Dear Donald,
> The FIR approach is aimed at characterizing the shape of the response (HDR) to the stimulus trial.
> limiting the number of "gaps" modeled separately
Indeed, the seminal papers suggest to go this way, but in the end it's open to interpretations whether the time bin predictors reflect activations for a certain time point or some interval, e.g. average activation within that time bin (and/or activations based on trials with onsets +/- 1/2 time bin units). In the latter case random onsets shouldn't be an issue as long as there are no fundamental differences between conditions with regard to that "randomness". The time bin resolution also does not have to correspond to one TR, with longer time bins the predictors already "average" across several TRs (or the certain time points within the different TRs). Depending on settings some of the regressors will then be based on e.g. two TRs and others on e.g. three. Marsbar seems to allow to go with time bins shorter than TRs, but I have not yet looked at the implementation, could also be just some interpolation.
> I'm not sure applying a suboptimal model
The reason why I came up with adjusted FIR predictors in the first place is the current application of FIR in literature. It is quite common to run a standard GLM based on the canonical HRF, find some clusters, forward these (for "additional information" ;-) or some a-priori regions into an FIR analysis. Usually these experiments seem to include some jitter with several steps, so likely they are not optimized for the FIR (in the original sense). Accepting the temporal blurring would be one option, considering some weigthing for different trials depending on onsets should be another, if one really wants to obtain activation estimates for certain time points in the context of (somewhat) random presentation. But I agree it's another issue to find a good solution.
Another observation, in most instances the plotted FIR curves look non-canonical, except that there is *some* rise at *some* point, which casts doubts on the standard GLM results which are often presented as well. It should rather be the opposite, conduct a FIR model, ideally on an independent data set, and then use the estimated BOLD response as a predictor for the other data set. I remember Tor Wager and colleagues discussing this issue several years ago in one of their papers on constructing predictors.
Best
Helmut
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