Three questions about 1/f

[MakotoMiyakoshi]

Here are three questions (or challenges?) from Eugen.

1. Is the 1/f dependence universal, or does it manifest itself in specific cases? How precise is it? Is it 1/f or 1/f ^alpha? How do filter settings affect the nature of the dependence?
2. Does the effect occur at the level of an individual neuron or through their interactions? A trial calculation showed that a Poisson stream of signals similar in shape to EPSPs and IPSPs generates a spectrum close to 1/f. Is this "colored noise" or something more complex?
3. How are 1/f components and periodic components (peaks in the spectrum) related?

[EugenMasherov]

Thank you for posting my questions!
And no, this isn't a challenge, but I'm genuinely trying to find answers to these questions as part of a more general problem: is the entire EEG truly generated by a single mechanism, or are we seeing the sum of signals from fundamentally different sources? It seems to me that there are at least three different mechanisms, one of which generates sinusoidal signals (sharp peaks in the frequency spectrum), a second produces a broad spectrum with a 1/f power decay, and a third is responsible for the peaks and sharp waves (I don't think I was the first to figure this out; there was something similar in a 1944 paper). This concept is published in a Russian paper (I can provide an English translation, but it's poorly edited).
https://www.med-alphabet.com/jour/article/view/4644

And a few clarifications on my questions.

1. If the power changes as 1/f, then the amplitude changes as 1/sqrt(f). Or if the amplitude changes as 1/f, then the power changes as 1/f^2. But perhaps we should consider a more general power-law relationship 1/f^alpha. In this case, the original signal was filtered, which could distort the relationship.
2. If there is a stream of pulses I(t), each generating a response g(tau), then the total signal will be Integral I(t-tau)*g(tau) d tau, and the Fourier transform of the total signal will be the product of the Fourier transforms of I(t) and g(tau). In particular, if the stream I(t) is Poisson, the spectrum of the output signal will match the spectrum g(tau), up to a factor of . Taking a function similar to the EPSP as , I obtained a spectrum close to 1/f.
3. It seems to me that the mechanisms generating the peaks in the spectrum and 1/f are dependent, but I am unaware of a mathematical framework that could reveal this dependence. I am relying on the collective wisdom of the group. Perhaps cepstral analysis could help here, but so far, attempts in this direction have been unsuccessful.

[angianno]

Hello all,

Some of my thoughts/intuitions about (part of) these intriguing research questions.

If we consider 1/f^a, then α can be different in different subranges (e.g. 1–20 vs 20–80 Hz). So when people say “1/f”, it’s usually a good approximation over a limited band (?), not an exact law across all frequencies. In practice, α can differ between subjects, brain regions, conditions (eyes closed vs open, task vs rest, etc.) So, it can be generally 1/f^α with α varying, and 1/f is a convenient shorthand, not a precise law.

About filters, they can have a large effect on the observed slope. If we consider different filtering options, we can say that:
(a) A high-pass filter (e.g. at 0.1 Hz, 0.5 Hz, 1 Hz) removes or attenuates low frequencies where the power is highest. On log–log spectrum, this bends the spectrum near the cut-off frequency and, thus, can make the fitted α smaller or larger depending on the frequency band we fit. If we fit α including the region near the high-pass cutoff, the slope will not reflect the true underlying 1/f^α but a mixture of brain + filter properties.
(b) A low-pass filter (e.g. at 40 Hz) removes high-frequency content and can flatten or steepen the slope near the cutoff. So, it can change α if our fitting band extends close to cutoff.
(c) Notch filters (e.g. 50/60 Hz) cut out a small frequency band, which can create dips in the spectrum.
(d) Band-pass filters (e.g. 8–12 Hz) distort the power-law picture completely, because we’re no longer seeing the broad background, only a narrow band.

So, practically, if we want to analyze the 1/f^α background, we should use broadband data with minimal distortions (mild high-pass, wide low-pass), as well as we should avoid fitting near filter cutoff frequencies. So, a design of the analysis to explicitly handle filter characteristics is essential using raw or minimally processed spectra.

[EugenMasherov]

If the power depends on frequency as 1/f , and this remains true over the entire frequency range up to infinity, then the total signal power is infinite, since the integral of 1/f diverges. Therefore, one or both of the following conditions must be met:

1. The power depends on frequency as 1/f ^ alpha , alpha>1
2. The dependence holds only over a certain range.

It seems to me that both statements are true, but experimental verification is needed. Judging by the literature, the exponent lies in the range 0.8<alpha<1.5 and is estimated for a limited band.
Filters cannot have a perfect frequency response, since the steeper it is, the longer the settling time (the uncertainty principle in signal processing). In real devices, a compromise occurs, and near the passband edge, the power is distorted.
To estimate such a dependence in the presence of peaks, robust regression could perhaps be used, which would treat the peaks as neglected outliers.If the power depends on frequency as 1/f , and this remains true over the entire frequency range up to infinity, then the total signal power is infinite, since the integral of 1/f diverges. Therefore, one or both of the following conditions must be met:

1. The power depends on frequency as 1/f ^ alpha , alpha>1
2. The dependence holds only over a certain range.

It seems to me that both statements are true, but experimental verification is needed. Judging by the literature, the exponent lies in the range 0.8<alpha<1.5 and is estimated for a limited band.
Filters cannot have a perfect frequency response, since the steeper it is, the longer the settling time (the uncertainty principle in signal processing). In real devices, a compromise occurs, and near the passband edge, the power is distorted.
To estimate such a dependence in the presence of peaks, robust regression could perhaps be used, which would treat the peaks as neglected outliers.