The causality measure of partial mutual information from mixed embedding (PMIME) revisited

The measure of partial mutual information from mixed embedding (PMIME) is an information theory-based measure to accurately identify the direct and directional coupling, termed Granger causality or simply causality, between the observed variables or subsystems of a high-dimensional dynamical and complex system, without any a priori assumptions about the nature of the coupling relationship. In its core, it is a forward selection procedure that aims to iteratively identify the lag-dependence structure of a given observed variable (response) to all the other observed variables (candidate drivers). This model-free approach is capable of detecting nonlinear interactions, abundantly present in real-world complex systems, and it was shown to perform well on multivariate time series of moderately high dimension. However, the PMIME presents some inefficiencies in its performance mainly when applied on strongly stochastic (linear or nonlinear) systems as it may falsely detect non-existent relationships. Moreover, and by construction, the measure cannot extract purely synergetic relationships present in a system. In the current work, the issue of false detections is addressed by introducing an improved resampling significance test and a procedure of rechecking the identified drivers (backward revision). Regarding the inability to detect synergetic relationships, the PMIME is further enhanced by checking pairs as candidate drivers for the response variable after having considered all drivers individually. The effects of these modifications are investigated in a systematic simulation study on properly designed systems involving strong stochasticity, regressor terms with synergetic effects, and a system dimension ranging from 3 to 30. The overall results of the simulations indicate that these modifications indeed improve the performance of PMIME and alleviate to a significant degree the issues of the original algorithm. Guidelines for balancing between accuracy and computational efficiency are also given, particularly relevant for real-world applications. Finally, the measure performance is investigated in the study of futures of various government bonds and stock market indices in the period around COVID-19 pandemic.

Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct ROMs on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. We show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Detecting Nonlinear Interactions in Complex Systems: Application in Financial Markets

Emerging or diminishing nonlinear interactions in the evolution of a complex system may signal a possible structural change in its underlying mechanism. This type of structural break may exist in many applications, such as in climate and finance, and standard methods for change-point detection may not be sensitive to it. In this article, we present a novel scheme for detecting structural breaks through the occurrence or vanishing of nonlinear causal relationships in a complex system. A significance resampling test was developed for the null hypothesis (H₀) of no nonlinear causal relationships using (a) an appropriate Gaussian instantaneous transform and vector autoregressive (VAR) process to generate the resampled multivariate time series consistent with H₀; (b) the modelfree Granger causality measure of partial mutual information from mixed embedding (PMIME) to estimate all causal relationships; and (c) a characteristic of the network formed by PMIME as test statistic. The significance test was applied to sliding windows on the observed multivariate time series, and the change from rejection to no-rejection of H₀, or the opposite, signaled a non-trivial change of the underlying dynamics of the observed complex system. Different network indices that capture different characteristics of the PMIME networks were used as test statistics. The test was evaluated on multiple synthetic complex and chaotic systems, as well as on linear and nonlinear stochastic systems, demonstrating that the proposed methodology is capable of detecting nonlinear causality. Furthermore, the scheme was applied to different records of financial indices regarding the global financial crisis of 2008, the two commodity crises of 2014 and 2020, the Brexit referendum of 2016, and the outbreak of COVID-19, accurately identifying the structural breaks at the identified times.

Bridging the gap: TMS-EEG from Lab to Clinic

The combination of transcranial magnetic stimulation (TMS) and electroencephalography (EEG) has reached technological maturity and has been an object of significant scientific interest for over two decades. Ιn parallel, accumulating evidence highlights the potential of TMS-EEG as a useful tool in the field of clinical neurosciences. Nevertheless, its clinical utility has not yet been established, partly because technical and methodological limitations have created a gap between an evolving scientific tool and standard clinical practice. Here we review some of the identified gaps that still prevent TMS-EEG moving from science laboratories to clinical practice. The principal and partly overlapping gaps include: 1) complex and laborious application, 2) difficulty in obtaining high-quality signals, 3) suboptimal accuracy and reliability, and 4) insufficient understanding of the neurobiological substrate of the responses. All these four aspects need to be satisfactorily addressed for the method to become clinically applicable and enter the diagnostic and therapeutic arena. In the current review, we identify steps that might be taken to address these issues and discuss promising recent studies providing tools to aid bridging the gaps.

Comparison of Causality Network Estimation in the Sensor and Source Space: Simulation and Application on EEG

The usage of methods for the estimation of the true underlying connectivity among the observed variables of a system is increasing, especially in the domain of neuroscience. Granger causality and similar concepts are employed for the estimation of the brain network from electroencephalogram (EEG) data. Also source localization techniques, such as the standardized low resolution electromagnetic tomography (sLORETA), are widely used for obtaining more reliable data in the source space. In this work, connectivity structures are estimated in the sensor and in the source space making use of the sLORETA transformation for simulated and for EEG data with episodes of spontaneous epileptiform discharges (ED). From the comparative simulation study on high-dimensional coupled stochastic and deterministic systems originating in the sensor space, we conclude that the structure of the estimated causality networks differs in the sensor space and in the source space. Moreover, different network types, such as random, small-world and scale-free, can be better discriminated on the basis of the data in the original sensor space than on the transformed data in the source space. Similarly, in EEG epochs containing epileptiform discharges, the discriminative ability of network topological indices was significantly better in the sensor compared to the source level. In conclusion, causality networks constructed at the sensor and source level, for both simulated and empirical data, exhibit significant structural differences. These observations indicate that further studies are warranted in order to clarify the exact relationship between data registered in the sensor and source space.

Evaluation of White Matter Integrity Utilizing the DELPHI (TMS-EEG) System

Objective

The aim of this study was to evaluate brain white matter (WM) fibers connectivity damage in stroke and traumatic brain injury (TBI) subjects by direct electrophysiological imaging (DELPHI) that analyzes transcranial magnetic stimulation (TMS)-evoked potentials (TEPs).

Methods

The study included 123 participants, out of which 53 subjects with WM-related pathologies (39 stroke, 14 TBI) and 70 healthy age-related controls. All subjects underwent DELPHI brain network evaluations of TMS-electroencephalogram (EEG)-evoked potentials and diffusion tensor imaging (DTI) scans for quantification of WM microstructure fractional anisotropy (FA).

Results

DELPHI output measures show a significant difference between the healthy and stroke/TBI groups. A multidimensional approach was able to classify healthy from unhealthy with a balanced accuracy of 0.81 ± 0.02 and area under the curve (AUC) of 0.88 ± 0.01. Moreover, a multivariant regression model of DELPHI output measures achieved prediction of WM microstructure changes measured by FA with the highest correlations observed for fibers proximal to the stimulation area, such as frontal corpus callosum (r = 0.7 ± 0.02), anterior internal capsule (r = 0.7 ± 0.02), and fronto-occipital fasciculus (r = 0.65 ± 0.03).

Conclusion

These results indicate that features of TMS-evoked response are correlated to WM microstructure changes observed in pathological conditions, such as stroke and TBI, and that a multidimensional approach combining these features in supervised learning methods serves as a strong indicator for abnormalities and changes in WM integrity.

Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings.

Evaluation of algorithms for correction of transcranial magnetic stimulation-induced artifacts in electroencephalograms

Transcranial magnetic stimulation combined with electroencephalography (TMS-EEG) is widely used to study the reactivity and connectivity of brain regions for clinical or research purposes. The electromagnetic pulse of the TMS device generates at the instant of administration an artifact of large amplitude and a duration up to tens of milliseconds that overlaps with brain activity. Methods for TMS artifact correction have been developed to remove the artifact and recover the underlying, immediate response of the cerebral cortex to the magnetic stimulus. In this study, four such algorithms are evaluated. Since there is no ground truth for the masked brain activity, pilot data formed from the superposition of the isolated TMS artifact on EEG brain activity are used to evaluate the performance of the algorithms. Different scenarios of TMS-EEG experiments are considered for the evaluation: TMS at resting state, TMS inducing epileptiform discharges, and TMS administered during epileptiform discharges. We show that a proposed gap filling method is able to reproduce qualitative characteristics and, in many cases, closely resemble the hidden EEG signal. Finally, shortcomings of the TMS correction algorithms as well as the pilot data approach are discussed. Graphical abstract The transcranial magnetic stimulation (TMS) artifact on the electroencephalogram (EEG) and its correction.

Introducing a Novel Approach for Evaluation and Monitoring of Brain Health Across Life Span Using Direct Non-invasive Brain Network Electrophysiology

Objective

Evaluation and monitoring of brain health throughout aging by direct electrophysiological imaging (DELPHI) which analyzes TMS (transcranial magnetic stimulation) evoked potentials.

Methods

Transcranial magnetic stimulation evoked potentials formation, coherence and history dependency, measured using electroencephalogram (EEG), was extracted from 80 healthy subjects in different age groups, 25–85 years old, and 20 subjects diagnosed with mild dementia (MD), over 70 years old. Subjects brain health was evaluated using MRI scans, neurocognitive evaluation, and computerized testing and compared to DELPHI analysis of brain network functionality.

Results

A significant decrease in signal coherence is observed with age in connectivity maps, mostly in inter-hemispheric temporal, and parietal areas. MD patients display a pronounced decrease in global and inter-hemispheric frontal connectivity compared to healthy controls. Early and late signal slope ratio also display a significant, age dependent, change with pronounced early slope, phase shift, between normal healthy aging, and MD. History dependent analysis demonstrates a binary step function classification of healthy brain vs. abnormal aging subjects mostly for late slope. DELPHI measures demonstrate high reproducibility with reliability coefficients of around 0.9.

Conclusion

These results indicate that features of evoked response, as charge transfer, slopes of response, and plasticity are altered during abnormal aging and that these fundamental properties of network functionality can be directly evaluated and monitored using DELPHI.

Which Reference Should We Use for EEG and ERP practice?

Which reference is appropriate for the scalp ERP and EEG studies? This unsettled problem still inspires unceasing debate. The ideal reference should be the one with zero or constant potential but unfortunately it is well known that no point on the body fulfills this condition. Consequently, more than ten references are used in the present EEG-ERP studies. This diversity seriously undermines the reproducibility and comparability of results across laboratories. A comprehensive review accompanied by a brief communication with rigorous derivations and notable properties (Hu et al. Brain Topogr, 2019. https://doi.org/10.1007/s10548-019-00706-y ) is thus necessary to provide application-oriented principled recommendations. In this paper current popular references are classified into two categories: (1) unipolar references that construct a neutral reference, including both online unipolar references and offline re-references. Examples of unipolar references are the reference electrode standardization technique (REST), average reference (AR), and linked-mastoids/ears reference (LM); (2) non-unipolar references that include the bipolar reference and the Laplacian reference. We show that each reference is derived with a different assumption and serves different aims. We also note from (Hu et al. 2019) that there is a general form for the reference problem, the 'no memory' property of the unipolar references, and a unified estimator for the potentials at infinity termed as the regularized REST (rREST) which has more advantageous statistical evidence than AR. A thorough discussion of the advantages and limitations of references is provided with recommendations in the hope to clarify the role of each reference in the ERP and EEG practice.

The Statistics of EEG Unipolar References: Derivations and Properties

In this brief communication, which complements the EEG reference review (Yao et al. in Brain Topogr, 2019), we provide the mathematical derivations that show: (1) any EEG reference admits the general form of a linear transformation of the ideal multichannel EEG potentials with reference to infinity; (2) the average reference (AR), the reference electrode standardization technique (REST), and its regularized version (rREST) are solving the linear inverse problems that can be derived from both the maximum likelihood estimate (MLE) and the Bayesian theory; however, REST is based on more informative prior/constraint of volume conduction than that of AR; (3) we show for the first time that REST is also a unipolar reference (UR), allowing us to define a general family of URs with unified notations; (4) some notable properties of URs are ‘no memory’, ‘rank deficient by 1’, and ‘orthogonal projector centering’; (5) we also point out here, for the first time, that rREST provides the optimal interpolating function that can be used when the reference channel is missing or the ‘bad’ channels are rejected. The derivations and properties imply that: (a) any two URs can transform to each other and referencing with URs multiple times will not accumulate artifacts; (b) whatever URs the EEG data was previously transformed with, the minimum norm solution to the reference problem will be REST and AR with and without modeling volume conduction, respectively; (c) the MLE and the Bayesian theory show the theoretical optimality of REST. The advantages and limitations of AR and REST are discussed to guide readers for their proper use.

A study for multiscale information transfer measures based on conditional mutual information

As the big data science develops, efficient methods are demanded for various data analysis. Granger causality provides the prime model for quantifying causal interactions. However, this theoretic model does not meet the requirement for real-world data analysis, because real-world time series are diverse whose models are usually unknown. Therefore, model-free measures such as information transfer measures are strongly desired. Here, we propose the multi-scale extension of conditional mutual information measures using MORLET wavelet, which are named the WM and WPM. The proposed measures are computational efficient and interpret information transfer by multi-scales. We use both synthetic data and real-world examples to demonstrate the efficiency of the new methods. The results of the new methods are robust and reliable. Via the simulation studies, we found the new methods outperform the wavelet extension of transfer entropy (WTE) in both computational efficiency and accuracy. The features and properties of the proposed measures are also discussed.

Automatic Seizure Detection Based on Morphological Features Using One-Dimensional Local Binary Pattern on Long-Term EEG

Epileptic neurological disorder of the brain is widely diagnosed using the electroencephalography (EEG) technique. EEG signals are nonstationary in nature and show abnormal neural activity during the ictal period. Seizures can be identified by analyzing and obtaining features of EEG signal that can detect these abnormal activities. The present work proposes a novel morphological feature extraction technique based on the local binary pattern (LBP) operator. LBP provides a unique decimal value to a sample point by weighing the binary outcomes after thresholding the neighboring samples with the present sample point. These LBP values assist in capturing the rising and falling edges of the EEG signal, thus providing a morphologically featured discriminating pattern for epilepsy detection. In the present work, the variability in the LBP values is measured by calculating the sum of absolute difference of the consecutive LBP values. Interquartile range is calculated over the preprocessed EEG signal to provide dispersion measure in the signal. For classification purpose, K-nearest neighbor classifier is used, and the performance is evaluated on 896.9 hours of data from CHB-MIT continuous EEG database. Mean accuracy of 99.7% and mean specificity of 99.8% is obtained with average false detection rate of 0.47/h and sensitivity of 99.2% for 136 seizures.

MATLAB Toolboxes for Reference Electrode Standardization Technique (REST) of Scalp EEG

Reference electrode standardization technique (REST) has been increasingly acknowledged and applied as a re-reference technique to transform an actual multi-channels recordings to approximately zero reference ones in electroencephalography/event-related potentials (EEG/ERPs) community around the world in recent years. However, a more easy-to-use toolbox for re-referencing scalp EEG data to zero reference is still lacking. Here, we have therefore developed two open-source MATLAB toolboxes for REST of scalp EEG. One version of REST is closely integrated into EEGLAB, which is a popular MATLAB toolbox for processing the EEG data; and another is a batch version to make it more convenient and efficient for experienced users. Both of them are designed to provide an easy-to-use for novice researchers and flexibility for experienced researchers. All versions of the REST toolboxes can be freely downloaded at http://www.neuro.uestc.edu.cn/rest/Down.html, and the detailed information including publications, comments and documents on REST can also be found from this website. An example of usage is given with comparative results of REST and average reference. We hope these user-friendly REST toolboxes could make the relatively novel technique of REST easier to study, especially for applications in various EEG studies.

Electroencephalography in the Diagnosis of Genetic Generalized Epilepsy Syndromes

Genetic generalized epilepsy (GGE) consists of several syndromes diagnosed and classified on the basis of clinical features and electroencephalographic (EEG) abnormalities. The main EEG feature of GGE is bilateral, synchronous, symmetric, and generalized spike-wave complex. Other classic EEG abnormalities are polyspikes, epileptiform K-complexes and sleep spindles, polyspike-wave discharges, occipital intermittent rhythmic delta activity, eye-closure sensitivity, fixation-off sensitivity, and photoparoxysmal response. However, admixed with typical changes, atypical epileptiform discharges are also commonly seen in GGE. There are circadian variations of generalized epileptiform discharges. Sleep, sleep deprivation, hyperventilation, intermittent photic stimulation, eye closure, and fixation-off are often used as activation techniques to increase the diagnostic yield of EEG recordings. Reflex seizure-related EEG abnormalities can be elicited by the use of triggers such as cognitive tasks and pattern stimulation during the EEG recording in selected patients. Distinct electrographic abnormalities to help classification can be identified among different electroclinical syndromes.

Is the Surface Potential Integral of a Dipole in a Volume Conductor Always Zero? A Cloud Over the Average Reference of EEG and ERP

Currently, average reference is one of the most widely adopted references in EEG and ERP studies. The theoretical assumption is the surface potential integral of a volume conductor being zero, thus the average of scalp potential recordings might be an approximation of the theoretically desired zero reference. However, such a zero integral assumption has been proved only for a spherical surface. In this short communication, three counter-examples are given to show that the potential integral over the surface of a dipole in a volume conductor may not be zero. It depends on the shape of the conductor and the orientation of the dipole. This fact on one side means that average reference is not a theoretical 'gold standard' reference, and on the other side reminds us that the practical accuracy of average reference is not only determined by the well-known electrode array density and its coverage but also intrinsically by the head shape. It means that reference selection still is a fundamental problem to be fixed in various EEG and ERP studies.

Efficient Computation of Functional Brain Networks: toward Real-Time Functional Connectivity

Functional Connectivity has demonstrated to be a key concept for unraveling how the brain balances functional segregation and integration properties while processing information. This work presents a set of open-source tools that significantly increase computational efficiency of some well-known connectivity indices and Graph-Theory measures. PLV, PLI, ImC, and wPLI as Phase Synchronization measures, Mutual Information as an information theory based measure, and Generalized Synchronization indices are computed much more efficiently than prior open-source available implementations. Furthermore, network theory related measures like Strength, Shortest Path Length, Clustering Coefficient, and Betweenness Centrality are also implemented showing computational times up to thousands of times faster than most well-known implementations. Altogether, this work significantly expands what can be computed in feasible times, even enabling whole-head real-time network analysis of brain function.

A Multimodal Imaging- and Stimulation-based Method of Evaluating Connectivity-related Brain Excitability in Patients with Epilepsy

Resting-state functional connectivity MRI (rs-fcMRI) is a technique that identifies connectivity between different brain regions based on correlations over time in the blood-oxygenation level dependent signal. rs-fcMRI has been applied extensively to identify abnormalities in brain connectivity in different neurologic and psychiatric diseases. However, the relationship among rs-fcMRI connectivity abnormalities, brain electrophysiology and disease state is unknown, in part because the causal significance of alterations in functional connectivity in disease pathophysiology has not been established. Transcranial Magnetic Stimulation (TMS) is a technique that uses electromagnetic induction to noninvasively produce focal changes in cortical activity. When combined with electroencephalography (EEG), TMS can be used to assess the brain's response to external perturbations. Here we provide a protocol for combining rs-fcMRI, TMS and EEG to assess the physiologic significance of alterations in functional connectivity in patients with neuropsychiatric disease. We provide representative results from a previously published study in which rs-fcMRI was used to identify regions with abnormal connectivity in patients with epilepsy due to a malformation of cortical development, periventricular nodular heterotopia (PNH). Stimulation in patients with epilepsy resulted in abnormal TMS-evoked EEG activity relative to stimulation of the same sites in matched healthy control patients, with an abnormal increase in the late component of the TMS-evoked potential, consistent with cortical hyperexcitability. This abnormality was specific to regions with abnormal resting-state functional connectivity. Electrical source analysis in a subject with previously recorded seizures demonstrated that the origin of the abnormal TMS-evoked activity co-localized with the seizure-onset zone, suggesting the presence of an epileptogenic circuit. These results demonstrate how rs-fcMRI, TMS and EEG can be utilized together to identify and understand the physiological significance of abnormal brain connectivity in human diseases.

Discrimination of coupling structures using causality networks from multivariate time series

Measures of Granger causality on multivariate time series have been used to form the so-called causality networks. A causality network represents the interdependence structure of the underlying dynamical system or coupled dynamical systems, and its properties are quantified by network indices. In this work, it is investigated whether network indices on networks generated by an appropriate Granger causality measure can discriminate different coupling structures. The information based Granger causality measure of partial mutual information from mixed embedding (PMIME) is used to form causality networks, and a large number of network indices are ranked according to their ability to discriminate the different coupling structures. The evaluation of the network indices is done with a simulation study based on two dynamical systems, the coupled Mackey-Glass delay differential equations and the neural mass model, both of 25 variables, and three prototypes of coupling structures, i.e., random, small-world, and scale-free. It is concluded that the setting of PMIME combined with a network index attains high level of discrimination of the coupling structures solely on the basis of the observed multivariate time series. This approach is demonstrated to identify epileptic seizures emerging during electroencephalogram recordings.

Detection of Intracranial Signatures of Interictal Epileptiform Discharges from Concurrent Scalp EEG

Interictal epileptiform discharges (IEDs) are transient neural electrical activities that occur in the brain of patients with epilepsy. A problem with the inspection of IEDs from the scalp electroencephalogram (sEEG) is that for a subset of epileptic patients, there are no visually discernible IEDs on the scalp, rendering the above procedures ineffective, both for detection purposes and algorithm evaluation. On the other hand, intracranially placed electrodes yield a much higher incidence of visible IEDs as compared to concurrent scalp electrodes. In this work, we utilize concurrent scalp and intracranial EEG (iEEG) from a group of temporal lobe epilepsy (TLE) patients with low number of scalp-visible IEDs. The aim is to determine whether by considering the timing information of the IEDs from iEEG, the resulting concurrent sEEG contains enough information for the IEDs to be reliably distinguished from non-IED segments. We develop an automatic detection algorithm which is tested in a leave-subject-out fashion, where each test subject’s detection algorithm is based on the other patients’ data. The algorithm obtained a [Formula: see text] accuracy in recognizing scalp IED from non-IED segments with [Formula: see text] accuracy when trained and tested on the same subject. Also, it was able to identify nonscalp-visible IED events for most patients with a low number of false positive detections. Our results represent a proof of concept that IED information for TLE patients is contained in scalp EEG even if they are not visually identifiable and also that between subject differences in the IED topology and shape are small enough such that a generic algorithm can be used.

Causality networks from multivariate time series and application to epilepsy

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. For this, realizations on high dimensional coupled dynamical systems are considered and the performance of the Granger causality measures is evaluated, seeking for the measures that form networks closest to the true network of the dynamical system. In particular, the comparison focuses on Granger causality measures that reduce the state space dimension when many variables are observed. Further, the linear and nonlinear Granger causality measures of dimension reduction are compared to a standard Granger causality measure on electroencephalographic (EEG) recordings containing episodes of epileptiform discharges.