchronicity in migraine. To test this hypothesis in future research, electrophysiological recordings should be combined with neuroimaging studies so that the temporal patterns of sensory processing in patients with migraine can be correlated with the accompanying anatomical and functional changes.

Nonlinear parametric model for Granger causality of time series

The notion of Granger causality between two time series examines if the prediction of one series could be improved by incorporating information of the other. In particular, if the prediction error of the first time series is reduced by including measurements from the second time series, then the second time series is said to have a causal influence on the first one. We propose a radial basis function approach to nonlinear Granger causality. The proposed model is not constrained to be additive in variables from the two time series and can approximate any function of these variables, still being suitable to evaluate causality. Usefulness of this measure of causality is shown in two applications. In the first application, a physiological one, we consider time series of heart rate and blood pressure in congestive heart failure patients and patients affected by sepsis: we find that sepsis patients, unlike congestive heart failure patients, show symmetric causal relationships between the two time series. In the second application, we consider the feedback loop in a model of excitatory and inhibitory neurons: we find that in this system causality measures the combined influence of couplings and membrane time constants.

Identification of network modules by optimization of ratio association

We introduce a novel method for identifying the modular structures of a network based on the maximization of an objective function: the ratio association. This cost function arises when the communities detection problem is described in the probabilistic autoencoder frame. An analogy with kernel k-means methods allows us to develop an efficient optimization algorithm, based on the deterministic annealing scheme. The performance of the proposed method is shown on real data sets and on simulated networks.

Causal information approach to partial conditioning in multivariate data sets.

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.

NEW REGULARIZATION SCHEME FOR PHASE UNWRAPPING

A new, to our knowledge, algorithm for the phase unwrapping (PU) problem that is based on stochastic relaxation is proposed and analyzed. Unlike regularization schemes previously proposed to handle this problem, our approach dispells the following two assumptions about the solution: a Gaussian model for noise and the magnitude of the true phase-field gradients being less than pi everywhere. We formulate PU as a constrained optimization problem for the field of integer multiples of 2pi, which must be added to the wrapped phase gradient to recover the true phase gradient. By solving the optimization problem using simulated annealing with constraints, one can obtain a consistent solution under difficult conditions resulting from noise and undersampling. Results from synthetic test images are reported.

Clustering data by inhomogeneous chaotic map lattices.

A new approach to clustering, based on the physical properties of inhomogeneous coupled chaotic maps, is presented. A chaotic map is assigned to each data point and short range couplings are introduced. The stationary regime of the system corresponds to a macroscopic attractor independent of the initial conditions. The mutual information between pairs of maps serves to partition the data set in clusters, without prior assumptions about the structure of the underlying distribution of the data. Experiments on simulated and real data sets show the effectiveness of the proposed algorithm.

Synergy and redundancy in the Granger causal analysis of dynamical networks

We analyze, by means of Granger causality (GC), the effect of synergy and redundancy in the inference (from time series data) of the information flow between subsystems of a complex network. While we show that fully conditioned GC (CGC) is not affected by synergy, the pairwise analysis fails to prove synergetic effects. In cases when the number of samples is low, thus making the fully conditioned approach unfeasible, we show that partially conditioned GC (PCGC) is an effective approach if the set of conditioning variables is properly chosen. Here we consider two different strategies (based either on informational content for the candidate driver or on selecting the variables with highest pairwise influences) for PCGC and show that, depending on the data structure, either one or the other might be equally valid. On the other hand, we observe that fully conditioned approaches do not work well in the presence of redundancy, thus suggesting the strategy of separating the pairwise links in two subsets: those corresponding to indirect connections of the CGC (which should thus be excluded) and links that can be ascribed to redundancy effects and, together with the results from the fully connected approach, provide a better description of the causality pattern in the presence of redundancy. Finally we apply these methods to two different real datasets. First, analyzing electrophysiological data from an epileptic brain, we show that synergetic effects are dominant just before seizure occurrences. Second, our analysis applied to gene expression time series from HeLa culture shows that the underlying regulatory networks are characterized by both redundancy and synergy.