Phase, ITC and cross-frequency coupling

Why this matters

Every complex time-frequency coefficient has an angle, and throwing it away is what a power map does. The angle answers questions power cannot: did the trials line up, and does the phase of one rhythm predict the amplitude of another. Both questions have a well-known statistic and both statistics have a well-known failure mode. Intertrial phase coherence is biased upward at small trial counts — with 40 trials, pure noise reads about 0.14, not 0 — and phase-amplitude coupling is produced, reliably and strongly, by a single non-sinusoidal rhythm with no second oscillator anywhere.

Concepts

Intertrial phase coherence

Take the complex coefficient at one frequency and one latency on every trial, discard its magnitude, keep the unit vector, and average:

ITC = | (1/N) · Σ_k exp( i·θ_k ) |

It is the mean resultant length of the trial phases: 1 when every trial agrees exactly, 0 when the phases are spread evenly around the circle. It is the same quantity that scaled the ERP in L4.1 — which is the useful way to hold it in mind. ITC is the factor by which trial averaging preserves activity at that frequency and latency. Writing trial k’s coefficient as a_k · e^{iθ_k}, the trial average has magnitude |(1/N)·Σ a_k e^{iθ_k}|, which reduces to a · ITC when the single-trial amplitudes are all equal — so evoked power is total power times ITC² in that case, and the two quantities are not independent findings.

Three properties decide how it can be used.

It is bounded above by 1 and it does not get there. Anything that adds noise to a trial perturbs its measured angle, so a perfectly phase-locked oscillation sitting in a realistic background does not read 1. On the widget’s synthetic fixture, a burst with zero phase jitter reads 0.971 ± 0.009 through the full pipeline — synthesise trials, transform, take the angle — and reads exactly 1 when the background is switched off or when the same phases are fed through an ideal phasor. The shortfall is the background, not the estimator.

Its floor is not 0. With N phases drawn uniformly at random the resultant is a two-dimensional random walk of N unit steps, whose length is Rayleigh-distributed. Its expectation is

E[ITC] ≈ √π / (2·√N)      — 0.1401 at N = 40

with an exact root-mean-square of 1/√N (0.1581 at N = 40) and a standard deviation of √((1 − π/4)/N) (0.0732 at N = 40). The floor falls only as 1/√N, so it halves when N quadruples. Any ITC has to be read against the floor for the N you had, and comparing ITC between conditions with different trial counts compares two different floors — the same problem, in the same direction, as comparing peak amplitudes at different trial counts (L3.5, pf-peak-amplitude-noise-bias).

There is a test, and there is a correction. The Rayleigh test gives a critical value of √(−ln α / N), which at N = 40 and α = .05 is 0.2737 — so at 40 trials an ITC of 0.25 is not significant, although it is nearly twice the “floor”. A bias-corrected estimate follows from the exact identity E[ITC²] = R² + (1 − R²)/N, giving R̂ = √( max(0, (N·ITC² − 1)/(N − 1)) ). The estimator of R² is unbiased; its square root is not, and it has to be clamped at zero, so the correction is a repair rather than a cure. Reporting N beside every ITC is not optional.

Phase resetting versus an additive response

Two accounts of how an evoked response arises:

• Additive. A stimulus-locked response is added to ongoing activity, which carries on unchanged.
• Phase reset. The stimulus resets the phase of an ongoing oscillation, which then aligns across trials without any new energy being added.

Both raise ITC. The additive model raises it because adding a fixed vector to a set of randomly oriented vectors pulls their angles toward it — no trial’s own oscillation has been touched. The widget makes this arithmetic visible: with phase jitter held at a full 360°, so the ongoing rhythm is not phase-locked at all, adding a fixed component to every trial raises ITC from 0.137 at 0 µV through 0.184 at 5 µV and 0.306 at 10 µV to 0.626 at 20 µV. No phase resetting happens anywhere in that sweep.

So a high ITC does not demonstrate a phase reset, and this is the single most common over-reading of the statistic. The usual argument for telling the accounts apart is that an additive response raises the mean single-trial power while a pure phase reset does not. The arithmetic of that argument is exact and the widget computes it. TODO(confirm): whether the power criterion is usable as evidence in real recordings is argued about in the literature rather than settled — trial-to-trial amplitude variation, amplitude modulation accompanying a reset, and the difficulty of estimating pre-stimulus power without the response leaking into it all complicate it — and the widget marks the inference as contested while marking the numbers as exact. Treat the two as competing accounts to be distinguished by design, not by a single statistic.

Cross-frequency coupling

Phase-amplitude coupling (PAC) asks whether the amplitude of a fast rhythm varies systematically with the phase of a slow one. The standard recipe: band-pass for the phase band and take the instantaneous phase; band-pass for the amplitude band and take the envelope (L4.3); then summarise their relationship — by binning the envelope over phase and measuring how far the result departs from uniform (the Tort modulation index), by the length of the mean vector of envelope-weighted phases, or by the correlation of the envelope with a phase-derived regressor.

Four confounds, and the first one is not a corner case:

1. Waveform shape. A rhythm that is not a sine wave has harmonics by definition — a sharp-trough alpha rhythm must have energy near 2f, 3f and so on, all phase-locked to the fundamental because they are the same event. Run PAC between the fundamental’s phase and the harmonic’s amplitude and you will find strong, significant coupling produced by one oscillator. This is pf-harmonics-as-pac, and the figure in that entry shows it on a real recording: a fitted 9.9 Hz peak with a second at 18.0 Hz, a modulation index 19 standard deviations above 200 phase-randomised surrogates, and every one of those surrogates beaten — with no second oscillator involved. (Aru et al., 2015) is the standard treatment of this and the other confounds.
2. Band choice. The amplitude band has to be wide enough to contain the sidebands the modulation creates — roughly the phase frequency on either side of the amplitude centre — or the coupling is filtered away before it is measured. A band too narrow finds nothing; a band chosen after looking finds whatever it looked for.
3. Non-stationarity and edges. Sharp transients, epoch boundaries and artefacts produce broadband energy at a specific phase of whatever slow rhythm happens to be running. A single blink can contribute coupling.
4. The statistic’s own bias. Modulation-index-type measures are biased by the number of cycles in the record, so a slow phase band — fewer cycles in the same data — gets a larger index for free. Comparing raw coupling values across phase frequencies, or across recordings of different lengths, compares biases.

The control that addresses most of this is a surrogate test that preserves what you are not testing and destroys what you are: shuffle trials, or time-shift the amplitude series relative to the phase series by more than the slow period, or phase-randomise while keeping the power spectrum, and compare the real statistic against the distribution of surrogate ones. Note what a phase-randomising surrogate does and does not control: it preserves the power spectrum — the harmonic included — while destroying waveform shape, so a real value far above such surrogates says the coupling is a property of the waveform, which is exactly the harmonic confound rather than a defence against it. The defences against confound 1 are to look at the waveform, to check whether the amplitude band sits at an integer multiple of the phase band, and to test coupling between bands that are not harmonically related.

The data behind this lesson

• The widget is synthetic, and says so: parameters from /data/widgets/w-phase-clock/fixtures.json, written by data/scripts/make_phase_fixtures.py. Same generative model as L4.1 — a 20 µV, 10 Hz, 7-cycle burst centred at 0.3 s on a 1/f^1.343, 19.8 µV background at 250 Hz, seed 20260918 — plus a fixed evoked component added identically to every trial for the additive account. ITC is read at 10 Hz with 7 cycles at the burst centre. Phases, transform and ITC all run in the browser.
• The jitter slider’s convention is a control. The fixture defines phase_jitter_deg as the full width of a uniform draw, so 360° is an exactly uniform draw; the widget defaults to that and offers the von Mises circular-standard-deviation reading as a named alternative, printing both.
• The notebook works on real data: ITC on ds-erpcore P3 trials, and phase-amplitude coupling with surrogate testing on ds-eegbci eyes-closed baseline. TODO(confirm): ds-erpcore’s licence is contested at source and the site ships derivatives under the strictest reading (CC BY-SA 4.0); the author reconciles it (§13 item 22).
• A positive control this lesson expected to use does not work, and that is itself worth reading. §6 names the 40 Hz auditory steady-state response in ds-aszed as the thing a correct ITC pipeline must find. nb-4-5 looked for it in 30 recordings and did not find it: 40 Hz power during the stimulus blocks is 0.586× its own 30–50 Hz neighbourhood rather than a peak, none of the 30 reaches twice its neighbourhood, and the decisive paired test — stimulus against silence within each recording, each normalised by its own neighbourhood — gives +0.259 dB, t(29) = 0.33, p = .745, with 15 of 30 above zero. Pooled ITC at 40 Hz is 0.036 against a null expectation of 0.077. The archive’s own protocol names the phase “Fixed Auditory Stimulus” and never states a modulation rate; the 40 Hz in this site’s catalog is an inference, not a documented fact. What is not claimed is that the dataset contains no steady-state response: this is one channel, one reference, 30 recordings, and a mains line that is enormous in this montage. TODO(confirm): the author should either establish the rate from the authors or drop the paradigm tag. The lesson’s point stands either way, and is sharper for it. A positive control that fails tells you something about your control, not only about your pipeline — which is why the notebook ships two controls that do work: the P3 itself (peak ITC 0.607, 0.733 and 0.732 at Pz for three subjects, against Rayleigh critical values near 0.29) and an exact synthetic phasor.

Explore

The widget opens on 40 trials with uniformly random phase and no evoked component. Read the ITC: it is not 0, and it is not the expectation either — one draw scatters around the expectation by about 0.07, which is the whole reason the exercise asks what you expect rather than what you see. Press Redraw phases half a dozen times and watch the value move; then turn on the null distribution and see where your draws sit in it. Drag the trial count and watch the floor fall as 1/√N against the closed form. Press 40 locked and confirm that the answer is not 1, then switch the background off and watch it become exactly 1. Finally go back to uniform phase and raise the added amplitude: ITC climbs with no phase resetting anywhere, and the mean single-trial power climbs with it.

Practice

The notebook computes ITC on real trials, checks the pipeline against two positive controls that do work — the P3 itself and an exact synthetic phasor — reports in full the steady-state control that does not, and runs phase-amplitude coupling with a surrogate test, with the bands, the trial counts and the surrogate scheme stated in one cell.

Exercises

Both questions ask for the value you should expect, not the value the widget happens to be showing. The opening draw is a single realisation: at 40 trials the null has a standard deviation of about 0.073, so a first look near 0.06 or near 0.22 is ordinary scatter rather than a defect.