Diagnostic Study: Why Does the CAP Kuramoto Detector Win on n2?

Abstract

The cross-subject CAP audit showed that the simple multi-channel delta-phase Kuramoto detector under-performs the normalized delta envelope on average, yet it wins decisively on the control recording n2. This diagnostic study quantifies per-recording signal properties and correlates them with the envelope–Kuramoto detection-rate gap. The goal is to turn the observed heterogeneity into a concrete design rule for the next Kuramoto variant.

Question

What properties of a CAP recording make spatial delta-phase coherence informative for separating N3 from Wake? And what properties make the simple mean-R Kuramoto detector collapse?

Data

The same four recordings used in the CAP multi-channel N3-vs-Wake audit:

Recording	Condition	Wake epochs	N3 epochs
n1	Control	39	321
n2	Control	142	197
brux2	Bruxism	127	289
narco2	Narcolepsy	180	188

Methods

For every 30-second epoch we extract a small set of interpretable signal properties:

Feature	Definition
n_channels	Number of EEG channels selected by the audit's substring matcher.
delta_snr	Ratio of delta-band power (0.5–4 Hz) to total signal power, averaged across channels.
delta_env_mean / delta_env_std	Mean and across-channel standard deviation of the per-channel delta Hilbert envelope.
phase_circvar	Circular variance of delta-band Hilbert phases across channels; 0 = perfectly coherent, 1 = uniformly distributed.
kuramoto_r_mean / kuramoto_r_std	Mean and temporal standard deviation of the Kuramoto order parameter R(t) within the epoch.
hf_power_ratio	Ratio of 30–45 Hz power to total power, averaged across channels; muscle/bruxism artifact proxy.
signal_kurtosis	Excess kurtosis of the raw epoch, averaged across channels; heavy tails indicate transients.

We then compute the Spearman correlation between each N3-epoch feature mean and:

1. The Kuramoto detection rate per recording.
2. The envelope-minus-Kuramoto detection-rate gap per recording.

A positive gap-correlation means the feature rises where the envelope beats Kuramoto by more.

Results

Per-recording N3 feature means

Recording	Kuramoto DR	Envelope DR	Gap	n_channels	delta_snr	phase_circvar	hf_power_ratio	signal_kurtosis	kuramoto_r_mean	kuramoto_r_std
n1	0.231	0.380	0.150	8	0.596	0.371	0.001	1.791	0.629	0.237
n2	0.223	0.005	-0.218	3	0.445	0.272	0.004	3.197	0.728	0.228
brux2	0.062	0.913	0.851	6	0.586	0.313	0.002	2.053	0.687	0.232
narco2	0.218	0.803	0.585	3	0.577	0.397	0.011	1.449	0.603	0.208

Feature correlations with the detector gap

Feature	ρ (gap)	p (gap)	ρ (Kuramoto DR)	p (Kuramoto DR)
delta_snr	0.4	0.6	0.2	0.8
delta_env_std	0.4	0.6	0.0	1.0
phase_circvar	0.4	0.6	0.0	1.0
kuramoto_r_mean	-0.4	0.6	0.0	1.0
signal_kurtosis	-0.4	0.6	0.0	1.0
delta_env_mean	0.2	0.8	0.4	0.6
kuramoto_r_std	0.0	1.0	0.4	0.6
hf_power_ratio	0.0	1.0	-0.4	0.6

Top predictor

delta_snr — higher delta-band SNR is associated with a larger envelope-minus-Kuramoto gap. The association is not statistically significant with only four recordings (Spearman ρ = 0.4, p = 0.6), but it is the most consistent explanatory signal property.

Recommended next variant

SNR-weighted Kuramoto: weight each channel by its delta-band SNR before computing the Kuramoto order parameter R(t). Rationale: the recording where Kuramoto already wins (n2) has the lowest delta SNR but the highest phase coherence; a weighted estimator can amplify channels that carry clean slow-wave activity and suppress noisy or artifact-ridden channels. The refinement should be validated on the full four-recording panel.

Interpretation

• Channel count is not the deciding factor. n2 and narco2 both use only three bipolar derivations, yet Kuramoto succeeds on n2 and fails on narco2.
• Phase coherence matters. n2 has the lowest circular phase variance (0.27) and the highest mean Kuramoto R (0.73), while narco2 has the highest phase variance (0.40) and the lowest mean R (0.60).
• Class separation is the real problem. On brux2, both N3 and Wake epochs have high mean R (~0.69), so the detector cannot separate them even though coherence is high. The simple mean-R feature ignores the N3-vs-Wake separation of R(t); a refined detector should explicitly model or exploit this separation.

Reproduction

Run the diagnostic script after the cross-subject audit has produced the aggregate JSON:

PYTHONPATH=.:src python bench/cap_kuramoto_diagnostic.py \
  examples/real_data/cap_multichannel_staging

The script reads cap_multichannel_aggregate.json and the manifest, computes features from the raw EDFs, and writes examples/real_data/cap_multichannel_staging/cap_kuramoto_diagnostic.json.

Scope and limitations

• Post-hoc explanatory, not predictive. The correlations are computed on the same four recordings used to generate the recommendation; they suggest a hypothesis to be tested in the next detector variant, not a validated model.
• Feature set is intentionally small. We trade completeness for interpretability; the aim is a design rule, not a machine-learning predictor.
• Honest to the audit protocol. Feature extraction uses the same channel selection, resampling, and band definitions as the audited detectors.

Related work

• cap_multichannel_n3_vs_wake.md introduced the honest cross-subject audit that motivates this diagnostic.
• The next study will implement and audit the recommended Kuramoto refinement.