Dear Chris,

>
> Basically, I want to see if the area 1 -> area 2 connectivity  
> varies as a
> function of activity in area 3, without any a priori assumptions about
> whether either areas 1 or 2 are directly connected to area 3. I'm  
> looking at
> habituation of inhibitory circuits, so I'd like to see that as  
> activity in
> area 3 decreases, the throughput from 1 to 2 goes up... without  
> making any
> assumptions about how area 3 is acting on that connection (i.e.  
> directly,
> indirectly, etc.). Any suggestions?

It sounds to me like you were spot on in your first mail. I think PPI is
a way to answer the question you have here.

PPI is "normally" read out as psycho-physiological interaction, meaning
that you look at how a "psychological state" (often defined through the
instructions to the subject such as "attend to this" or "now it might
hurt" or something) changes the connectivity between two areas. This
means that one of the functions, f, is derived as an adjusted
time-series from some VOI in region A in your data. This is the
physiological parameter.

Let us say we were now to put this into your design matrix, along with
any other regressors modelling task effects etc such that your new
design was X = [X f]
Any voxels with non-zero parameter estimates for f could then be said to
correlate with A over and above what can be explained by a common
correlation with some external stimulus.

We would then have a "psychological function", which is typically some
state or context function with the value one indicating one state (e.g.
"concentrating on the color of the dots") and zero indicating another
state (e.g. "concentrating on the speed of the dots"). This is therefore
a "known" function with no uncertainty associated with it.

A PPI is then a regression on the interaction (or element-wise product)
of these two functions.

To make it really concrete. Let's say our (short) time series from VOI A
is f=[1 3 1 3 1 3 1 3]. Note how it seems to be more "active" every
other scan. Furthermore, let's say the subject was told for the first
half of the experiment that he was "safe", and that in the second half
he might get a painful shock at any time (though god forbid we actually
did that). The (mean-corrected) "fear"-regressor would then look like
g=[-1 -1 -1 -1 1 1 1 1], and the resulting interaction regressor fxg=[-1
-3 -1 -3 1 3 1 3].

When we then put these into our design we get X = [X f g fxg] where our
PPI regressor will pick up anything that is "more positively correlated
with A during "fear" than during "non-fear"".

In the simple example above I have disregarded that we observe the brain
through the bold effect, and we should really have 
X = [X f C(g) fxC(g)], where C() denotes convolution with the HRF.

So, this is the underlying principle of PPI. There is still a little
catch though. The "communication" between the A and the other regions of
the brain is defined on the neuronal level (the brain doesn't talk
through the BOLD effect). So, strictly speaking fxg at the neuronal
level should really be fxg = iC(f)xg, where iC() denotes deconvolution
with the HRF (i.e. trying to deduce the underlying neuronal firing from
a BOLD time-series. The fxg that we observe, and hence should put into
the design matrix is C(fxg) = C(iC(f)xg).

However, for designs of the type that I have described above where we
have a known psychological variable/function that is typically varied
on/off in a block fashion there tend to be little difference between
fxC(g) (which is easy to calculate) and C(iC(f)xg) (which is trickier to
calculate). Therefore, I would say that mostly it doesn't really matter.

Your case is different, and is really a "physio-physiological
interaction". You have two regions A and B, and you want to know how the
activity in B affects the connectivity between A and other parts of the
brain. The reasoning is exactly the same as before with the
psychological function g replaced by the time-series, g, from region B.
Since g is already convolved with the HRF (by the brain) so now all you
would need to do is fxg = fxg (i.e. elementwise multiplication of the
two time-series).

HOWEVER, as before, the correct thing to do is fxg = C(iC(f)xiC(g)). And
this time there is potentially a quite substantial difference between
fxg=fxg and fxg=C(iC(f)xiC(g)). Sorry that notation is getting a little
confused.

So, in order to implement your specific design you would ideally need a
tool that allows you do deconvolve an fMRI time-series with the HRF.
Without blowing the noise out of proportion.

Now, I hesitate to say this, but SPM (shudder) has such a tool. I would
therefore in your case recommend you to have a look at using SPM for
creating your PPI regressor. After which I expect you to stick it right
back into FSL!!

Good Luck Jesper