I have read several threads (and a couple papers) on the use of temporal and dispersion derivatives, but would like to clarify how derivatives are included in/applied to a model.
1. Are there multiple HRFs tested at each voxel (e.g.., the canonical HRF, time-modulated HRFs and/or dispersion-modulated HRFs)? If so, is it possible that different HRFs would be retained for different voxels (e.g., the canonical HRF for voxel A and a time-modulated HRF for voxel B)?
2. Or are the derivatives regressors in the model for the HRF, i.e., variables that can account for and thus reduce error/variance, much like a covariate? This would imply that only one HRF is modeled at each voxel, but time- and dispersion-related variance between voxels is taken into account in the model, correct?
3. If derivatives are error-reducing regressors, why are they represented as independent columns (1 for time, 1 for dispersion, for each condition) in the design matrix? Relatedly, when F-contrasting 2 conditions (say, A and B), why is it necessary to explicitly include the derivatives in the contrast specification (e.g.: 1 1 1 -1 -1 -1 for the A minus B contrast)?
4. Lastly, is there any reason derivatives should NOT be included in a 1st-level analysis (testing for fixed effects in a small group of subjects, n=5)?