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.

[MakotoMiyakoshi]

Hi @EugenMasherov,

1. This is a good basic point to confirm.
2. I think this is the basic principle of Gao et al. (2017) which is simple but still eyes-opening (in English, people say "duh" in this case). But it is still surprising to me that the resultant colored power distribution is valid across spatial scales from a single neuron to scalp EEG.
3. It is independent. Read mgy42's post. He authored a paper to empirically prove it. Three must-read papers for understanding EEG 1/f-ness #4

[EugenMasherov]

2. I think this is precisely what's so appealing about the 1/f model—self-similarity. A concept that encompasses both the medieval philosophy of "Microcosm equals macrocosm" and modern fractals.
   Let f(x)=ax^(-k)
   Then f(cx)=c^(-k)*f(x)
   Power law exist in hydrology (Horton's law), linguistics (Zips's law), economy (Pareto's law) and other.
   But I do not have sufficient knowledge to assert that this is a manifestation of a general great law, and not a convenient empirical device for describing results.

[dieloz]

Eugen you're right regarding the pervasivity of the 1/f in many fronts. Indeed, there's a recent paper pointing towards a property of non-linear systems when you drive them with noise

Mark A. Kramer, Catherine J. Chu; A General, Noise-Driven Mechanism for the 1/f-Like Behavior of Neural Field Spectra. Neural Comput 2024; 36 (8): 1643–1668. doi: https://doi.org/10.1162/neco_a_01682 (please find preprint and code of the same paywalled paper)

"Under the general framework considered here, the range of aperiodic exponents reflects different types of noise. The observable dynamics (e.g., variable X1 in equation 2.1) depend directly on the stochastic drives to the observable variable, and indirectly on the stochastic drives to the latent vari-able(s). The latent dynamics introduce correlations in the uncorrelated la-tent noise process before this noise reaches the observable dynamics. There-fore, the observable dynamics depend on both uncorrelated and correlated noise inputs, and we may interpret the aperiodic exponent in terms of these different types of noise; an aperiodic exponent near β = −4 indicates that correlated noise inputs dominate the high-frequency observable dynam-ics, while β = −2 indicates that uncorrelated noise dominates the high-frequency observable dynamics. These general results are independent of a specific biophysical mechanism, and a biophysical interpretation of the relationship between different types of noise and the aperiodic exponent depends on the specific model choice"

[EugenMasherov]

Zipf law, sorry.

[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.

[MakotoMiyakoshi]

Hi @angianno @EugenMasherov

Here is a SPEX plot of 20,000 'brain' independent component collected from n=820 subjects during eyes open and closed resting combined (62 and 64 channels). The intercept of PSDs are set to 0, and mean of each SPEX bin are plotted. I believe these numbers refer to the alpha in 1/f^alpha. I used FOOOF with single and relatively narrow peak fitting. Taken from a draft of a paper I've been working on, in which I used SPEX just for artifact rejection (i.e., flat SPEX == poorly brain). It's an unpublished data, so do not redistribute it (despite it is openly available here!)

[angianno]

Hi @MakotoMiyakoshi,

Thanks a lot for sharing the SPEX plot, this is really informative. The way you grouped ~20,000 ICs by aperiodic exponent bins makes the 1/f structure very clear, especially after normalizing PSD intercepts. If I’m understanding correctly, the SPEX values correspond directly to the fitted α from FOOOF’s aperiodic component (with the single narrow peak model), so each curve effectively represents the mean 1/f^α background for ICs falling into that exponent range.

What stands out to me is how systematically the spectra steepen as α increases, and conversely how the “flatter” components (α < 0.1) behave more like artifact/EMG-dominated ICs. The high-frequency portion (30–45 Hz) reflects that the flatter slopes retain power in that range, whereas the higher-α ICs drop off dramatically, which is exactly what we’d expect if those components reflect genuine cortical sources rather than broadband muscle noise.

So this figure proves that the “1/f” in EEG is really a distribution of 1/f^α behaviors, not a universal single exponent, and that α varies widely across ICs (and subjects).

I would be happy to know how you handled low-freq flattening, filter settings, or alternative parameterizations of the aperiodic component.

Thanks again for sending this over.