Luiz A. Baccalá

Partial directed coherence: a new concept in neural structure determination.

Abstract. This paper introduces a new frequency-domain approach to describe the relationships (direction of information flow) between multivariate time series based on the decomposition of multivariate partial coherences computed from multivariate autoregressive models. We discuss its application and compare its performance to other approaches to the problem of determining neural structure relations from the simultaneous measurement of neural electrophysiological signals. The new concept is shown to reflect a frequency-domain representation of the concept of Granger causality.

Using partial directed coherence to describe neuronal ensemble interactions

This paper illustrates the use of the recently introduced method of partial directed coherence in approaching how interactions among neural structures change over short time spans that characterize well defined behavioral states. Central to the method is its use of multivariate time series modelling in conjunction with the concept of Granger causality. Simulated neural network models were used to illustrate the techniques power and limitations when dealing with neural spiking data. This was followed by the analysis of multi-unit activity data illustrating dynamical change in the interaction of thalamo-cortical structures in a behaving rat.

Comparison of different cortical connectivity estimators for high-resolution EEG recordings

The aim of this work is to characterize quantitatively the performance of a body of techniques in the frequency domain for the estimation of cortical connectivity from high‐resolution EEG recordings in different operative conditions commonly encountered in practice. Connectivity pattern estimators investigated are the Directed Transfer Function (DTF), its modification known as direct DTF (dDTF) and the Partial Directed Coherence (PDC). Predefined patterns of cortical connectivity were simulated and then retrieved by the application of the DTF, dDTF, and PDC methods. Signal‐to‐noise ratio (SNR) and length (LENGTH) of EEG epochs were studied as factors affecting the reconstruction of the imposed connectivity patterns. Reconstruction quality and error rate in estimated connectivity patterns were evaluated by means of some indexes of quality for the reconstructed connectivity pattern. The error functions were statistically analyzed with analysis of variance (ANOVA). The whole methodology was then applied to high‐resolution EEG data recorded during the well‐known Stroop paradigm. Simulations indicated that all three methods correctly estimated the simulated connectivity patterns under reasonable conditions. However, performance of the methods differed somewhat as a function of SNR and LENGTH factors. The methods were generally equivalent when applied to the Stroop data. In general, the amount of available EEG affected the accuracy of connectivity pattern estimations. Analysis of 27 s of nonconsecutive recordings with an SNR of 3 or more ensured that the connectivity pattern could be accurately recovered with an error below 7% for the PDC and 5% for the DTF. In conclusion, functional connectivity patterns of cortical activity can be effectively estimated under general conditions met in most EEG recordings by combining high‐resolution EEG techniques, linear inverse estimation of the cortical activity, and frequency domain multivariate methods such as PDC, DTF, and dDTF. Hum. Brain Mapp, 2007.

Thalamic bursting in rats during different awake behavioral states

Thalamic neurons have two firing modes: tonic and bursting. It was originally suggested that bursting occurs only during states such as slow-wave sleep, when little or no information is relayed by the thalamus. However, bursting occurs during wakefulness in the visual and somatosensory thalamus, and could theoretically influence sensory processing. Here we used chronically implanted electrodes to record from the ventroposterior medial thalamic nucleus (VPM) and primary somatosensory cortex (SI) of awake, freely moving rats during different behaviors. These behaviors included quiet immobility, exploratory whisking (large-amplitude whisker movements), and whisker twitching (small-amplitude, 7- to 12-Hz whisker movements). We demonstrated that thalamic bursting appeared during the oscillatory activity occurring before whisker twitching movements, and continued throughout the whisker twitching. Further, thalamic bursting occurred during whisker twitching substantially more often than during the other behaviors, and a neuron was most likely to respond to a stimulus if a burst occurred ≈120 ms before the stimulation. In addition, the amount of cortical area activated was similar to that during whisking. However, when SI was inactivated by muscimol infusion, whisker twitching was never observed. Finally, we used a statistical technique called partial directed coherence to identify the direction of influence of neural activity between VPM and SI, and observed that there was more directional coherence from SI to VPM during whisker twitching than during the other behaviors. Based on these findings, we propose that during whisker twitching, a descending signal from SI triggers thalamic bursting that primes the thalamocortical loop for enhanced signal detection during the whisker twitching behavior.

Assessing cortical functional connectivity by partial directed coherence: simulations and application to real data

The aim of this paper is to test a technique called partial directed coherence (PDC) and its modification (squared PDC; sPDC) for the estimation of human cortical connectivity by means of simulation study, in which both PDC and sPDC were studied by analysis of variance. The statistical analysis performed returned that both PDC and sPDC are able to estimate correctly the imposed connectivity patterns when data exhibit a signal-to-noise ratio of at least 3 and a length of at least 27 s of nonconsecutive recordings at 250 Hz of sampling rate, equivalent, more generally, to 6750 data samples

Frequency domain connectivity identification: An application of partial directed coherence in fMRI

Functional magnetic resonance imaging (fMRI) has become an important tool in Neuroscience due to its noninvasive and high spatial resolution properties compared to other methods like PET or EEG. Characterization of the neural connectivity has been the aim of several cognitive researches, as the interactions among cortical areas lie at the heart of many brain dysfunctions and mental disorders. Several methods like correlation analysis, structural equation modeling, and dynamic causal models have been proposed to quantify connectivity strength. An important concept related to connectivity modeling is Granger causality, which is one of the most popular definitions for the measure of directional dependence between time series. In this article, we propose the application of the partial directed coherence (PDC) for the connectivity analysis of multisubject fMRI data using multivariate bootstrap. PDC is a frequency domain counterpart of Granger causality and has become a very prominent tool in EEG studies. The achieved frequency decomposition of connectivity is useful in separating interactions from neural modules from those originating in scanner noise, breath, and heart beating. Real fMRI dataset of six subjects executing a language processing protocol was used for the analysis of connectivity. Hum Brain Mapp, 2009.

Overcoming the limitations of correlation analysis for many simultaneously processed neural structures.

Publisher Summary Modern methods in molecular biology, neuroanatomy, functional imaging, and monitoring electric signals from neuronal depolarization remains important when evaluating the functional aspects of both normal and pathological neural circuitry. Correlation methods ranks popular and are extensively used to analyze the functional interaction in the electroencephalogram (EEG), magnetoencephalogram, local field potentials, and recorded single- and multi-unit activity of many structures. A host of analytical techniques emerged, some employing information theoretic rationales by assessing mutual information or interdependence between signal pairs, while others are extensions of spectral analysis/coherence analysis. A large fraction of neuroscientists rely on the cross-correlation between the activities of pairs of neural structures to infer their functionality. The effective structural inference is possible, if simultaneous signals from many structures are jointly analyzed. To handle simultaneous structures, the recently introduced notion of partial directed coherence (PDC) is employed. This approach for simultaneous multichannel data analysis is based on Granger causality that employs multivariate auto-regressive (MAR) models for computational purposes. By analyzing linear toy models, PDCs superior performance over other commonly used methods specially cross-correlation and classical coherence, directed transfer function (DTF) analysis provides complementary information whose analysis is less clear than PDCs.

Information theoretic interpretation of frequency domain connectivity measures

In order to provide adequate multivariate measures of information flow between neural structures, modified expressions of partial directed coherence (PDC) and directed transfer function (DTF), two popular multivariate connectivity measures employed in neuroscience, are introduced and their formal relationship to mutual information rates are proved.

A new blind time-domain channel identification method based on cyclostationarity

A blind channel identification scheme based on oversampling the channel output and employing a (discrete) cyclic correlation function similar to Gardner (1991) is proposed. It is shown that the known results on blind identifiability follow directly. Further, this connection suggests a new and efficient time-domain algorithm for blind channel identification that is applied to some representative examples.<<ETX>>

Unified asymptotic theory for all partial directed coherence forms

This paper presents a unified mathematical derivation of the asymptotic behaviour of the three main forms of partial directed coherence (PDC). Numerical examples are used to contrast PDC, gPDC (generalized PDC) and iPDC (information PDC) as to meaning and applicability and, more importantly, to show their essential statistical equivalence insofar as connectivity inference is concerned.