Ising model with conserved magnetization on the human connectome: Implications on the relation structure-function in wakefulness and anesthesia.

Dynamical models implemented on the large scale architecture of the human brain may shed light on how a function arises from the underlying structure. This is the case notably for simple abstract models, such as the Ising model. We compare the spin correlations of the Ising model and the empirical functional brain correlations, both at the single link level and at the modular level, and show that their match increases at the modular level in anesthesia, in line with recent results and theories. Moreover, we show that at the peak of the specific heat (the critical state), the spin correlations are minimally shaped by the underlying structural network, explaining how the best match between the structure and function is obtained at the onset of criticality, as previously observed. These findings confirm that brain dynamics under anesthesia shows a departure from criticality and could open the way to novel perspectives when the conserved magnetization is interpreted in terms of a homeostatic principle imposed to neural activity.

Synergy as a warning sign of transitions: The case of the two-dimensional Ising model.

We consider the formalism of information decomposition of target effects from multisource interactions, i.e., the problem of defining redundant and synergistic components of the information that a set of source variables provides about a target, and apply it to the two-dimensional Ising model as a paradigm of a critically transitioning system. Intuitively, synergy is the information about the target variable that is uniquely obtained by taking the sources together, but not considering them alone; redundancy is the information which is shared by the sources. To disentangle the components of the information both at the static level and at the dynamical one, the decomposition is applied respectively to the mutual information and to the transfer entropy between a given spin, target, and a pair of neighboring spins (taken as the drivers). We show that a key signature of an impending phase transition (approached from the disordered size) is the fact that the synergy peaks in the disordered phase, both in the static and in the dynamic case: The synergy can thus be considered a precursor of the transition. The redundancy, instead, reaches its maximum at the critical temperature. The peak of the synergy of the transfer entropy is far more pronounced than those of the static mutual information. We show that these results are robust with respect to the details of the information decomposition approach, as we find the same results using two different methods; moreover, with respect to previous literature rooted in the notion of global transfer entropy, our results demonstrate that considering as few as three variables is sufficient to construct a precursor of the transition, and provide a paradigm for the investigation of a variety of systems prone to crisis, such as financial markets, social media, or epileptic seizures.

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 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 well-known transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

Phase diagram of a generalized Winfree model.

We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling k be a function of the Kuramoto order parameter r through an exponent z such that z=1 corresponds to the standard Winfree model, z<1 strengthens the coupling at low r (low amount of synchronization), and at z>1 , the coupling is weakened for low r . Using both analytical and numerical approaches, we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing z<1 . We also find that the original Winfree model is a rather special case; indeed, the partial locked behavior disappears for z>1 . At fixed k and varying gamma , the stability boundary of the locked phase corresponds to a transition that is continuous for z<1 and first order for z>1 . This change in the nature of the transition is in accordance with a previous study of a similarly modified Kuramoto model.

Leave-one-out prediction error of systolic arterial pressure time series under paced breathing.

In this paper, we consider systolic arterial pressure time series from healthy subjects and chronic heart failure patients, undergoing paced respiration, and show that different physiological states and pathological conditions may be characterized in terms of predictability of time series signals from the underlying biological system. We model time series by the regularized least-squares approach and quantify predictability by the leave-one-out error. We find that the entrainment mechanism connected to paced breath, that renders the arterial blood pressure signal more regular and thus more predictable, is less effective in patients, and this effect correlates with the seriousness of the heart failure. Using a Gaussian kernel, so that all orders of nonlinearity are taken into account, the leave-one-out error separates controls from patients (probability less than 10(-7)), and alive patients from patients for whom cardiac death occurred (probability less than 0.01).

Phase shifts of synchronized oscillators and the systolic-diastolic blood pressure relation.

We study the phase-synchronization properties of systolic and diastolic arterial pressure in healthy subjects. We find that delays in the oscillatory components of the time series depend on the frequency bands that are considered, in particular we find a change of sign in the phase shift going from the very low frequency band to the high frequency band. This behavior should reflect a collective behavior of a system of nonlinear interacting elementary oscillators. We prove that some models describing such systems, e.g., the Winfree and the Kuramoto models, offer a clue to this phenomenon. For these theoretical models there is a linear relationship between phase shifts and the difference of natural frequencies of oscillators and a change of sign in the phase shift naturally emerges.

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.

Expanding the transfer entropy to identify information subgraphs in complex systems.

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 an high value will be associated to informational circuits present in the system, with an informational character (synergetic or redundant) which can be associated to the sign of the contribution. We also present preliminary results on fMRI and EEG data sets.

Expanding the transfer entropy to identify information circuits in complex systems.

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.

Lack of effects of low frequency repetitive transcranial magnetic stimulation on alpha rhythm phase synchronization in migraine patients.

The study aimed to test the modulation induced by 1 Hz repetitive Transcranial Magnetic Stimulation (rTMS) of the occipital cortex on the alpha phase synchronization under repetitive flash stimuli in 15 migraine without aura patients compared to 10 controls. The EEG was recorded by 7 channels, while flash stimuli were delivered at 9, 18, 21 and 24 Hz in basal, rTMS (15 min of 1 Hz stimulation of the occipital cortex) and sham conditions. Migraine patients displayed increased alpha-band phase synchronization under visual stimulation, while an overall desynchronizing effect was evident in controls. The rTMS resulted in a slight increase of synchronization index in migraine patients, which did not cause significant differences in respect to the basal and sham conditions. The synchronizing-desynchronizing changes of alpha rhythm under repetitive flash stimulation, seem independent from the state of occipital cortex excitability. Other mechanisms beyond cortical excitability may contribute to explain migraine pathogenesis.

Redundant variables and Granger causality.

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: (i) for a given target, the remaining variables are grouped so as to maximize the total causality and (ii) 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.

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.

Phase ordering in chaotic map lattices with conserved dynamics.

Dynamical scaling in a two-dimensional lattice model of chaotic maps, in contact with a thermal bath at temperature T, is numerically studied. The model here proposed is equivalent to a conserved Ising model with couplings that fluctuate over the same time scale as spin moves. When coupling fluctuations and thermal fluctuations are both important, this model does not belong to the class of universality of a Langevin equation known as model B; the scaling exponents are continuously varying with T and depend on the map used. The universal behavior of model B is recovered when thermal fluctuations are dominant.

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.

Steady-state visual evoked potentials and phase synchronization in migraine patients.

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.