An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias

Abstract

Phase-synchronization is a manifestation of interaction between neuronal groups measurable from LFP, EEG or MEG signals, however, volume conduction can cause the coherence and the phase locking value to spuriously increase. It has been shown that the imaginary component of the coherency (ImC) cannot be spuriously increased by volume-conduction of independent sources. Recently, it was proposed that the phase lag index (PLI), which estimates to what extent the phase leads and lags between signals from two sensors are nonequiprobable, improves on the ImC. Compared to ImC, PLI has the advantage of being less influenced by phase delays. However, sensitivity to volume-conduction and noise, and capacity to detect changes in phase-synchronization, is hindered by the discontinuity of the PLI, as small perturbations turn phase lags into leads and vice versa. To solve this problem, we introduce a related index, namely the weighted phase lag index (WPLI). Differently from PLI, in WPLI the contribution of the observed phase leads and lags is weighted by the magnitude of the imaginary component of the cross-spectrum. We demonstrate two advantages of the WPLI over the PLI, in terms of reduced sensitivity to additional, uncorrelated noise sources and increased statistical power to detect changes in phase-synchronization. Another factor that can affect phase-synchronization indices is sample-size bias. We show that, when directly estimated, both PLI and the magnitude of the ImC have typically positively biased estimators. To solve this problem, we develop an unbiased estimator of the squared PLI, and a debiased estimator of the squared WPLI.

Introduction

Oscillatory neuronal activity has been implied in numerous functions (Buzsáki and Draguhn, 2004, Fries, 2009, Gray et al., 1989, Pesaran et al., 2002, Salinas and Sejnowski, 2001), such as attention, spatial navigation, perceptual binding and memory. Oscillatory activity in different areas can be phase-coupled, i.e., display systematic phase-delays, a phenomenon called phase-synchronization, which has been hypothesized to be an important mechanism for creating a flexible communication structure between brain areas (Engel et al., 2001, Fries, 2005, Varela et al., 2001). In support of this hypothesis, correlations between cognitive functions and long-range phase-synchronization have been demonstrated in many different areas and species, e.g. (Benchenane et al., 2010, Buschman and Miller, 2009, Gregoriou et al., 2009, Pesaran et al., 2008, Roelfsema et al., 1997, Siapas et al., 2005, von Stein et al., 2000, Womelsdorf et al., 2007).

Traditionally, spectral coherence has been used to quantify phase-synchronization for electrophysiological data (EEG and MEG) (Adey et al., 1961, Mitra and Pesaran, 1999, Nunez and Srinivasan, 2006, Walter, 1963). Since coherence merely indicates linear correlation between signals, intermingling phase and amplitude correlations, Lachaux et al. (1999) proposed to use only the relative phase between signals to index phase-synchronization, resulting in an index called the phase locking value (PLV).

However, it is well-known that indexing phase-synchronization can be complicated by four problems: (i) the presence of a common reference, (ii) volume-conduction of source activity, (iii) the presence of noise sources, and (iv) sample-size bias. Volume-conduction of source activity, and, in case of EEG (but not MEG) data, the use of a common reference, can spuriously inflate phase-synchronization indices (Fein et al., 1988, Nolte et al., 2004, Nunez and Srinivasan, 2006, Stam et al., 2007). The problem of volume-conduction is especially large for scalp EEG and MEG data, because of their low spatial resolution. However, it can still be significant when the EEG is measured intracranially (ECoG) or from electrode tips within the tissue (Local Field Potentials — LFPs) if a common reference is used and/or the cross-spectrum is defined over data from spatially close sensors.

To overcome these problems, Nolte et al. (2004) proposed the imaginary component of the coherency (ImC) as a conservative index of phase-synchronization, and showed that volume conduction of uncorrelated sources cannot ‘create’ a non-zero ImC, based on the conventional assumption that, for typical EEG and MEG frequencies of interest, the quasi-stationary description of the Maxwell equations holds (Maxwell, 1865, Plonsey and Heppner, 1967, Stinstra and Peters, 1998), implying that the conducted electric activity of a single source affects spatially separate sensors with negligible time delay.

Stam et al. (2007) argued that one disadvantage of the ImC lies in the fact that it can be strongly influenced by the phase of the coherency, so that it is most effective in detecting synchronization with a phase lag corresponding to a quarter cycle, and breaks down if the two sources of interest are in phase or in phase opposition. As a potential improvement on the ImC, Stam et al. (2007) therefore proposed the phase lag index (PLI). The PLI estimates, for a particular frequency, to what extent the phase leads and lags between signals from two sensors are nonequiprobable, irrespective of the magnitude of the phase leads and lags. In simulations, the PLI performed better than the ImC in detecting true changes in phase-synchronization, and was less sensitive to the addition of volume-conducted noise sources (Stam et al., 2007). Still, PLI's sensitivity to noise and volume conduction may be hindered by the discontinuity in this index as small perturbations turn phase lags into leads and vice versa, a problem that becomes more serious for synchronization effects of small magnitude.

To increase the capacity to detect true changes in phase-synchronization, to reduce the influence of common noise sources (and for EEG data, a common reference) and to reduce the influence of changes in the phase of the coherency, we add a new member to the family of phase-synchronization indices that are based on the imaginary component of the cross-spectrum, namely the weighted phase lag index (WPLI). The WPLI extends the PLI in that it weights the contribution of observed phase leads and lags by the magnitude of the imaginary component of the cross-spectrum; in this way it alleviates the discontinuity mentioned above. We will demonstrate two main advantages of the WPLI over the PLI, in terms of their sensitivity to additional, uncorrelated noise sources and their capacity to detect true changes in phase-synchronization.

To bridge the analysis in terms of population parameters to the practical case of data obtained under limited sampling, we will address a final confounding factor that complicates the estimation of phase-synchronization, namely sample-size bias, usually increasing with smaller sample sizes. We will show that the direct PLI estimator (Stam et al., 2007) is positively biased, and that the direct estimator of the ImC's magnitude is typically positively biased, but can be negatively biased as well. To solve this problem, we will introduce an unbiased estimator of the squared PLI and a debiased estimator of the squared WPLI.

The paper is organized as follows. We commence by introducing a conventional linear mixture model (The linear mixture model section), and existing indices of phase-synchronization (Existing indices of phase-synchorinization section). This sets the stage for the definition of the WPLI (The weighted phase-lag index section). The comparison of the WPLI with previous statistics will proceed as follows. First, for two correlated sources of interest, we transition from the case of ideal sensors (i.e., neglecting the reference) without volume-conduction to the more realistic case of ideal sensors where the two sources of interest are volume-conducted (Volume-conducting correlated sources of interest section). Second, we continue with a more realistic case where uncorrelated, volume-conducted noise sources have been added (Addition of uncorrelated, volume-conducted noise reduction section). Third, we study the effect of changes in the coherency between the activities of two sources of interest (Detecting changes in the coherency between sources' activities section). After this comparison, we continue with the practical case where population parameters have to be estimated from small sample sizes, (The problem of sample-size bias section), and finish our theoretical section by comparing the statistical powers of the WPLI and PLI estimators (Comparison between the statistical power of the squared PLI and WPLI estimators section). Finally, we apply the proposed techniques to actual LFP data recorded from the rat orbitofrontal cortex (OFC) (Application to experimental LFP data: methods, Application to experimental LFP data: results sections).