Estimating Granger causality after stimulus onset: A cautionary note
Introduction
Invasively-recorded local field potential (LFP) and scalp-recorded electroencephalography (EEG) are widely used electrophysiological measures for investigating the neural mechanisms of cognition. In an event-related paradigm, single trial LFPs or EEGs are traditionally modeled as the linear superposition of a stimulus-locked waveform called event-related potential (ERP) and ongoing activity. Ensemble average is performed to obtain the average ERP which we will henceforth refer to as AERP. Subtracting the AERP from each single trial yields the ongoing brain activity which is further analyzed by time-frequency and functional connectivity methods (Kalcher and Pfurtscheller, 1995). The problem with this simple approach is that it does not account for the influence of latency and amplitude variability of the evoked potential across individual trials. In particular, removal of the AERP from individual trials leaves a mixture of ongoing and evoked activities, and the analysis of those ongoing activities in the frequency range overlapping with that of the evoked potential is significantly affected. Truccolo et al. (Truccolo et al., 2002) have shown that statistical measures such as power spectral density and coherence evaluated as a function of time by a sliding window approach can exhibit temporal modulations that are artifacts due to the trial-to-trial variability of cortical evoked responses.
Granger causality has emerged in recent years as a useful tool to investigate the directions of neuronal interactions (Hesse et al., 2003, Brovelli et al., 2004, Roebroeck et al., 2005, Seth, 2005, Lungarella and Sporns, 2006). It can yield insights not possible with other techniques. For example, a time–frequency analysis of Granger causality promises to shed light on the debate regarding whether stimulus–response association is mediated by a pure feed-forward process (Fabre-Thorpe et al., 2001, Thorpe and Fabre-Thorpe, 2001) or by an elaborate reciprocal computation involving multiple cortical areas (Ullman, 1995, Grossberg, 1999, Liang et al., 2000). To achieve a temporal function of Granger causality we fit multivariate autoregressive (MVAR) models to ongoing activity time series in short sliding windows (Ding et al., 2000). One requirement is that the time series in each window be generated by a zero-mean stationary stochastic process. Here zero-mean is defined with respect to an ensemble of realizations (trials). Since the AERP is the ensemble mean of the observed multi-trial data, the removal of AERP from single trial data meets the zero-mean requirement. However, the difference between the individual trial's ERP and the AERP remains in the data, and this difference is not uniform in time (Truccolo et al., 2002), leading to the violation of the stationarity assumption. In this paper, we show that Granger causality, like power and coherence, is adversely affected by the trial-to-trial variability of cortical evoked responses. In particular, without being cognizant of such adverse effects, the analysis result can easily be misconstrued as lending support to the view that stimulus processing in the brain involves reciprocal computation (Liang et al., 2000).
The main conclusion is established by several lines of evidence. First, based on the statistical meaning of Granger causality, we present arguments that predict the outcome of combining the simple AERP removal with the sliding window method in the presence of trial-to-trial evoked response variability. Second, numerical simulations mimicking actual neurophysiological recordings are created. We test the prediction on the simulated data where the correct answers are known. Third, we test the prediction on LFP data acquired from a monkey performing a visuomotor task. A novel single trial analysis method named Analysis of Single-trial Evoked response and Ongoing activity (ASEO) is used to separate the evoked response from the ongoing activity on a trial-by-trial basis (Xu et al., in press). This allows us to demonstrate that the observed temporal modulation of Granger causality may not reflect temporally resolved feed-forward and feedback stimulus processing but may be the result of incomplete removal of evoked activity which varies from trial to trial.
Section snippets
MVAR model and Granger causality estimation
Granger causality is based on the idea of prediction. For two simultaneously measured jointly stationary time series, one series can be called causal to the other if we can better predict the second by incorporating past knowledge of the first (Wiener, 1956, Granger, 1969). Granger causality analysis can be performed in the time-domain as well as in the frequency-domain. We summarize the basic steps here. For more details see Ding et al. (2006). Let Wt = [Xt, Yt]′ be a two dimensional stationary
Theoretical consideration
To establish the specific manner in which trial-to-trial variability of event-related potentials adversely affects the time–frequency analysis of Granger causality, we consider a simple conceptual model. The goal is to generate predictions that can be tested on both simulated and experimental data. Figs. 1(a) and (b) show event-related potentials simulated by sinusoids from two channels. By construction, channel 2 (Fig. 1(b)) lags behind channel 1 (Fig. 1(a)) by 20 ms, and the amplitudes of the
Discussion
The classical approach to cognitive neuroscience views the event-related potential as the signal and the ongoing activity as noise to be eliminated with averaging. Extensive research over the past two decades shows that this view is overly simplistic. Ongoing processes, rich in oscillation and synchronized activity, provide another powerful index of cognitive operation (Singer, 1993, Herrmann and Knight, 2001, Liang et al., 2002). In light of this revelation, how to recover the ongoing activity
Acknowledgment
This research was supported by NIH grants MH071620, MH070498, and MH079388.
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