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My read of "Human brain networks function in connectome-specific harmonic waves" (Atasoy 2016) seems to be that you can model brain activity in terms of standing waves of the connectome (functional connectivity from fMRI). But my understanding is that functional connectivity is itself inferred from brain activity. This seems like a chicken-and-egg problem. What is the difference between modeling brain activity with spectral graph theory on the connectome, vs Fourier analysis of the BOLD signal directly?

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Interestingly enough, I myself have been reading papers like these for some weeks. Although I think your question does not have a correct answer (since it is a little bit open) I can try to give my perspective, but please be aware that is may not be 100% accurate.

Atasoy, et al. 2016 provide a really fascinating observation which is that functional networks (i.e. Default Mode Network) can be recovered from a discrete (and rather small) set of combination of structural connections, which they refer to as the "connectome harmonics". Here I point you to Fig.3 of the paper.

Furthermore, they solve a certain (supposedly realistic) mathematical model for brain dynamics which use diffusion equations. The laplacian harmonics appear naturally in that formalism, so they conclude that functional activity derives from structural connections (and not the other way around). In my opinion, this is reasonable (and one might even say obvious), since a network without structural connections can not exhibit any kind of functional activity.

Having said this, it is true that their observation is not particularly useful in terms of methods to analize functional activity. It requires, by definition, a structural network, which normally you don't have when dealing with BOLD signals. Even more, analizing fMRI signals through spatial connections seems a bit of a painful detour. So, to really give an answer to your inquiry, I don't think it's reasonable to compare the two approaches. The findings in Atasoy, et al. 2016 fall in the "theoretical" side while directly performing Fourier Analysis on your BOLD signals should still be good and useful.

As a side note, I am not sure if by Fourier Analysis you mean analizing fMRI by means of Laplacian eigendecomposition of the graph or not. But I would say that this is the way to go instead of just doing Fast Fourier Transform on the BOLD signal if you want to do community detection. There is a rather big number of papers on this matter.

Hope I provided a reasonable point of view on this matter.

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