Correct interpretation of coherence (physics vs engineering)

[Joel F]

Hi Mike,

I have a conceptual question (with implications for EEG). There seems to be a bit of a mismatch between how physicists talk about coherence vs how engineers talk about coherence. I know that in engineering and digital signal processing, the magnitude squared coherence is a measure of phase correlation between two signals defined as the cross-spectral density squared over the product of both auto-spectral densities. This is analogous to the Pearson correlation coefficient (covariance squared over the product of each standard deviation).

Now, how do we interpret this? Coherence as a quantitative concept seems to come from physics. If we consider the plain language definition of coherence from Wikipedia, "two wave sources are perfectly coherent if they have a constant phase difference and the same frequency, and the same waveform." It seems that this physics interpretation is incompatible with the engineering definition. When we take the cross-spectrum to derive magnitude squared coherence, we are really taking the FFT of the cross-correlation function. FFT uses sine waves and cosine waves as basis functions, so we know we have the same waveform at each frequency.

Given this fact, how can the phase difference between two sinusoids of the same frequency ever change? By definition, they each have the same d phi / d t, where phi is phase and t is time. And because the magnitude squared coherence is considering sinusoids at fixed frequencies, it seems impossible that these sinusoids of the same frequency might not have a constant phase difference. Furthermore, we know that coherence assumes stationary signals, so their frequency content must stay fixed over time.

What is the correct way to talk about coherence in EEG, then? Can we use the intuitive concept of coherent waves from physics if we are talking about magnitude squared coherence from EEG? I hope this is forum is an appropriate place for this question. Thanks in advance for your time!

Best,

Joel

[Mike X Cohen]

Hi Joel. Great question. There is a signal-processing answer and there is a neuroscience answer. I'll start with the signal-processing answer.

The key insight is that neural signals are highly non-stationary (with some constraints). That's why we talk about frequency bands instead of precise frequencies. It makes sense to talk about "alpha" as energy between ~8 and ~12 Hz, but it doesn't really make sense to talk about energy at 11.23442 Hz. So, if you would extract a single Fourier coefficient from two channels, then over time, the phase difference is trivially stationary, as you noted. But a single Fourier coefficient is not really interpretable in brain data. Therefore, you would filter from 8-12 Hz, and then there will be frequency fluctuations over time between the two channels (or, more generally, two oscillators). Those two oscillators can be synchronized, desynchronized, or transiently synchronized. Even if you are considering only a single Fourier coefficient, because of temporal nonstationarities, it doesn't make sense to compute one Fourier coefficient over an entire recording of, say, 10 minutes. Instead, you would compute the FFT over 2-second intervals, and compare the phase differences between two channels at that Fourier coefficient across the distribution of 2-second intervals. In this case, synchronization is definitely not guaranteed, because the signal is changing over time. 

I hope that explanation makes sense. The neuroscience explanation is more theoretical, but is grounded in >150 years of thinking and empirical data. The idea is that oscillations are used to group assemblies of neurons at multiple spatial scales, from local (dozens or hundreds of neurons within, say, 1 mm) to long-range (many mm's to cm's). This is attributable to oscillations reflecting mass-subthreshold potentials, and thus neurons that are "tuned" into an oscillation will all be relatively hyperpolarized or relatively depolarized at the same time, which means that the probability of action potentials is modulated by a circular clock. At a larger spatial scale, neural networks that are phase-synchronized should be able to transfer information better. You can find more detailed explanations of these ideas in review papers, e.g,. by Pascal Fries, Ole Jensen, Gyorgy Buzsaki, and many others.

For this reason, phase synchronization is generally assumed to be a measure of narrowband functional connectivity. It's also important to keep in mind that the signal-processing and statistics aspects of coherence (like with most other analyses applied in neuroscience) are fairly well mapped out, while the exact neurophysiological interpretation remains open, debated, and likely to change over time with new data, experiments, and theories.

Mike

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[Joel Frohlich]

Hi Mike,

Thanks so much for your thorough reply, very helpful. I have two follow up questions, if you don't mind:

1) I believe my original confusion is resolved by the fact that the phase relation can indeed change across FFT windows. However, this seems to assume that signals are nonstationary on time scales longer than a single window (otherwise the phase relation would not change from one window to the next). An often cited limitation of coherence is that it assumes stationarity. Why is this the case when coherence appears to exploit the nonstationarity across windows?

2) Is there any sense in converting the magnitude squared coherence to a t-statistic the way one would with a Pearson correlation coefficient to then obtain a p-value? I'm assuming this is probably not meaningful because with enough FFT windows, any two EEG signals from the same subject will show significant coherence when statistical significance is measured this way.

Best,

Joel

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[Mike X Cohen]

Hi Joel. Indeed, you need to assume stationarity for the duration of the time window for the short-time FFT or the width of the taper for a wavelet analysis. 

As for t-values, oftentimes the best way to do inferential statistics is via permutation testing. I talk about both of these issues in the book.

Mike