Duration Regressors with fMRI

This subject was brought to my attention by a colleague who wanted to know whether parametric modulation or duration modulation was a better way to account for RT effects. While it can depend on the question you are trying to answer, often duration modulation (referred to here as a "variable epoch model") works best. The following highlights the different approaches for modeling trials which involve a period of decision-making or an interval between presentation of a stimulus and the resulting response.

Over the past few years there has been a renewed interest in modeling duration in fMRI data. In particular, a methods paper by Grinband and colleagues (2008) compared the effects of modeling the duration of a trial - as measured by its reaction time (RT) - against models which used RT as a parametric modulator and against models which did not use RT at all. The argument against using RT-modulated regressors was that, at short time intervals (i.e., less than four seconds), using an impulse function was a good approximation to the resulting BOLD signal (cf. Henson, 2003).

Figure depicting relationship between stimulus onset, BOLD response, and underlying neural activity. Red lines: activity associated with punctate responses (e.g., light flashed at the subject). Blue lines: Activity associated with trials of unequal duration (e.g., decision-making).

However, for a few investigators, such assumptions were not good enough. To see whether different models of RT led to noticeable differences in BOLD signal, Grinband et al (2008) examined four types of modeling:
  1. Convolving the onset of a condition or response with the canonical HRF (constant impulse model);
  2. Separately modeling both the main effect of the condition as well as a mean-centered parametric modulator - in this case RT (variable impulse model);
  3. Binning each condition onset into a constant amount of time (e.g., 2 seconds) and convolving with the canonical HRF (constant epoch model); and
  4. Modeling each event as a boxcar function equal to the length of the subject's RT (variable epoch model).

Graphical summary of models from Grinband et al (2008). Top: Duration of cognitive process as indexed by reaction time. Constant Impulse:  Onset of each event treated as a punctate response. Constant Epoch: Onset of each event convolved with boxcar function of constant duration. Variable Impulse: Punctate response functions modulated by mean-centered parameter (here, RT). Variable Epoch: Each event modeled by boxcar of duration equal to that event's RT.

Each of these models was then compared using data from a decision-making task in which subjects determined whether a line was long or short. If this sounds uninteresting to you, you have obviously never done a psychology experiment before.

The authors found that the variable epoch model - in other words, convolving each event with a boxcar equal to the length of the subject's RT for that trial - captured more of the variability in the BOLD response, in addition to reducing false positives as compared to the other models. The variable epoch model also dramatically increased sexual drive and led to an unslakeable thirst for mindless violence. Therefore, these simulations suggest that for tasks requiring time - such as decision-making tasks - convolution with boxcar regressors is a more faithful representation of the underlying neuronal dynamics (cf. the drift-diffusion model of Ratcliff & McKoon, 2008). The following figures highlight the differences between the impulse and epoch models:

Comparison of impulse models and epoch models as depicted in Grinband et al (2008). A) For impulse models, the shape remains constant while the amplitude varies; for epoch models, increasing the duration of a trial leads to changes in both shape and amplitude. B) Under the impulse model, increasing the duration of a stimulus or cognitive process (as measured by RT) leads to a reduction in explained variance.

Figure from Grinband et al (2008) showing differential effects of stimulus intensity and stimulus duration. Left: Increasing stimulus intensity has no effect on the time to peak of the BOLD response. Right: Increasing stimulus duration (or the duration of the cognitive process) leads to a linear increase in the time for the BOLD response to peak.

One caveat: note well that both parametric modulation and convolution with boxcar functions will account for RT-related effects in your data; and although the Grinband simulations establish the supremacy of boxcar functions, there may be occasions that warrant parametric modulation. For example, one may be interested in the differences of RT modulation for certain trial types as compared to others; and the regressors generated by parametric modulation will allow the researcher to test them against each other directly.

Duration Modulation in AFNI

Modulating signal by duration - usually reaction time (RT) - is increasingly common in fMRI data analysis, particularly in light of recent studies examining how partialing out RT can reduce or even eliminate effects in certain regions of the brain (e.g., anterior cingulate; see Grinband et al, 2010; Yarkoni et al, 2009). In light of these findings, it appears as though parametrically modulating regressors by RT, or the duration of a given condition, is an important factor in any analysis where this data is available.

When performing this type of analysis, therefore, it is important to know how your analysis software processes duration modulation data. With AFNI, the default behavior of their duration modulation basis function (dmBLOCK) used to scale everything to 1, no matter how long the trial lasted. This may be useful for comparison to other conditions which have also been scaled the same way, but is not an appropriate assumption for conditions lasting only a couple of seconds or less. The BOLD response tends to saturate over time when exposed to the same stimulation for an extended period (e.g., block designs repeatedly presenting visual or auditory stimulation), and so it is reasonable to assume that trials lasting only a few hundred milliseconds will have less of a ramping up effect in the BOLD response than trials lasting for several seconds.

The following simulations were generated with AFNI's 3dDeconvolve using a variation of the following command:

3dDeconvolve -nodata 350 1 -polort -1 -num_stimts 1 -stim_times_AM1 q.1D 'dmBLOCK' -x1D stdout: | 1dplot -stdin -thick -thick

Where the file "q.1D" contains the following:

10:1 40:2 70:3 100:4 130:5 160:6 190:7 220:8 250:9 280:30

In AFNI syntax, this means that event 1 started 10 seconds into the experiment, with a duration of 1 second; event 2 started 40 seconds into the experiment with a duration of 2 seconds, and so on.

The 3dDeconvolve command above is a good way to generate simulation data, through the "-nodata" option which tells 3dDeconvolve that there is no functional data to process. The command tells 3dDeconvolve to use dmBLOCK as a basis function, convolving each event with a boxcar function the length of the specified duration.

Running this command as is generates the following graph:

As is expected, trials that are shorter are scaled less, while trials lasting longer are scaled more, with a saturation effect occurring around 8-9 seconds.

Running 3dDeconvolve with a basis function scaling the signal change in each to 1 is done with the following:

3dDeconvolve -nodata 350 1 -polort -1 -num_stimts 1 -stim_times_AM1 q.1D 'dmBLOCK(1)' -x1D stdout: | 1dplot -stdin -thick -thick

And generates the following output:

Likewise, the ceiling on the basis function can be set to any arbitrary number, e.g.:

3dDeconvolve -nodata 350 1 -polort -1 -num_stimts 1 -stim_times_AM1 q.1D 'dmBLOCK(10)' -x1D stdout: | 1dplot -stdin -thick -thick

However, the default behavior of AFNI is to scale events differently based on different duration (and functions identically to the basis function dmBLOCK(0)). This type of "tophat" function makes sense, because unlimited signal increase as duration also increases would lead to more and more bloodflow to the brain, which, taken to its logical conclusion, would mean that if you showed someone flashing checkerboards for half an hour straight their head would explode.

As always, it is important to look at your data to see how well your model fits the timecourse of activity in certain areas. While it is reasonable to think that dmBLOCK(0) is the most appropriate basis function to use for duration-related trials, this may not always be the case.

These last two figures show the same subject analyzed with both dmBLOCK(0) and dmBLOCK(1). The underlying beta values for each do not differ significantly, although there is some variability in how much they differ in distinct cortical areas, and small but consistent changes in variability can lead to relatively large effects at the second level.

The image on the left hasn't been masked, but the underlying beta estimates should be the same in either case.