The gold standard of a certain class of psychological experiment is the double dissociation.  [NB: they are not without their detractors: see here and here.] The idea is that if you can separate your subjects into two classes, where one class has X and the other that has Y, then X and Y must be independent functions.  Wikipedia quotes the following explanation:

… If on the other hand you have two TV sets, one without sound and one without a picture you can conclude that these must be two independent functions (double dissociation). [A.J. Parkin: Explorations in Cognitive Neuropsychology. Blackwell, Oxford, 1996.]

And, that’s unarguably true, except that we don’t really know what an “independent function” is.

The TV example probably dates from the ancient days of analog television sets.  In such a set, you basically have a branching pipeline:

1. An antenna and maybe an RF amplifier,
   
2. A mixer that takes the input signal and reduces its frequency to a more convenient (and fixed) value.   Recall that old-style pre-digital TV signals were multiplexed by sending each channel in a different frequency band.  It was technically hard to make amplifiers that worked well at all frequencies, so it was easiest to convert whichever channel was desired to a common frequency (typically 10.7 MHz) where one could conveniently build electronics that would amplify and do all the needed signal processing.
3. The signal is demodulated (see description of the NTSC standard).   At this point, the video and audio signals start on separate paths.
4. (a) the video signal is processed, (b) the audio signal is processed.
5. (a) the video signal drives the picture tube, (b) the audio signal drives the speakers.

So, a double dissociation merely shows that the pipeline branches: steps 4a and 5a for video signals are executed on different hardware from 4b and 5b for analog signals.  A double dissociation can happen even if there are a lot of common steps (1, 2, 3) that are shared by both video and audio.  In principle, the common steps could even comprise almost all of the system, and the dissociation could be proven by a single step.

TV sets are (or used to be, at least) composed of independent modules.   Brains are not.   [Of course, in more modern systems, you can have independent software modules that can share almost all the hardware with each other.  The processor can be shared, even much of the working memory.  All a software module needs to call its own is a few memory locations to store the code and any data that needs persistent storage.  There’s no reason to think a brain is closer to a TV set than a computer.  See “Coarse and Fine”, below.]

What does it mean to get a double dissociation in a brain?  In practice, if often means that you look at the brain with fMRI and you find an area of activation (call it “a”) that is associated with stimulus A, and a different one (“b”) associated with stimulus B.  Or, more commonly, stimulus A makes areas “a” and “c” light up, and stimulus B activates “b” and “c”.  There is almost always a common region (“c”) and it is often large.  You can see the analogy to TV sets here.  The area of the brain activated by both stimuli corresponds to steps 1, 2, 3, and the dissociation area “a” corresponds to TV steps 4a & 5a, et cetera.

So, the double dissociation proves that the response “pipeline” isn’t quite the same for the two stimuli, but not necessarily much more than that.

Multiple Dissociations

A human geographical analogy is the following: suppose we have two populations of humans, labeled “M” and “F”, for convenience.  We find a double dissociation: M humans go into certain rooms that are almost never visited by F humans, and vice versa.  What does that prove?  Obviously, that M and F humans aren’t identical.

Now, you may be thinking that M=male and F=female, and the rooms are bathrooms/loos.  That’s one example, and it’s an example where the double dissociation says something about the structure of M and F humans.   But, that particular double dissociation happens for almost any two groups of humans.  Consider:

• M=plumbers and F=lawyers.   The relevant rooms are plumbing supply stores and courtrooms.
• M=Catholics and F=Jews.  The relevant rooms are churches and synagogues.
• M=Parents and F=childless couples.  The relevant rooms are schools and fancy restaurants.fMRI
• M=Londoners and F=New Yorkers.
• et cetera.

In this case — if you have dissociations everywhere — you would have to be careful in interpreting them.  Differences between plumbers and lawyers certainly exist, but they have to be seen in the context that every person has their own individual pattern of room use.  In fact, for almost any possible division of the human population into groups, you can find rooms that are heavily used by one group, but essentially never entered by the other.  I wonder if that’s the case for words in fMRI images?  To what extent are they dissociated by part of speech, or other group properties, versus each word simply having its own activation pattern?

Coarse and Fine Dissociations.

Dissociations on fMRI images are on a fairly coarse scale, compared to individual neurons.  fMRI machines can see changes in the average activity of voxels (regions of the brain) that are a few millimeters across.  The neurons that make up your brain are much smaller: about 10 microns across (1/100 of a millimeter) so there are a million or more within a fMRI system’s voxel.  That means there can be a lot of dissociations that we just cannot see on fMRI.  Especially so, when you realize that when a fMRI system says a region is active, it doesn’t mean that all the neurons are firing, just that there is substantially more activity than normal.

So, you could imagine numbering the neurons in a voxel from one to a million, and associate all the even-numbered ones to one task, and the odd-numbered ones to another task.  The two tasks would be completely dissociated, but they would look — to an fMRI machine — as if they share the same regions of the brain.

And, finally, any time that the subject’s behavior differs between two stimuli, you know there is a fine-scale dissociation in the brain, somewhere.  For if the behavior differs, it is the consequence of different muscles being activated, which are excited by motor neurons, which are controlled by the motor cortex.  So, if stimulus A leads to the subject pushing button A, and stimulus B leads to B, you know that there are different sets of neurons involved, and therefore there is a double dissociation somewhere in the brain.  FMRI, then, is not so much about proving that a dissociation exists as finding where it is.