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Many different analysis techniques have been developed and applied to EEG recordings that allow one to investigate how different brain areas interact. One particular class of methods, based on the linear parametric representation of multiple interacting time series, is widely used to study causal connectivity in the brain. However, the results obtained by these methods should be interpreted with great care. The goal of this paper is to show, both theoretically and using simulations, that results obtained by applying causal connectivity measures on the sensor (scalp) time series do not allow interpretation in terms of interacting brain sources. This is because (1) the channel locations cannot be seen as an approximation of a source’s anatomical location and (2) spurious connectivity can occur between sensors. Although many measures of causal connectivity derived from EEG sensor time series are affected by the latter, here we will focus on the well-known time domain index of Granger causality (GC) and on the frequency domain directed transfer function (DTF). Using the state-space framework and designing two simulation studies we show that mixing effects caused by volume conduction can lead to spurious connections, detected either by time domain GC or by DTF. Therefore, GC/DTF causal connectivity measures should be computed at the source level, or derived within analysis frameworks that model the effects of volume conduction. Since mixing effects can also occur in the source space, it is advised to combine source space analysis with connectivity measures that are robust to mixing.

Recent studies in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017) showed that a union of partially ordered orthants in Rn can be decomposed only into the largest and the second largest chains. This allows us to calculate the probability of the union of such events in a recursive manner. If the vertices of such orthants designate p-level efficient points, i.e., the multivariate quantile or the multivariate value-at-risk (MVaR) in Rn, then the number of them, say N, is typically very large, which makes it almost impossible to calculate the multivariate conditional value-at-risk (MCVaR) introduced by Prékopa (Ann Oper Res 193(1):49–69, 2012). This is because it takes O(2N) in case of N MVaRs in Rn to find the exact value of MCVaR. In this paper, upon the basis of ideas in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017), together with proper adjustments, we study efficient methods for the calculation of the MCVaR without resorting to an approximation. In fact, the proposed methods not only have polynomial time complexity but also computes the exact value of MCVaR. We also discuss additional benefits MCVaR has to offer over its univariate counter part, the conditional value-at-risk, by providing numerical results. Numerical examples are presented with computing time in both cases of given population and sample data sets.

This study examines the effects of long-run civil wars on healthcare, which is an important component of human capital development and their causality nexus in Afghanistan using the MVAR (modified vector autoregressive) approach and the Granger non-causality model covering data period 2002Q3-2020Q4. The primary results support a significant long-run relationship between variables, while the results of the MVAR model indicate the per capita cost of war, per capita GDP, and age dependency ratio have significantly positive impacts on per capita health expenditures, whereas child mortality rate and crude death rate have negative impacts. The results of the Granger non-causality approach demonstrate that there is a statistically significant bidirectional causality nexus between per capita health expenditure, per capita cost of war, per capita GDP, child mortality rate, crude death rate, and age dependency ratio, while it also supports the existence of strong and significant interconnectivity and multidimensionality between per capita cost of war and per capita health expenditure, with a significantly strong feedback response from the control variables. Important policy implications sourced from the key findings are also discussed.

Measures of interaction ( ) of the EEG are at the forefront of current neuroscientific research. Unfortunately, test-retest reliability can be very low, depending on the measure and its estimation, the EEG-frequency of interest, the length of the signal, and the population under investigation. In addition, artifacts can hamper the continuity of the EEG signal, and in some clinical situations it is impractical to exclude artifacts. We aimed to examine factors that moderate test-retest reliability of measures of interaction. The study involved 40 patients with a range of neurological diseases and memory impairments (age median: 60; range 21-76; 40% female; 22 mild cognitive impairment, 5 subjective cognitive complaints, 13 temporal lobe epilepsy), and 20 healthy controls (age median: 61.5; range 23-74; 70% female). We calculated 14 measures of interaction based on the multivariate autoregressive model from two EEG-recordings separated by 2 weeks. We characterized test-retest reliability by correlating the measures between the two EEG-recordings for variations of data length, data discontinuity, artifact exclusion, model order, and frequency over all combinations of channels and all frequencies, individually for each subject, yielding a correlation coefficient for each participant. Excluding artifacts had strong effects on reliability of some measures, such as classical, real valued coherence (~0.1 before, ~0.9 after artifact exclusion). Full frequency directed transfer function was highly reliable and robust against artifacts. Variation of data length decreased reliability in relation to poor adjustment of model order and signal length. Variation of discontinuity had no effect, but reliabilities were different between model orders, frequency ranges, and patient groups depending on the measure. Pathology did not interact with variation of signal length or discontinuity. Our results emphasize the importance of documenting reliability, which may vary considerably between measures of interaction. We recommend careful selection of measures of interaction in accordance with the properties of the data. When only short data segments are available and when the signal length varies strongly across subjects after exclusion of artifacts, reliability becomes an issue. Finally, measures which show high reliability irrespective of the presence of artifacts could be extremely useful in clinical situations when exclusion of artifacts is impractical.

Time-varying connectivity analysis based on sources reconstructed using inverse modeling of electroencephalographic (EEG) data is important to understand the dynamic behaviour of the brain. We simulated cortical data from a visual spatial attention network with a time-varying connectivity structure, and then simulated the propagation to the scalp to obtain EEG data. Distributed EEG source modeling using sLORETA was applied. We compared different dipole (representing a source) selection strategies based on their time series in a region of interest. Next, we estimated multivariate autoregressive (MVAR) parameters using classical Kalman filter and general linear Kalman filter approaches followed by the calculation of partial directed coherence (PDC). MVAR parameters and PDC values for the selected sources were compared with the ground-truth. We found that the best strategy to extract the time series of a region of interest was to select a dipole with time series showing the highest correlation with the average time series in the region of interest. Dipole selection based on power or based on the largest singular value offer comparable alternatives. Among the different Kalman filter approaches, the use of a general linear Kalman filter was preferred to estimate PDC based connectivity except when only a small number of trials are available. In the latter case, the classical Kalman filter can be an alternative.

Parkinson’s disease (PD) is characterized by abnormal brain oscillations that can change rapidly. Tracking neural alternations with high temporal resolution electrophysiological monitoring methods such as EEG can lead to valuable information about alterations observed in PD. Concomitantly, there have been advances in the high-accuracy performance of deep neural networks (DNNs) using few-patient data. In this study, we propose a method to transform resting-state EEG data into a deep latent space to classify PD subjects from healthy cases. We first used a general orthogonalized directed coherence (gOPDC) method to compute directional connectivity (DC) between all pairwise EEG channels in four frequency bands (Theta, Alpha, Beta, and Gamma) and then converted the DC maps into 2D images. We then used the VGG-16 architecture (trained on the ImageNet dataset) as our pre-trained model, enlisted weights of convolutional layers as initial weights, and fine-tuned all layer weights with our data. After training, the classification achieved 99.62% accuracy, 100% precision, 99.17% recall, 0.9958 F1 score, and 0.9958 AUC averaged for 10 random repetitions of training/evaluating on the proposed deep transfer learning (DTL) network. Using the latent features learned by the network and employing LASSO regression, we found that latent features (as opposed to the raw DC values) were significantly correlated with five clinical indices routinely measured: left and right finger tapping, left and right tremor, and body bradykinesia. Our results demonstrate the power of transfer learning and latent space derivation for the development of oscillatory biomarkers in PD.

Neural information modeling and analysis often requires a measurement of the mutual influence among many signals. A common technique is the conditional Granger causality (cGC) which measures the influence of one time series on another time series in the presence of a third. Geweke has translated this condition into the frequency domain and has explored the mathematical relationships between the time and frequency domain expressions. Chen has observed that in practice, the expressions may return (meaningless) negative numbers, and has proposed an alternative which is based on a partitioned matrix scheme, which we call partitioned Granger causality (pGC). There has been some confusion in the literature about the relationship between cGC and pGC; some authors treat them as essentially identical measures, while others have noted that some properties (such as the relationship between the time and frequency domain expressions) do not hold for the pGC. This paper presents a series of matrix equalities that simplify the calculation of the pGC. In this simplified expression, the essential differences and similarities between the cGC and the pGC become clear; in essence, the pGC is dependent on only a subset of the parameters in the model estimation, and the noise residuals (which are uncorrelated in the cGC) need not be uncorrelated in the pGC. The mathematical results are illustrated with a simulation, and the measures are applied to an EEG dataset.

Directional connectivity in the brain has been typically computed between scalp electroencephalographic (EEG) signals, neglecting the fact that correlations between scalp measurements are partly caused by electrical conduction through the head volume. Although recently proposed techniques are able to identify causality relationships between EEG sources rather than between recording sites, most of them need a priori assumptions about the cerebral regions involved in the EEG generation. We present a novel methodology based on multivariate autoregressive (MVAR) modeling and Independent Component Analysis (ICA) able to determine the temporal activation of the intracerebral EEG sources as well as their approximate locations. The direction of synaptic flow between these EEG sources is then estimated using the directed transfer function (DTF), and the significance of directional coupling strength evaluated with surrogated data. The reliability of this approach was assessed with simulations manipulating the number of data samples, the depth and orientation of the equivalent source dipoles, the presence of different noise sources, and the violation of the non-Gaussianity assumption inherent to the proposed technique. The simulations showed the superior accuracy of the proposed approach over other traditional techniques in most tested scenarios. Its validity was also evaluated analyzing the generation mechanisms of the EEG-alpha rhythm recorded from 20 volunteers under resting conditions. Results suggested that the major generation mechanism underlying EEG-alpha oscillations consists of a strong bidirectional feedback between thalamus and cuneus. The precuneus also seemed to actively participate in the generation of the alpha rhythm although it did not exert a significant causal influence neither on the thalamus nor on the cuneus. All together, these results suggest that the proposed methodology is a promising non-invasive approach for studying directional coupling between mutually interconnected neural populations.

The dynamic nature of functional brain networks is being increasingly recognized in cognitive neuroscience, and methods to analyse such time-varying networks in EEG/MEG data are required. In this work, we propose a pipeline to characterize time-varying networks in single-subject EEG task-related data and further, evaluate its validity on both simulated and experimental datasets. Pre-processing is done to remove channel-wise and trial-wise differences in activity. Functional networks are estimated from short non-overlapping time windows within each \"trial,\" using a sparse-MVAR (Multi-Variate Auto-Regressive) model. Functional \"states\" are then identified by partitioning the entire space of functional networks into a small number of groups/symbols via -means clustering.The multi-trial sequence of symbols is then described by a Markov Model (MM). We show validity of this pipeline on realistic electrode-level simulated EEG data, by demonstrating its ability to discriminate \"trials\" from two experimental conditions in a range of scenarios. We then apply it to experimental data from two individuals using a Brain-Computer Interface (BCI) via a P300 oddball task. Using just the Markov Model parameters, we obtain statistically significant discrimination between target and non-target trials. The functional networks characterizing each 'state' were also highly similar between the two individuals. This work marks the first application of the Markov Model framework to infer time-varying networks from EEG/MEG data. Due to the pre-processing, results from the pipeline are orthogonal to those from conventional ERP averaging or a typical EEG microstate analysis. The results provide powerful proof-of-concept for a Markov model-based approach to analyzing the data, paving the way for its use to track rapid changes in interaction patterns as a task is being performed. MATLAB code for the entire pipeline has been made available.

We propose a new integrative approach to characterize the structure of seizures in the space, time, and frequency domains. Such characterization leads to a new technical development of ictal source analysis for the presurgical evaluation of epilepsy patients. The present new ictal source analysis method consists of three parts. First, a three-dimensional source scanning procedure is performed by a spatio-temporal FINE source localization method to locate the multiple sources responsible for the time evolving ictal rhythms at their onsets. Next, the dynamic behavior of the sources is modeled by a multivariate autoregressive process (MVAR). Lastly, the causal interaction patterns among the sources as a function of frequency are estimated from the MVAR modeling of the source temporal dynamics. The causal interaction patterns indicate the dynamic communications between sources, which are useful in distinguishing the primary sources responsible for the ictal onset from the secondary sources caused by the ictal propagation. The present ictal analysis strategy has been applied to a number of seizures from five epilepsy patients, and their results are consistent with observations from either MRI lesions or SPECT scans, which indicate its effectiveness. Each step of the ictal source analysis is statistically evaluated in order to guarantee the confidence in the results.