1. What is jittering?

It's the practice of varying the timing of your TR relative to your stimulus presentation. It's also often connected to, or even identified as, the practice of varying your inter-trial interval.

The idea in both of these practices is the same. If your TR is 2 seconds, and your stimulus is always presented exactly at the beginning of a TR and always 10 seconds long, then you'll sample the same point in your subject's BOLD response many times - but you might miss points in between those sampling points. Those in-between points might be the peak of your HRF, or an inflection point, or simply another point that will help you characterize the shape of your HRF.

If you made your TR 2.5 seconds, you'd automatically get to sample several other points in your response, at the expense of sampling each of them fewer times. That's "jittering" your TR.

Alternatively, you might keep your TR at two seconds, and make the time between your 10-second trials (your inter-stimulus interval, or ISI) vary at random between 0 and 4 seconds. You'd accomplish the same effect - sampling many more points of your HRF than you would with a fixed ISI. You'd also get an added benefit - you'd "uncover" a whole chunk of your HRF (the chunk between 10 seconds and 14 seconds) that you wouldn't sample at all with a fixed ISI.

That lower portion can help you better determine the shape of your whole HRF and find a good baseline from which to evaluate your peaks. This added benefit is why most people go the second route in trying to "jitter" their experiment - varying your ISI gets you all the benefits of an offset TR, plus more.

2. What are the pros and cons of jittering / variable-ISI experiments? When is it a good/bad idea?

Variable-ISI experiments are a way of making the tradeoff between power and efficiency in an experiment.

Fixed-ISI designs are extremely limited in their potential efficiency. They can have highpower by clustering the stimuli together: this is a standard block design. But in order to get decent efficiency in an experiment, you need to sample many points of the HRF, and that means variable ISI. Any experiment that needs high efficiency - say, a mental chronometry experiment, or one where you're explicitly looking for differences in HRF shape between regions - necessarily should be using a variable-ISI design. By contrast, if you're using a brand new paradigm and aren't even sure if you can get any activation at all with it, you're probably better off using a block design and a fixed ISI to maximize your detection power.

From a psychological standpoint as well, the big advantage of variable-ISI designs is that they seem far more "random" to subjects. With a fixed ISI, anticipation effects can become quite substantial in subjects just before a stimulus appears, as they catch on to the timing of the experiment. Variable ISIs can decrease this anticipation effect to a greater or lesser degree, depending on how variable they are. Liu explores the concept of "conditional entropy" as a measure of randomness in how an experiment "seems," and it is, predictably, intertwined with the power/efficiency tradeoff.

3. How do I decide how much to jitter, or what my mean ISI should be?

Great question. Depends a lot on what your experimental paradigm is - how long your trials are, what psychological factors you'd like to control - as well as what type of effect you're looking for. Dale (JitterPapers) lays out some fairly intelligible math for calculating the potential efficiency of your experiment. Probably even easier, though, is to use something like Tom Liu's experimental design toolbox (JitterLinks) or, even better, AFNI's 3dDeconvolve -noinput option, which will take a given experimental paradigm and calculate how good it is in terms of power and efficiency. Once you're within the ballpark for the type of paradigm you like, these tools can be an invaluable way to optimize your design's jitter / ISI variation, and are highly recommended for use.

4. Should the jitter be a multiple of the TR in an event-related fMRI design?

FAQ on   https://www.researchgate.net/post/Should_the_jitter_be_a_multiple_of_the_TR_in_an_event-related_fMRI_design

5. randomized the order of the trials or randomized the ITI

As will be explained below, in order to be sensitive to differences between trials close together in time (e.g, less than 20s), one either 1) uses a fixed ITI but varies the order of different trial-types (conditions), or 2) constrains their order but varies the ITI. Thus a design in which two trials alternate every 4s is very inefficient for detecting the difference between them (as explained below). One could either randomise their order (keeping the ITI fixed at 4s), or vary their ITI (keeping the alternating order). Note that, in this context, blocks can be viewed as runs of trials of the same type, and a blocked design corresponds to a varying-ITI design in which there is bimodal distribution of ITIs: a short ITI corresponding to the ITI within blocks, and a long ITI corresponding to the ITI between the last trial of one block and the first of the next (as elaborated below).