Causality or driving in electrophysiological signal analysis

Partial directed coherence: twenty years on some history and an appraisal

Here while we reminisce about how partial directed coherence was proposed, its motivation and evolution, we take the opportunity to relate it to some of its kin quantities and some of its offspring. Emphasis is placed on our development of asymptotic criteria to place it as a reliable investigation tool, where the connectivity detection problem is completely solved as opposed to what we call the characterization problem. We end by musing over some points now on our wishlist.

Defective corticomuscular connectivity during walking in patients with Parkinson's disease

Gait disturbances are common in individuals with Parkinson's disease (PD). Although the basic patterns of walking are thought to be controlled by the brainstem and spinal networks, recent studies have found significant corticomuscular coherence in healthy individuals during walking. However, it still remains unknown how PD affects the cortical control of muscles during walking. As PD typically develops in older adults, it is important to investigate the effects of both aging and PD when examining disorders in patients with PD. Here, we assessed the effects of PD and aging on corticomuscular communication during walking by investigating corticomuscular coherence. We recorded electroencephalographic and electromyographic signals in 10 individuals with PD, 9 healthy older individuals, and 15 healthy young individuals. We assessed the corticomuscular coherence between the motor cortex and two lower leg muscles, tibialis anterior (TA) and medial gastrocnemius, during walking. Older and young groups showed sharp peaks in muscle activation patterns at specific gait phases, whereas the PD group showed prolonged patterns. Smaller corticomuscular coherence was found in the PD group compared with the healthy older group in the α band (8-12 Hz) for both muscles, and in the β band (16-32 Hz) for TA. Older and young groups did not differ in the magnitude of corticomuscular coherence. Our results indicated that PD decreased the corticomuscular coherence during walking, whereas it was not affected by aging. This lower corticomuscular coherence in PD may indicate lower-than-normal corticomuscular communication, although direct or indirect communication is unknown, and may cause impaired muscle control during walking.NEW & NOTEWORTHY Mechanisms behind how Parkinson's disease (PD) affects cortical control of muscles during walking remain unclear. As PD typically develops in the elderly, investigation of aging effects is important to examine deficits regarding PD. Here, we demonstrated that PD causes weak corticomuscular synchronization during walking, but aging does not. This lower-than-normal corticomuscular communication may cause impaired muscle control during walking.

Mapping the Information Trace in Local Field Potentials by a Computational Method of Two-Dimensional Time-Shifting Synchronization Likelihood Based on Graphic Processing Unit Acceleration

The local field potential (LFP) is a signal reflecting the electrical activity of neurons surrounding the electrode tip. Synchronization between LFP signals provides important details about how neural networks are organized. Synchronization between two distant brain regions is hard to detect using linear synchronization algorithms like correlation and coherence. Synchronization likelihood (SL) is a non-linear synchronization-detecting algorithm widely used in studies of neural signals from two distant brain areas. One drawback of non-linear algorithms is the heavy computational burden. In the present study, we proposed a graphic processing unit (GPU)-accelerated implementation of an SL algorithm with optional 2-dimensional time-shifting. We tested the algorithm with both artificial data and raw LFP data. The results showed that this method revealed detailed information from original data with the synchronization values of two temporal axes, delay time and onset time, and thus can be used to reconstruct the temporal structure of a neural network. Our results suggest that this GPU-accelerated method can be extended to other algorithms for processing time-series signals (like EEG and fMRI) using similar recording techniques.

Nonparametric directionality measures for time series and point process data

The need to determine the directionality of interactions between neural signals is a key requirement for analysis of multichannel recordings. Approaches most commonly used are parametric, typically relying on autoregressive models. A number of concerns have been expressed regarding parametric approaches, thus there is a need to consider alternatives. We present an alternative nonparametric approach for construction of directionality measures for bivariate random processes. The method combines time and frequency domain representations of bivariate data to decompose the correlation by direction. Our framework generates two sets of complementary measures, a set of scalar measures, which decompose the total product moment correlation coefficient summatively into three terms by direction and a set of functions which decompose the coherence summatively at each frequency into three terms by direction: forward direction, reverse direction and instantaneous interaction. It can be undertaken as an addition to a standard bivariate spectral and coherence analysis, and applied to either time series or point-process (spike train) data or mixtures of the two (hybrid data). In this paper, we demonstrate application to spike train data using simulated cortical neurone networks and application to experimental data from isolated muscle spindle sensory endings subject to random efferent stimulation.

Identification of directed interactions in networks

Multichannel data collection in the neurosciences is routine and has necessitated the development of methods to identify the direction of interactions among processes. The most widely used approach for detecting these interactions in such data is based on autoregressive models of stochastic processes, although some work has raised the possibility of serious difficulties with this approach. This article demonstrates that these difficulties are present and that they are intrinsic features of the autoregressive method. Here, we introduce a new method taking into account unobserved processes and based on coherence. Two examples of three-process networks are used to demonstrate that although coherence measures are intrinsically non-directional, a particular network configuration will be associated with a particular set of coherences. These coherences may not specify the network uniquely, but in principle will specify all network configurations consistent with their values and will also specify the relationships among the unobserved processes. Moreover, when new information becomes available, the values of the measures of association already in place do not change, but the relationships among the unobserved processes may become further resolved.

Measuring interdependences in dissipative dynamical systems with estimated Fokker-Planck coefficients

We propose a data-driven approach to measure interdependences between dissipative dynamical systems under the influence of noise. We estimate drift and diffusion coefficients of a Fokker-Planck equation and derive measures that allow one to quantify the asymmetry in coupling in a fully automated and computationally inexpensive and simple way. Our approach makes it possible to discriminate between interdependences in the deterministic and stochastic parts of the dynamics. We report results of numerical studies of exemplary time series from coupled stochastic and deterministic model systems and of an application to electroencephalographic recordings from epilepsy patients.

Comments on 'Is partial coherence a viable technique for identifying generators of neural oscillations?': Why the term 'Gersch Causality' is inappropriate: common neural structure inference pitfalls

To aid prospective neural connectivity inference analysts and hoping to preclude misconception spread, we exploit the didatic value of some of the issues raised by Albo et al. (Biol Cybern 90: 318-326, 2004) who claim that signal-to-noise ratio (SNR) values can lead to mistakes in structural inference when using partial coherence in connection to Gersch's 1970 method for spotting signal sources (Gersch in Math Biosci 14: 177- 196, 1972). We show theoretically that Gersch's method is able only to spot which measurement of some common underlying factor has the least amount of additive noise and that this has nothing to do with any reasonable notion of 'causality' as suggested by Albo et al. (Biol Cybern 90: 318-326, 2004). We also show that despite the inherent structural ambiguity of the model used by Albo et al. (Biol Cybern 90: 318-326, 2004) to back their claim, its data can nonetheless furnish the correct time precedence hierarchy between the activities in its measured structures, both when simple (correlation) and more sophisticated methods are used (partial directed coherence) (Baccala and Sameshima in Biol Cybern 84:463-474, 2001a) in a true depiction of time series causality.

Partial phase synchronization for multivariate synchronizing systems

Graphical models applying partial coherence to multivariate time series are a powerful tool to distinguish direct and indirect interdependencies in multivariate linear systems. We carry over the concept of graphical models and partialization analysis to phase signals of nonlinear synchronizing systems. This procedure leads to the partial phase synchronization index which generalizes a bivariate phase synchronization index to the multivariate case and reveals the coupling structure in multivariate synchronizing systems by differentiating direct and indirect interactions. This ensures that no false positive conclusions are drawn concerning the interaction structure in multivariate synchronizing systems. By application to the paradigmatic model of a coupled chaotic Roessler system, the power of the partial phase synchronization index is demonstrated.

Determination of transmission patterns in multichannel data

The power of today's laboratory equipment allows more and more data channels to be easily recorded. However, some misunderstandings about processing such multivariate data may still be found in the literature. The typical mistake is to treat a multichannel system as comprising pairs of channels; this approach does not use all the collected information about the investigated system, and may lead to erroneous conclusions. In this paper, the differences between single- and multichannel approaches will be briefly summarized and some examples of problems will be described.

Nonlinear multivariate analysis of neurophysiological signals

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependence between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Statistical assessment of nonlinear causality: application to epileptic EEG signals

In this study an information-theoretic test for general Granger causality is used to identify couplings and information transport between different brain areas during epileptic activities. This method can distinguish information that is actually exchanged between two systems from that due to the response to a common signal or past history. This is achieved by an appropriate conditioning of probabilities. Statistical assessment of causality is made from a nonparametric bootstrap test, whereas nonlinearity is assessed by a comparison with a linearized version of the causality index. The framework proposed here provides a useful and model free test to characterize interactions in intracranial electroencephalography (EEG) signals.

Anterior thalamic mediation of experimental seizures: selective EEG spectral coherence

PURPOSE

Physiological evidence has shown that the anterior thalamus (AN) and its associated efferents/afferents constitute an important propagation pathway for pentylenetetrazol (PTZ)-mediated generalized seizures in rodents. Previous work demonstrated metabolic, physical, chemical, and electrical stimulation data supporting a role for AN in the expression of PTZ seizures. We now extend these observations through examination of neuroelectric signal indicators during seizure epochs. We show that the EEG recorded from AN is highly coherent with surface cortical (CTX) EEG during the immediate preconvulsant period and during the ictal stateough.

METHODS

Awake rats were continuously infused with PTZ until clonic seizures were recorded by using both subcortical AN, posterior thalamus (PT), or hippocampal (HPC) bipolar electrodes and cortical EEG. Through the signal-analysis techniques of ordinary and partial coherence, it was possible to focus selectively on signal correlations between AN and CTX (AN/CTX) by removing the effects of unaffiliated regions such as PT and HPC.

RESULTS

Coherence of PT/CTX was observed to be modest, and partial coherence of PT/CTX with the effects of AN/CTX removed did not improve the signal coherence of PT/CTX (PT/CTX-AN). In contrast, AN/CTX coherence was observed to be high, with undiminished correlation when PT/CTX influence was removed (AN/CTX-PT). The most robust band of AN/CTX coherence was centered around the spike-wave clonic frequency of 1-3 Hz. Partial multiple coherence-analysis techniques were used to remove the possible signal contributions from hippocampus in addition to PT. The AN/CTX coherence remained fully preserved in the low-frequency bands.

CONCLUSIONS

These data provide electrophysiologic evidence supporting the special role of the anterior thalamus in the propagation of seizure activity between subcortex and cortex.

Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex

Information processing in the visual cortex depends on complex and context sensitive patterns of interactions between neuronal groups in many different cortical areas. Methods used to date for disentangling this functional connectivity presuppose either linearity or instantaneous interactions, assumptions that are not necessarily valid. In this paper a general framework that encompasses both linear and non-linear modelling of neurophysiological time series data by means of Local Linear Non-linear Autoregressive models (LLNAR) is described. Within this framework a new test for non-linearity of time series and for non-linearity of directedness of neural interactions based on LLNAR is presented. These tests assess the relative goodness of fit of linear versus non-linear models via the bootstrap technique. Additionally, a generalised definition of Granger causality is presented based on LLNAR that is valid for both linear and non-linear systems. Finally, the use of LLNAR for measuring non-linearity and directional influences is illustrated using artificial data, reference data as well as local field potentials (LFPs) from macaque area TE. LFP data is well described by the linear variant of LLNAR. Models of this sort, including lagged values of the preceding 25 to 60 ms, revealed the existence of both uni- and bi-directional influences between recording sites.

Identification of synaptic connections in neural ensembles by graphical models

A method for the identification of direct synaptic connections in a larger neural net is presented. It is based on a conditional correlation graph for multivariate point processes. The connections are identified via the partial spectral coherence of two neurons, given all others. It is shown how these coherences can be calculated by inversion of the spectral density matrix. In simulations with GENESIS, we discuss the relevance of the method for identifying different neural ensembles including an excitatory feedback loop and networks with lateral inhibitions.

Spectral decomposition in multichannel recordings based on multivariate parametric identification

A method of spectral decomposition in multichannel recordings is proposed, which represents the results of multivariate (MV) parametric identification in terms of classification and quantification of different oscillating mechanisms. For this purpose, a class of MV dynamic adjustment (MDA) models in which a MV autoregressive (MAR) network of causal interactions is fed by uncorrelated autoregressive (AR) processes is defined. Poles relevant to the MAR network closed-loop interactions (cl-poles) and poles relevant to each AR input are disentangled and accordingly classified. The autospectrum of each channel can be divided into partial spectra each relevant to an input. Each partial spectrum is affected by the cl-poles and by the poles of the corresponding input; consequently, it is decomposed into the relevant components by means of the residual method. Therefore, different oscillating mechanisms, even at similar frequencies, are classified by different poles and quantified by the corresponding components. The structure of MDA models is quite flexible and can be adapted to various sets of available signals and a priori hypotheses about the existing interactions; a graphical layout is proposed that emphasizes the oscillation sources and the corresponding closed-loop interactions. Application examples relevant to cardiovascular variability are briefly illustrated.

Spectral analysis of a thalamus-to-cortex seizure pathway

Physiological evidence has shown that the anterior thalamus (AN) and its associated efferents/afferents constitute an important propagation pathway for one animal model of generalized tonic-clonic epileptic seizures. In this study we extend and confirm the support for AN's role by examining neuroelectric signal indicators during seizure episodes. We show that the electroencephalogram (EEG) recorded from AN is highly coherent with the EEG derived from the cortex (CTX). By removing the effects of another thalamic nucleus, posterior thalamus (PT)-unaffiliated with the tract linking AN to cortex-partial coherence analysis leaves the CTX/AN coherence undiminished. The most robust band of strong CTX-AN coherence is centered around the spike-wave pacing frequency of 1-3 Hz. Partial-multiple coherence analysis techniques are used to remove the possible signal contribution from hippocampus in addition to PT. The CTX-AN coherence still remains undiminished in the low-frequency bands. Conclusive evidence from coherence studies and other spectral measures reaffirm the special role of the AN in the propagation of seizure activity from subcortex to cortex.

A general statistical framework for frequency-domain analysis of EEG topographic structure

A wide variety of rhythmic electrophysiological phenomena--including driven, induced, and endogenous activities of cortical neuronal masses--lend themselves naturally to analysis using frequency--domain techniques applied to multichannel recordings that discretely sample the overall spatial pattern of the rhythmic activity. For such cases, a large but so far poorly utilized body of statistical theory supports a third major approach to topographic analysis, complementing the more familiar mapping and source-recovery techniques. These methods, many of which have only recently become computationally feasible, collectively provide general solutions to the problem of detecting and characterizing systematic differences that arise--not only in the spatial distribution of the activity, but also in its frequency-dependent between-channel covariance structure--as a function of multiple experimental conditions presented in conformity with any of the conventional experimental designs. This application-oriented tutorial review provides a comprehensive outline of these resources, including: (1) real multivariate analysis of single-channel spectral measures (and measures of between-channel relationships such as coherence and phase), (2) complex multivariate analysis based on multichannel Fourier transforms, and (3) complex multivariate analysis based on multichannel parametric models. Special emphasis is placed on the potential of the multichannel autoregressive model to support EEG (and MEG) studies of perceptual and cognitive processes.

Topographic analysis of coherence and propagation of EEG activity during sleep and wakefulness

Overnight sleep EEG recorded from 21 derivations was studied in 8 healthy subjects. The vector autoregressive model was fitted to all 21 channels simultaneously. Ordinary, multiple and partial coherences and directed transfer functions were estimated for sleep stages and wakefulness. Ordinary coherences give rather trivial information that coherence decreases with distance. Partial coherences revealed specific structure that was well repeatable for the subjects studied. Differences in coherence patterns between sleep stages were found by means of statistical tests. An increase of coherence was found for sleep stages 2, 3 and 4. Directed transfer function made possible the identification of the main centers from which EEG activity is spreading during sleep and wakefulness. During sleep the influence of subcortical structures was manifested by propagation of activity from the fronto-central region. The range of this interaction was highest in sleep stages 3 and 4. An EEG analysis, based on the approach of treating time series as a realization of one process and on the simultaneous (not pair-wise) evaluation of signals offers new possibilities in the investigation of synchronization and functional relations in the brain.

Interhemispheric interactions in seizures of focal onset: data from human intracranial recordings

Interactions between the two hemispheres were studied during seizures recorded in 8 epileptic patients having chronic intracranial electrodes. Seven had temporal lobe foci and one a fronto-central focus. Strength of interaction was measured by the coherence between the EEGs from symmetrical contralateral locations. Time delays of a few milliseconds between discharges were computed by the coherence and phase method. The evolution of interactions was followed from the time a seizure of focal onset had become bilateral to its end. It was found that interhemispheric coherence was generally low throughout seizures, highest values being reached early in the seizure at the time of spread, or at the very end. Time delays most often indicated a lead from the side of onset, whether they were measured early or late in the seizure. Exceptions to these results were found in 2 of 3 patients with bilateral independent onsets: interhemispheric coherence was higher and time leads were always from the same side, independently of the side of onset. If it is assumed that high interhemispheric coherence is mediated by direct connections such as corpus callosum and anterior commissure, then these results can be interpreted as follows: the major commissures do not play an important role in contralateral spread of temporal lobe seizures although they are sometimes active, particularly at initial spread and at seizure end. Time leads from the side of onset indicate that the focus retains an influence over the contralateral discharge throughout the seizure. A different situation may exist with independent bitemporal foci.

Spatial distribution of generators of alpha activity

Source determination of alpha activity was studied using the relative power contribution analysis (RPCA) method which allows determination of the relative contributions of different areas to the power of a certain area at different frequencies. In 20 normal subjects, EEGs were recorded from F3, F4, C3, C4, P3, P4, O1 and O2, each referenced to a linked ear. An 8-dimensional autoregressive model was fitted to the EEGs of 10.24 sec. Based on the model, RPCA was performed. For each area, alpha activity was divided into two parts: one originating in its own area (endogenous) and another in the other areas (exogenous). Endogenous alpha activity increased as the area was more posterior. In the anterior regions (frontal and central), endogenous alpha power (power of endogenous alpha activity) was small, while exogenous alpha power was large. In the posterior regions (parietal and occipital), the amount of endogenous alpha power did not differ markedly from that of exogenous alpha power. The posterior regions, which generate more endogenous alpha activity, can be considered to play a dominant role in alpha generating mechanisms. In some subjects, alpha generators with a different frequency from that of the occipital areas were observed.

Experimental data on the nature of postresuscitation alpha frequency activity

Synchronized activity of alpha frequency recorded from a number of dog brain formations in the initial stages of postischaemic (postresuscitation) restoration was studied. Using the methods of destruction and elimination it was shown that the amygdaloid nuclei were the main source of such activity. Analysis of the general and particular coherence functions at the frequencies of alpha-like activity has shown that subcortical formations (thalamus, nucleus caudatus) become secondarily active sources of the generation of the bursts and of their generalization in the brain. Investigations and available literature lay the basis for the hypothesis that the activity observed in experiments and the activity described during postresuscitation alpha-pattern coma in humans have similar mechanisms of development and are, probably, identical.

Measurement of small time differences between EEG channels: method and application to epileptic seizure propagation

Small time differences between the EEG activities of two channels were measured by a method based on the use of coherence and phase spectra over a certain frequency range. In many cases this method allowed to establish that time differences of the order of 5-50 msec were actually present between two channels which appeared synchronous on visual inspection. The method was applied to seizure activity from the penicillin and kindling models in the cat and to seizures recorded from scalp and intracerebral electrodes in epileptic patients. When the seizure activity was widespread but was known to be related to an epileptic focus, it was found that the area of the focus had a consistent time lead over the other recording sites. It was concluded that the method could frequently allow to: (1) assess the presence of an epileptic focus even when only widespread seizure activity could be recorded; (2) make inferences about the possible routes of propagation of seizure activity.