Decomposing stimulus and response component waveforms in ERP.

Event-related potentials (ERPs) are evoked brain potentials that are averaged across many trial repetitions with individual trials aligned (i.e. time-locked) to a specific behavioral event, typically the onset of the stimulus (s-lock) or the onset of the behavioral response (r-lock). These evoked potential averages may reflect brain activities during the stimulus encoding/analyzing stage (stimulus component waveform, or 'S-component'), during the response preparation/production stage (response component waveform, or 'R-component'), or a combination thereof. In the stimulus-locked average of the ensemble of the recorded waveforms (i.e. in the s-locked ERP), the contribution of an R-component will be convoluted, due to the trial-by-trial variance in reaction time (RT): so will an S-component in the r-locked ERP. It is shown here that the knowledge of (1) the s-locked and r-locked ERP waveforms constructed from the same ensemble of trials and (2) the RT distribution of this ensemble allows us to determine whether the recorded potential results from a single S-component, a single R-component, or a single intermediate ('decisional' or D-) component related to the transition of the two stochastically independent stages. If it can be assumed that the evoked potential is the result of a linear summation of an S-component and an R-component, then there is a unique recovery into these two components, such that the reconstructed waveform on an individual trial is a superposition of the two components with their relative offset determined by the RT of that trial and the ensemble average is the experimentally obtained s-locked and r-locked ERP waveforms. Two independent methods can be used to recover those components, one based on Fourier transform techniques which was first proposed by Hansen (1983) in the context of ERP component isolation and the other based on a recursive iteration approach through which the contamination of the R or S-component is successively removed from the s-locked or r-locked ERP waveforms, respectively. The iterative procedure is analytically proven to converge to the Fourier-based solution, demonstrating the equivalence of the two approaches. Finally, if the condition of a single intermediate D-component is satisfied, then one can recover this component waveform along with the probability distributions of the relative durations of the two underlying linear stages; however, there is always an equivalent pair of S- and R-component which also satisfy the same data set (s-locked and r-locked ERP waveforms and the overall RT distribution). In this case, the S/R-component assumption and the D-component assumption cannot be distinguished solely on the ground of the available data set. The technique developed here outlines the assumptions and the boundary conditions upon which ensemble ERP waveforms are to be analyzed and interpreted in terms of processing mechanisms related to stimulus, to response, or to the transition between the two.