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I guess this question is for experts of the domain, but anyway, maybe someone can answer it.

So in EEG recording ERD and ERS (event related (de)synchronizition) designate short amplitude attenuation (or augmentation) of rythms within the alpha band..

So, if they are not phase locked it means that when the event appears, the time they actually get trigered by it must vary ?

On the contrary, for Bereitschaftspotential (BP), when the event appears, we know exactly when BP will be trigered ?

Are theses good explanations of the phase-lock and not phase-lock definition ?

Thank you

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Phase-locked/time-locked EEG components are expected when you have some coordinated neuronal response at that moment in time; for example, if a population of cells are depolarized in response to some stimulus, that will show up as a time-locked/additive component, also called an "evoked" response. Averaging many trials in the time domain will emphasize evoked responses.

However, there is lots of "ongoing" activity in the brain. It is possible to modulate the amplitude of this ongoing activity without an additive component. David et al (2006) refer to this as "structural" changes or "induced". A familiar example might be the volume control on a music player. The pattern of the sound waves doesn't change when you shift the volume. If you increase the volume at various random points in a song and average what comes next, it will eventually average to zero: there's no evoked component time-locked to the change, you just change the amplitude of what is already ongoing. However, if you plot a power spectrum, you'd see a clear broadband amplitude increase (of course, in the brain you may see induced responses limited to particular frequency bands, rather than broadband like a volume knob). You can isolate induced components by first subtracting any evoked response (that is, the time domain average of many trials).

You can also have responses that are evoked in nature but come with some latency variation. These responses will be blurred in the time domain if you average many trials, and may also appear like induced responses as overall changes in power without a clear time-locked component.

With only EEG, it's difficult to further identify the neuronal sources of different components of the response or to definitively state whether you have induced or jittered evoked responses. I've written a paper about these issues (Krause & Banks 2013, https://doi.org/10.1152/jn.00187.2013) and there are several other useful sources cited in that paper, but I'd recommend starting with the David et al 2006 paper below:

David, O., Kilner, J. M., & Friston, K. J. (2006). Mechanisms of evoked and induced responses in MEG/EEG. Neuroimage, 31(4), 1580-1591.

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