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I am performing time-frequency analysis on electrophysiological data with complex morlet wavelets. Each decomposition in the attached figure shows total power following the onset of a stimulus (top row) or preceding an eye movement (bottom row). There is a broadband, transient increase in power following stimulus onset.

I am interested in determining whether there are frequency-specific changes in power (i.e., 'oscillations') during the 0-200 ms epoch as well; therefore, I have parsed the total power into phase-locked and non-phase locked components. Because variability in response latencies may bleed into the estimate of non-phase locked power (David et al. 2006),

I want to compute adjusted power as recommended by these authors (pg 1589). I am unsure of how to implement this orthogonalization in Matlab-- it is slightly beyond my mathematical savvy.

I attached a snapshot of the related paragraph in the paper as well.

Is anybody familiar with this computation?

Total Power

Adjusted Power Computation

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The notation and terminology in this and the closely related paper could have been more simplified I think.

To understand what's going on there, we need to start from page 1583. Specifically:

  • $y(t)$ is basically your measured signal (the output from each neuronal population). A scaled version of that is what you would sense via some modality (in an ideal world).
  • $s(\omega, t)$ is the frequency-time representation of a $y(t)$. This is complex.
  • Given that this is about Event Related Responses (ERR):
    • $g(\omega, t)_T$ is the average absolute value of $s(\omega,t)$ after a number of trials. This is real valued.
    • $g(\omega, t)_e$ is the average of $s$ times the average of the conjugate of $s$ (both of which will be complex, the result of which however will be real).
    • These look similar but their magnitude is not.
  • Then the explanation of the breakdown of that "power" follows. What this means is that, based on the assumptions in the paper, we think that the neuronal activity is due to three things: a) Background activity, b) Activity because of connections between neuronal populations and c) Induced activity by the stimulus. Furthermore, these three components are assumed orthogonal. This means that if one "happens", the rest do not. This is important for later on.
  • WHERE/WHEN this holds then we could estimate the background activity by looking at the spectrum BEFORE the stimulus onset, we could estimate the activity due to the stimulus (the part that is phase locked to the stimulus), because we know the "shape" of the stimulus and therefore what is left should be due to structural connections.
    • These are captured in $g(\omega, t))_i$ and $g(\omega,t)_T$.
    • Notice here that the baseline is only $g(\omega)$
  • So, when you arrive at equation 23, what you see is the explanation of that "adjustment" where:
    • $g(\omega,t)_T$ is the frequency-time representation of the averaged stimulus response after $T$ trials as before.
    • $g(\omega,t)_e$ is the frequency-time representation "power of average" as before
    • $\hat{\eta}$ now denotes the proportion of the frequency-time representation of $g(\omega,t)$ that is due to $g(\omega,t)_e$. The "trick" here is this, you have already assumed that the three components are orthogonal. WHERE/WHEN this holds, the "pseudoinverse" ($g(\omega, t)^+$) of a given $g(\omega, t)$ multiplied by the "shape" of the stimulus response $g(\omega,t)$ over the trials would be zero and you would be left with the "shape" of the stimulus response. So, that $\hat{\eta}$ business there (and the subtraction) is what allows you to do that "separation".

Perhaps it is now clear how to produce that $g(\omega,t)^+$. Provided that you have a given $g(\omega,t)_e$, invert it with pinv (or anything else equivalent to a pinv in another platform).

But, I wonder how effective would that be for what you are trying to do. Irrespectively of this specific technique, a "pulse" or a step or any abrupt change of a signal in the time domain usually has a broad representation in the frequency domain (whether that is Fourier or Wavelet, the point is that you need more components to synthesize the step). You are looking for "oscillations" in a 0-200ms interval. What is the frequency of the oscillations you are looking for?

Hope this helps

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  • $\begingroup$ I am getting closer to understanding. I have the total, evoked, and induced power components obtained. Total power is a real valued matrix obtained following averaging time-frequency matrices across trials, evoked power is a real valued matrix obtained following time-frequency decomposition of the time-domain trial average, and induced power is obtained through subtracting evoked from total. I am still having a hard time seeing how g(w, t) [no subscript] relates to these three components. Where does it come from? $\endgroup$
    – Kevin
    May 17, 2018 at 14:05
  • $\begingroup$ Regarding the last question-- there is commonly an increase in beta band coherence between areas during this epoch. I am only recording from one area; however, want to see if a power modulation is buried beneath the transient. This was very helpful. $\endgroup$
    – Kevin
    May 17, 2018 at 14:14
  • $\begingroup$ @Kevin That's one point where he gets a bit "naughty" because $g(\omega, t)$ has a specific "meaning" throughout and then on page 1589 this meaning changes to now basically become a "stack" of $g(\omega,t)_e$. Please see equation 23 where the matrix is getting defined. Sounds like a challenging task (what you are trying to do). 200ms at the low end of beta will contain ~2.5 periods and ~6 periods at the high end. Perhaps modify the experiment a bit too (?) to give you some more "exposure" to what you are trying to measure. $\endgroup$
    – A_A
    May 18, 2018 at 7:02
  • $\begingroup$ @Kevin Glad to hear you found the post helpful, you can upvote and/or accept the answer from the controls on the left. Accepting the answer will also stop circulating it in the board as "still open". $\endgroup$
    – A_A
    May 18, 2018 at 7:03
  • $\begingroup$ Where do those 1's come from in the stack? And I believe the t1 to tT refers to time. I don't see how that naughty g(w, t) stack differs from the evoked power matrix. And, excuse my newbie behavior. Thanks again. $\endgroup$
    – Kevin
    May 18, 2018 at 11:11

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