Polarity is a geometric property of the neural field on the SPD manifold. It is the first confirmed member of an invariant hierarchy. Higher-order invariants — cross-frequency coupling and manifold trajectory geometry — are predicted but the circuit architectures for measuring them are still under active investigation.
For an M-channel EEG window of N samples, the spatial covariance matrix Σ = (1/N)XXT is a point on the SPD(M) manifold. Its eigendecomposition Σ = QΛQT describes the principal axes of neural activity. The A-Gate circuit measures this point and produces a polarity observable.
The key empirical finding: P1 is invariant within each patient across seizures and across window sizes from 1s to 30s. The same patient always inverts (or doesn’t). Drag the slider to see polarity stability across window durations.
Bandpass-filter the EEG into frequency bands f. The cross-frequency PLV between bands f1 and f2 measures how the phase of one band relates to the phase of another — a form of phase-amplitude or phase-phase coupling.
Each (f1, f2) pair produces an M×M PLV matrix — still SPD after regularization. Each has its own polarity: P2(Σf1,f2) = sgn(⟨Z⟩f1,f2). Click a frequency pair to see the predicted PLV matrix and polarity.
θ × α cross-frequency PLV matrix. Does polarity sign persist, differ, or vanish across frequency couplings?
The sequence of SPD matrices across consecutive windows {Σ(t1), Σ(t2), …, Σ(tk)} traces a curve on SPD(M). The geometric properties of this trajectory — velocity, curvature, acceleration — are higher-order invariants.
Interictal trajectory: slow, smooth movement on the SPD manifold. The neural field is near equilibrium.
QNFM identifies the class of invariants. Polarity is confirmed as the first member. Higher-order members are predictions with specific falsification criteria.
| Order | Observable | Status | Falsification Criterion |
|---|---|---|---|
| 1st | P1 = sgn(⟨Z⟩) from single-band PLV | CONFIRMED | Would be falsified by within-patient polarity inconsistency |
| 2nd | P2 = sgn(⟨Z⟩f1,f2) from cross-frequency PLV | PREDICTED | Test: run A-Gate on cross-frequency PLV matrices for all 22 patients |
| 3rd | κ(t), v(t) from SPD trajectory | PREDICTED | Test: compare trajectory curvature distributions ictal vs interictal |
External validation: Barachant (Melbourne 2016) and Hills (2014) independently found that cross-frequency coherence matrices and auto-correlation matrices with log-spaced delays carry seizure-discriminative information — empirical evidence that these higher-order geometric objects are information-bearing.