calculating frequency interaction (now with the complete message)

Robert Oostenveld r.oostenveld at FCDONDERS.RU.NL
Mon Aug 15 09:55:20 CEST 2005

```Hi Thomas,

On 14-aug-2005, at 17:05, Thomas Thesen wrote:

> Calculating the interaction as above could then result in erroneous
> estimates of the integration effect:
>
> A = 3 units; squared = 9
> V = 3 units; squared = 9
> AV = 6 units; squared = 36
>
> integration effect = 6^2-(3^2+3^2) = 18
>
> even though it is quite evident that the neuronal response to AV is
> a direct
> summation of A and V.
>

The effect of summation of the two signals on the estimated power (or
amplitude) of their sum depends on the phase relation between the two
signals.
1) If the two signals are in perfect phase alignment in each trial
(i.e. zero deg phase difference), they will add up as you described.
2) If they are in perfect antiphase (180 deg), they will cancel out.
3) If they have a random phase with respect to each other, i.e. the
phase difference is different in each trial, they will add up "a
little".
In case 1, the amplitude (i.e. sqrt of the power) depends linearly on
the amplitude of the two signals. In case 3, the power depends
linearly on the power of the two signals.

Please try playing with the following lines of code

real_pow1 = 3;
real_pow2 = 3;
t = linspace(0, 2*pi, 1000);

for trl=1:100
s1(trl,:) = sqrt(real_pow1)*sqrt(2)*sin(t);
phasediff = 2*pi*rand(size(t));     % CHANGE THIS TO SEE THE EFFECT
s2(trl,:) = sqrt(real_pow2)*sqrt(2)*sin(t + phasediff);
end

% add the two signals for each trial
s3 = s1+s2;

% estimate the single trial power
pow_s1 = sum(s1.^2,2)/length(t);
pow_s2 = sum(s2.^2,2)/length(t);
pow_s3 = sum(s3.^2,2)/length(t);

% estimate the power and amplitude
pow1  = sum(pow_s1)./100
pow2  = sum(pow_s2)./100
pow3  = sum(pow_s3)./100

ampl1 = sqrt(pow1)
ampl2 = sqrt(pow2)
ampl3 = sqrt(pow3)

I hope that this clarifies it. Of course, it does not yet help you
deciding how to test for the interaction, since the additive effect
(which you expect under the null hypothesis) can be either on
amplitude or on power. To choose the right test, you will have to
consider the sources of the two signals that are being mixed on the
channel level, i.e. are they coming from one source, or from two
sources, and in the latter case, are the two sources strongly
coherent or not?

best regards,
Robert

=======================================================
Robert Oostenveld, PhD
F.C. Donders Centre for Cognitive Neuroimaging