We present a new approach to the analysis of brain evoked electromagnetic potentials and fields. Multivariate version of the matching pursuit algorithm (MMP) performs an iterative, exhaustive search for waveforms, which optimally fit to signal structures, persistent in all the responses (trials) with the same time of occurrence, frequency, phase, and time width, but varying amplitude. The search is performed in a highly redundant time--frequency dictionary of Gabor functions, i.e., sines modulated by Gaussians. We present the feasibility of such a single-trial MMP analysis of the auditory M100 response, using an illustrative dataset acquired in a magnetoencephalographic (MEG) measurement with auditory stimulation with sinusoidal 1-kHz tones. We find that the morphology of the M100 estimate obtained from simple averaging of single trials can be very well explained by the average reconstruction with a few Gabor functions that parametrize those single trials. The M100 peak amplitude of single-trial reconstructions is observed to decrease with repetitions, which indicates habituation to the stimulus. This finding suggests that certain waveforms fitted by MMP could possibly be related to physiologically distinct components of evoked magnetic fields, which would allow tracing their dynamics on a single-trial level.

Abstract-The standard procedure to determine the brain response from a multitrial evoked magnetoencephalography (MEG) or electroencephalography (EEG) data set is to average the individual trials of these data, time locked to the stimulus onset. When the brain responses vary from trial-to-trial this approach is false. In this paper, a maximum-likelihood estimator is derived for the case that the recorded data contain amplitude variations. The estimator accounts for spatially and temporally correlated background noise that is superimposed on the brain response.The model is applied to a series of 17 MEG data sets of normal subjects, obtained during median nerve stimulation. It appears that the amplitude of late component (30-120 ms) shows a systematic negative trend indicating a weakening response during stimulation time. For the early components (20-35 ms) no such a systematic effect was found. The model is furthermore applied on a MEG data set consisting of epileptic spikes of constant spatial distribution but varying polarity. For these data, the advantage of applying the model is that positive and negative spikes can be processed with a single model, thereby reducing the number of degrees of freedom and increasing the signal-to-noise ratio.

MEG and EEG studies of event-related responses often involve comparisons of grand averages, requiring homogeneity of the variances. Here, we examine the possibility, implied by the nature of neural sources and the measuring principles involved, that the M100 component of auditory-evoked magnetic fields of different subjects, hemispheres, to different stimuli, and at different sensors differs by scaling factors. Such a multiplicative model predicts a linear increase in the standard deviation with the mean, and thus would have important implications for averaging and comparing such data. Our analyses, at the sensor and the source level, clearly show that the multiplicative model applies. We therefore propose geometric, rather than arithmetic, averaging of the M100 component across subjects and suggest a novel and superior normalization procedure. Our results question the justification of the common practice of subtracting arithmetic grand averages.

Stationarity is a crucial yet rarely questioned assumption in the analysis of time series of magneto-(MEG) or electroencephalography (EEG). One key drawback of the commonly used tests for stationarity of encephalographic time series is the fact that conclusions on stationarity are only indirectly inferred either from the Gaussianity (e.g. the Shapiro-Wilk test or Kolmogorov-Smirnov test) or the randomness of the time series and the absence of trend using very simple time-series models (e.g. the sign and trend tests by Bendat and Piersol). We present a novel approach to the analysis of the stationarity of MEG and EEG time series by applying modern statistical methods which were specifically developed in econometrics to verify the hypothesis that a time series is stationary. We report our findings of the application of three different tests of stationarity-the Kwiatkowski-Phillips-Schmidt-Schin (KPSS) test for trend or mean stationarity, the Phillips-Perron (PP) test for the presence of a unit root and the White test for homoscedasticity-on an illustrative set of MEG data. For five stimulation sessions, we found already for short epochs of duration of 250 and 500 ms that, although the majority of the studied epochs of single MEG trials were usually mean-stationary (KPSS test and PP test), they were classified as nonstationary due to their heteroscedasticity (White test). We also observed that the presence of external auditory stimulation did not significantly affect the findings regarding the stationarity of the data. We conclude that the combination of these tests allows a refined analysis of the stationarity of MEG and EEG time series.