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With regard to the design, in essence, there seem to be three inherent drawbacks:
1) high correlation between task regressor and the constant term (this is due to long blocks and very short rest periods in between, adding the 5 min rest at the beginning of the session somewhat reduces the correlation, but it is still very high)
2) signal extending into the low frequency range (this is due to the long blocks / long and fixed interval between successive onsets)
3) possibly, inappropriate predictors (does the assumption of linear summation still hold for trials of 2 min length, personally I doubt so, does learning occur within the trials, affecting neural activation and BOLD response)

Unfortunately there's no solution, as this is inherent to the design. Concerning 1), as another example, think of a Go/No-go task with much more Go trials than No-go trials, and densely packed stimuli. It is impossible to properly determine activation for Go trials, as the corresponding regressor is close to a flat line, but you can look at effects for the No-go predictor, thus signal deviating from the "background noise" / "implicit baseline" in a certain temporal pattern (as specified in the predictor). In your case, you should not be able to properly detect e.g. motor activation (or any other activation inherent to the task trials). This might be the reason why changing the HPF does not seem to have any effect (given the durations/intervals it should actually have an effect, but this might also result from 3), if the predictor is just inappropriate then changing HPF will just catch up different types of noise).

Concerning 2), as shown some while in that Excel sheet, when testing for linear changes in activation over trials the corresponding regressor seems to be strongly affected by the HPF. It might also correlate highly with realignment parameters (if subjects move slowly along one dimension over time, ressembling a linear drift, this regressor might correlate highly with the parametric modulator). Another aspect is whether there's really some linear change on neural level, or a more complex learning pattern (e.g. inverse U-shaped), and whether it's similar across subjects or not.

Best

Helmut