Dear John,

> Although I agree in general with your response, I think that you 
> overstate the benefits of temporal basis functions and the dangers
> of selective averaging.
> 
> First the dangers of selective averaging. The problem here is one of
> sampling.  If the data are collected at 1-second fixed intervals the
> sampling is almost certainly adequate sampling for the averaged
> response to accurately represent the hemodynamic response.  If the TR
> is five, the sampling is sure to be inadequate.  If one uses the same
> definiton of the Nyquist criterion as used in image processing, one
> would conclude that you need two samples per FWHM.  The FWHM of of the
> peak of the hemodynamic response is 5-6 seconds, so this says that the
> maximum TR would be 2.5 to 3 seconds.  This doesn't mean that we can
> assume that a TR of 2.5 seconds is without problems. The phone company
> oversamples by 50% to get good voice quality.  However, I wouldn't
> expect major errors at TRs of less than 2.5 seconds.

I agree entirely. If your studies use surface coils or small volumes
with short TRs selective averaging is quite tenable.

> Temporal basis functions (by which I mean bases other than the sum of
> delta functions) completely overcome these sampling issues if the
> stimulus presentation intervals are randomly varied.  However, this
> comes at the cost of a very strong assumption - that every hemodynamic
> response will be nearly the same and that these responses will be
> nearly equal to the impulse response.  This assumes that neuronal
> firing is either very brief or that it is known and can therefore be
> modeled by modifying the temporal basis functions.  It is not hard to
> think of complex behavioral paradigms where the experimenter cannot
> predict precisely when or for how long neurons will fire.  If the
> response is poorly modeled by the temproral basis functions (and many
> responses won't be because most of the bases proposed do not span the
> space of possible functions), it is likely to be completely missed
> Selective averaging combined with an F-test does not require any
> assumptions about the response timing or shape.
> 
> My conclusion is that one must take an agnostic approach to this
> problem. In experiments where temporal sampling is poor, temporal basis
> functions must be used.  In studies where the neuronal response may be
> unpredictable, selective averaging should be used.  In most
> experiments, the tradeoffs are hard to quantify because we don't have
> enough data yet.

I think we have to distinguish between two distinct issues here: (i)
What is the most appropriate basis set for any inference and (ii) is
'selective averaging' a useful approach.  Remember that selective
averaging is just a special case of temporal basis function analysis,
that necessarily imposes constraints on experimental design to provide
a discrete (and therefore biased) sampling of the inter-stimulus
interval.  In this sense I am not claiming any 'benefits of temporal
basis functions'.  I am simply observing that the experimental design
enforced by 'selective averaging' procedures is suboptimal.  The
assumption under which both approaches operate are identical (I am
assuming that 'selective averaging' is implemented using a conventional
least squares finite impulse response (FIR) estimation).

Arguments about being able to model highly variable voxel-specific
responses pertain to the choice of the basis set, not to whether one
uses 'selective averaging' or not.  Note that using a distributed
sampling of the inter-stimulus interval allows one to use MORE basis
functions than could be used in 'selective averaging'.

In short I cannot think of any principled motivation for using
'selective averaging' but fully accept the point that the choice of a
suitable basis set is critical.

In relation to Eric's point I am using the term 'basis' set in
reference to a set of vectors that span a subspace of all possible
responses defined at discrete intervals of peristimulus time (we use
100ms in constructing our sets).  This basis may, or may not, be an
ortho[normal] basis set.  If one considered the ultimate basis set of
all possible responses one would end up in Hilbert space (I do not
really know what a Hilbert space is but I know Eric does!).


With best wishes - Karl



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