Silvia Erla

Measuring Connectivity in Linear Multivariate Processes: Definitions, Interpretation, and Practical Analysis

This tutorial paper introduces a common framework for the evaluation of widely used frequency-domain measures of coupling (coherence, partial coherence) and causality (directed coherence, partial directed coherence) from the parametric representation of linear multivariate (MV) processes. After providing a comprehensive time-domain definition of the various forms of connectivity observed in MV processes, we particularize them to MV autoregressive (MVAR) processes and derive the corresponding frequency-domain measures. Then, we discuss the theoretical interpretation of these MVAR-based connectivity measures, showing that each of them reflects a specific time-domain connectivity definition and how this results in the description of peculiar aspects of the information transfer in MV processes. Furthermore, issues related to the practical utilization of these measures on real-time series are pointed out, including MVAR model estimation and significance assessment. Finally, limitations and pitfalls arising from model mis-specification are discussed, indicating possible solutions and providing practical recommendations for a safe computation of the connectivity measures. An example of estimation of the presented measures from multiple EEG signals recorded during a combined visuomotor task is also reported, showing how evaluation of coupling and causality in the frequency domain may help describing specific neurophysiological mechanisms.

A framework for assessing frequency domain causality in physiological time series with instantaneous effects.

We present an approach for the quantification of directional relations in multiple time series exhibiting significant zero-lag interactions. To overcome the limitations of the traditional multivariate autoregressive (MVAR) modelling of multiple series, we introduce an extended MVAR (eMVAR) framework allowing either exclusive consideration of time-lagged effects according to the classic notion of Granger causality, or consideration of combined instantaneous and lagged effects according to an extended causality definition. The spectral representation of the eMVAR model is exploited to derive novel frequency domain causality measures that generalize to the case of instantaneous effects the known directed coherence (DC) and partial DC measures. The new measures are illustrated in theoretical examples showing that they reduce to the known measures in the absence of instantaneous causality, and describe peculiar aspects of directional interaction among multiple series when instantaneous causality is non-negligible. Then, the issue of estimating eMVAR models from time-series data is faced, proposing two approaches for model identification and discussing problems related to the underlying model assumptions. Finally, applications of the framework on cardiovascular variability series and multichannel EEG recordings are presented, showing how it allows one to highlight patterns of frequency domain causality consistent with well-interpretable physiological interaction mechanisms.

Inferior frontal gyrus links visual and motor cortices during a visuomotor precision grip force task.

Coordination between vision and action relies on a fronto-parietal network that receives visual and proprioceptive sensory input in order to compute motor control signals. Here, we investigated with magnetoencephalography (MEG) which cortical areas are functionally coupled on the basis of synchronization during visuomotor integration. MEG signals were recorded from twelve healthy adults while performing a unimanual visuomotor (VM) task and control conditions. The VM task required the integration of pinch motor commands with visual sensory feedback. By using a beamformer, we localized the neural activity in the frequency range of 1-30Hz during the VM compared to rest. Virtual sensors were estimated at the active locations. A multivariate autoregressive model was used to estimate the power and coherence of estimated activity at the virtual sensors. Event-related desynchronisation (ERD) during VM was observed in early visual areas, the rostral part of the left inferior frontal gyrus (IFG), the right IFG, the superior parietal lobules, and the left hand motor cortex (M1). Functional coupling in the alpha frequency band bridged the regional activities observed in motor and visual cortices (the start and the end points in the visuomotor loop) through the left or right IFG. Coherence between the left IFG and left M1 correlated inversely with the task performance. Our results indicate that an occipital-prefrontal-motor functional network facilitates the modulation of instructed motor responses to visual cues. This network may supplement the mechanism for guiding actions that is fully incorporated into the dorsal visual stream.

k-Nearest neighbour local linear prediction of scalp EEG activity during intermittent photic stimulation

The characterization of the EEG response to photic stimulation (PS) is an important issue with significant clinical relevance. This study aims to quantify and map the complexity of the EEG during PS, where complexity is measured as the degree of unpredictability resulting from local linear prediction. EEG activity was recorded with eyes closed (EC) and eyes open (EO) during resting and PS at 5, 10, and 15 Hz in a group of 30 healthy subjects and in a case-report of a patient suffering from cerebral ischemia. The mean squared prediction error (MSPE) resulting from k-nearest neighbour local linear prediction was calculated in each condition as an index of EEG unpredictability. The linear or nonlinear nature of the system underlying EEG activity was evaluated quantifying MSPE as a function of the neighbourhood size during local linear prediction, and by surrogate data analysis as well. Unpredictability maps were obtained for each subject interpolating MSPE values over a schematic head representation. Results on healthy subjects evidenced: (i) the prevalence of linear mechanisms in the generation of EEG dynamics, (ii) the lower predictability of EO EEG, (iii) the desynchronization of oscillatory mechanisms during PS leading to increased EEG complexity, (iv) the entrainment of alpha rhythm during EC obtained by 10 Hz PS, and (v) differences of EEG predictability among different scalp regions. Ischemic patient showed different MSPE values in healthy and damaged regions. The EEG predictability decreased moving from the early acute stage to a stage of partial recovery. These results suggest that nonlinear prediction can be a useful tool to characterize EEG dynamics during PS protocols, and may consequently constitute a complement of quantitative EEG analysis in clinical applications.

An identifiable model to assess frequency-domain granger causality in the presence of significant instantaneous interactions

We present a new approach for the investigation of Granger causality in the frequency domain by means of the partial directed coherence (PDC). The approach is based on the utilization of an extended multivariate autoregressive (MVAR) model, including instantaneous effects in addition to the lagged effects traditionally studied, to fit the observed multiple time series prior to PDC computation. Model identification is performed combining standard MVAR coefficient estimation with a recent technique for instantaneous causal modeling based on independent component analysis. The approach is first validated on simulated MVAR processes showing that, in the presence of instantaneous effects, only the extended model is able to interpret the imposed Granger causality patterns, while the traditional MVAR approach may yield strongly biased PDC estimates. The subsequent application to multichannel EEG time series confirms the potentiality of the approach in real data applications, as the importance of instantaneous effects led to significant differences in the PDC estimated after traditional and extended MVAR identification.

Detecting nonlinear causal interactions between dynamical systems by non-uniform embedding of multiple time series

This study introduces a new approach for the detection of nonlinear Granger causality between dynamical systems. The approach is based on embedding the multivariate (MV) time series measured from the systems X and Y by means of a sequential, non-uniform procedure, and on using the corrected conditional entropy (CCE) as unpredictability measure. The causal coupling from X to Y is quantified as the relative decrease of CCE measured after allowing the series of X to enter the embedding procedure for the description of Y. The ability of the approach to quantify nonlinear causality is assessed on MV time series measured from simulated dynamical systems with unidirectional coupling (the Rössler-Lorenz deterministic system) and bidirectional coupling (two coupled stochastic systems). The method is then applied to real magnetoencephalographic data measured during a visuo-tactile cognitive experiment, showing values of causal coupling consistent with the hypothesis of a cross-processing of different sensory modalities.

Studying Brain Visuo-Tactile Integration through Cross-Spectral Analysis of Human MEG Recordings

An important aim in cognitive neuroscience is to identify the networks connecting different brain areas and their role in executing complex tasks. In this study, visuo-tactile tasks were employed to assess the functional correlation underlying the cooperation process between visual and tactile regions. MEG data were recorded from eight healthy subjects while performing a visual, a tactile, and a visuo-tactile task. To define regions of interest (ROIs), event-related fields (ERFs) were estimated from MEG data related to visual and tactile areas. The ten channels with the highest increase in ERF variance, moving from rest to task, were selected. Cross-spectral analysis was then performed to assess potential changes in the activity of the involved regions and quantify the coupling between visual and tactile ROIs. A significant decrease (p<0.01) in the power spectrum was observed during performing the visuo-tactile task compared to rest, both in alpha and beta bands, reflecting the activation of both visual and tactile areas during the execution of the corresponding tasks. Compared to rest, the coherence between visual and tactile ROIs increased during the visuo-tactile task. These observations seem to support the binding theory assuming that the integration of spatially distributed information into a coherent percept is based on transiently formed synchronized functional networks.

Compensating for instantaneous signal mixing in transfer entropy analysis of neurobiological time series

The transfer entropy (TE) has recently emerged as a nonlinear model-free tool, framed in information theory, to detect directed interactions in coupled processes. Unfortunately, when applied to neurobiological time series TE is biased by signal cross-talk due to volume conduction. To compensate for this bias, in this study we introduce a modified TE measure which accounts for possible instantaneous effects between the analyzed time series. The new measure, denoted as compensated TE (cTE), is tested on simulated time series reproducing conditions typical of neuroscience applications, and on real magnetoencephalographic (MEG) multi-trial data measured during a visuo-tactile cognitive experiment. Simulations show that cTE performs similarly to TE in the absence of signal cross-talk, and prevents false positive detection of information transfer in the case of instantaneous mixing of uncoupled signals. When applied to MEG data, cTE detects significant information flow from the visual cortex to the somatosensory area during task execution, suggesting the activation of mechanisms of multisensory integration.

Quantifying Changes in EEG Complexity Induced by Photic Stimulation

OBJECTIVES This study aims to characterize EEG complexity, measured as the prediction error resulting from nonlinear prediction, in healthy humans during photic stimulation. METHODS EEGs were recorded from 15 subjects with eyes closed (EC) and eyes open (EO), during the baseline condition and during stroboscopic photic stimulation (PS) at 5, 10, and 15 Hz. The mean squared prediction error (MSPE) resulting from nearest neighbor local linear prediction was taken as complexity index. Complexity maps were generated interpolating the MSPE index over a schematic scalp representation. RESULTS Statistical analysis revealed that: i) EEG shows good predictability in all conditions and seems to be well explained by a linear stochastic process; ii) the complexity is lower with EC than with EO and increases significantly during PS, to a lesser extent during 10 Hz stimulation; iii) significant differences of EEG complexity are detectable between anterior-central and posterior scalp regions. CONCLUSIONS Changes in EEG complexity during PS can be successfully assessed using nonlinear prediction. The observed modifications in the patterns of complexity seem to reflect neurophysiological behaviors and suggest future applicability of the method in clinical settings.

Robust estimation of Partial Directed Coherence by the vector optimal parameter search algorithm

We propose a method for the accurate estimation of Partial Directed Coherence (PDC) from multichannel time series. The method is based on multivariate vector autoregressive (MVAR) model identification performed through the recently proposed Vector Optimal Parameter Search (VOPS) algorithm. Using Monte Carlo simulations generated by different MVAR models, the proposed VOPS algorithm is compared with the traditional Vector Least Squares (VLS) identification method. We show that the VOPS provides more accurate PDC estimates than the VLS (either overall and single-arc errors) in presence of interactions with long delays and missing terms, and for noisy multichannel time series.