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Edited by: Ramon Guevara Erra, Laboratoire Psychologie de la Perception (CNRS), France

Reviewed by: Anna Korzeniewska, Johns Hopkins University, USA; Julià L. Amengual, Institute du Cerveau et de la Moelle Epiniere, France

*Correspondence: Katarzyna J. Blinowska

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper shortly reviews the measures used to estimate neural synchronization in experimental settings. Our focus is on multivariate measures of dependence based on the Granger causality (G-causality) principle, their applications and performance in respect of robustness to noise, volume conduction, common driving, and presence of a “weak node.” Application of G-causality measures to EEG, intracranial signals and fMRI time series is addressed. G-causality based measures defined in the frequency domain allow the synchronization between neural populations and the directed propagation of their electrical activity to be determined. The time-varying G-causality based measure Short-time Directed Transfer Function (SDTF) supplies information on the dynamics of synchronization and the organization of neural networks. Inspection of effective connectivity patterns indicates a modular structure of neural networks, with a stronger coupling within modules than between them. The hypothetical plausible mechanism of information processing, suggested by the identified synchronization patterns, is communication between tightly coupled modules intermitted by sparser interactions providing synchronization of distant structures.

In recent years a substantial effort has been directed toward elucidating the role of synchronization in mechanisms of neural population coupling. The kind of measure applied to estimate connectivity patterns plays a crucial role in the understanding of this synchronization. A multitude of methods have been devised for estimation of connectivity between neural populations: linear and non-linear, bivariate and multivariate, directed and undirected. It is impossible to describe all of the measures of synchronization in this mini review, but they are described in the review by Blinowska (

Among the most frequently used connectivity measures defined in the time domain, namely cross-correlation, Mutual Information, Transfer Entropy (TE) and Granger Causality Index (GCI), the last two indicate the directedness of information flow. For time-metric methods a contribution of different rhythms may be estimated by means of filtering; however, methods operating in the frequency domain are more convenient for synchronization assessment. In the frequency domain we can distinguish several measures of functional connectivity: coherence, pair-wise measures based on phase information such as the phase lag index (Stam et al.,

The common input problem is a source of a serious pitfalls, corrupting all bivariate measures and leading to the creation of spurious connections. Namely, if signals propagating from a given source are measured at

Volume conduction—a factor limiting the spatial resolution of synchronization measures—is connected to propagation of the electromagnetic field. Since the electromagnetic field propagates at the speed of light, it does not produce phase differences on the electrodes; hence, methods based on phase differences (among them DTF and PDC) are hardly influenced by volume conduction (Kaminski and Blinowska,

Non-linear methods of connectivity are much more affected by noise than the linear ones. Moreover, they are prone to systematic errors (Pereda et al.,

The notion of causality in time series, based on Wiener's idea (Wiener,

In the identification of casual relations one should try to incorporate all possible variables of the process. However, that may be difficult because of the influence of exogenous (environmental) and latent (unmeasured) variables. The problem of eliminating these confounding inputs was confronted by Eichler (

GC was successfully used e.g., to evaluate the directional influences in large-scale sensorimotor cortical networks (Brovelli et al.,

G-causality measures are usually computed in the multivariate autoregressive (MVAR) model framework defined by:

where

The transformation to the frequency domain yields:

To get a proper MVAR fit the number of data points must be larger (at least about an order of magnitude) than the number of model parameters: _{s} ≫ ^{2} (_{s} the number of data points in the window)r. This requires a compromise between _{s}. Alternatively, G-causality measures may be calculated by a non-parametric spectral method (Dhamala et al.,

Directed Transfer Function is defined in the form (Kamiński and Blinowska, _{ij} is an element of the transfer matrix of the MVAR model. DTF describes the causal influence of channel

The non-normalized DTF:

The direct Directed Transfer Function (dDTF) was introduced (Korzeniewska et al.,

ffDTF is a modification of DTF where the denominator is integrated over frequencies, which makes it independent on frequency.

DC—directed coherence (Baccala et al., _{ij}(_{j}(

PDC operates in the frequency domain. However, its spectrum weakly depends on frequency and does not have a direct correspondence to the power spectra of the channels of a process. Unlike DTF, PDC value shows a ratio between transmission from channel

Considering the dependence of PDC on a signal's dynamic ranges, Baccala et al. (

PDC found application e.g., in the study of epileptic seizures (Takahashi et al.,

Astolfi et al. (

Fasoula et al. (

The good spectral resolution of DTF and its robustness to noise makes it the proper measure for revealing synchronization between brain structures. Information processing in the brain involves short-time changes in electrical activity, and DTF is the only measure for which a time-varying version (SDTF) was developed and extensively used. In cases where there are multiple recordings of an experiment available, we may use the repetitions to effectively increase the statistical significance of estimates. In order to follow the dynamics we divide the data into shorter, presumably stationary, overlapping data windows (of length _{S}). Within each window the data covariance matrix ^{(r)} is calculated for every repetition separately (index (_{T} is the number of the repetitions), and then the resulting model is estimated based on the averaged matrix

Another possible solution for estimation of time-varying connectivity is an adaptive approach (Kalman filter, recursive least squares algorithm Hesse et al.,

The effect of transients (event related potentials, ERPs) may disturb connectivity values when estimating time varying transmission. Subtraction of the ERP may be the solution (Kamiński et al.,

In an experiment concerning a motor task and its imagination (Ginter et al.,

In the Continuous Attention Test different geometrical images were presented. The subject had to press a switch when two identical images (target condition) appeared and withhold the reaction for different images (non-target). We integrated flows (significantly differing from the resting state) in the 25–45 Hz frequency band and constructed animations representing dynamically changing propagation patterns (Figure

We also found, by means of SDTF, the main centers of EEG propagation in the frontal and parietal regions during a WM task involving memorization of relations (Blinowska et al.,

Animations (Blinowski et al.,

Application of a network formalism based on assortative mixing (Newman,

Multivariate G-causality based measures provide a useful framework for establishing causal relations between neural populations. They have been successfully applied for finding interactions at subcortical and cortical levels GC measures have been used extensively for intracranial signals (e.g., Brovelli et al.,

Application of G-causality measures to fMRI data is still controversial because of the low sampling rate, long delays of fMRI series in respect to neural activity, and the complex relation between neural activity and blood oxygenation level. The issue is currently under debate (e.g., Bressler and Seth,

DTF and PDC have been widely used for identification of causal relations in EEG. The results of DTF concerning e.g., synchronization mechanisms in sleep (Kaminski et al.,

In comparison with different methods of connectivity analysis, multivariate measures based on Granger principle provide information on causal frequency-specific coupling in neural assemblies, moreover they are robust in respect to noise and volume conduction. Additionally they offer possibilities to follow dynamical changes of interaction between brain structures. In summary, G-causality based measures provide a valuable tool for investigation of the large-scale neural synchronization and its dynamics.

All of the authors, MK, AB, JK, and KB: contributed to the design of the work and interpretation of the data; took part in drafting the paper; critically revised the manuscript; approved the final version; agreed to be accountable for all aspects of the work.

This material is based upon work supported by the National Science Centre in Poland under grants no. 2014/13/B/HS6/03155 and 2011/03/B/HS6/04458, Statutory Grant of Polish Ministry of Science and Higher Education to Faculty of Physics of University of Warsaw and Statutory Grant of Polish Ministry of Science and Higher Education to Institute of Biocybernetics and Biomedical Engineering.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.