Convolution models for induced electromagnetic responses

In Kilner et al. [Kilner, J.M., Kiebel, S.J., Friston, K.J., 2005. Applications of random field theory to electrophysiology. Neurosci. Lett. 374, 174–178.] we described a fairly general analysis of induced responses—in electromagnetic brain signals—using the summary statistic approach and statistical parametric mapping. This involves localising induced responses—in peristimulus time and frequency—by testing for effects in time–frequency images that summarise the response of each subject to each trial type. Conventionally, these time–frequency summaries are estimated using post‐hoc averaging of epoched data. However, post‐hoc averaging of this sort fails when the induced responses overlap or when there are multiple response components that have variable timing within each trial (for example stimulus and response components associated with different reaction times). In these situations, it is advantageous to estimate response components using a convolution model of the sort that is standard in the analysis of fMRI time series. In this paper, we describe one such approach, based upon ordinary least squares deconvolution of induced responses to input functions encoding the onset of different components within each trial. There are a number of fundamental advantages to this approach: for example; (i) one can disambiguate induced responses to stimulus onsets and variably timed responses; (ii) one can test for the modulation of induced responses—over peristimulus time and frequency—by parametric experimental factors and (iii) one can gracefully handle confounds—such as slow drifts in power—by including them in the model. In what follows, we consider optimal forms for convolution models of induced responses, in terms of impulse response basis function sets and illustrate the utility of deconvolution estimators using simulated and real MEG data.