A Generalized Framework of Antisymmetric Polyspectral Indices for Identifying High-Order Neural Interactions

Abstract: Cross-frequency interactions are fundamental brain mechanisms for integrating information across temporal scales. However, accurate identification of these couplings is hindered by complex multi-frequency nonlinearities and by spurious, zero-lag artifacts caused by volume conduction. To our knowledge, conventional metrics lack a robust framework to characterize genuine interactions among multiple time series where a frequency of interest $f_N$ arises from the combination of $N-1$ components such that $f_N = \sum_{i=1}^{N-1} f_i$. We introduce a general family of antisymmetric cross-polyspectral indices designed to quantify these harmonic dependencies while being intrinsically robust to instantaneous mixing. We derive the theoretical properties of these quantities and validate them through simulations of cubic nonlinearities. As a proof of concept, we apply the indices to empirical EEG recordings; the results reveal significant higher-order dependencies that elude standard analytical approaches. We further discuss how these indices can inform novel, personalized multi-site transcranial magnetic stimulation (mTMS) protocols by enabling the selective monitoring and modulation of specific multi-frequency network interactions.

• The paper introduces a generalized antisymmetric cross-polyspectral framework that detects N-way harmonic interactions while exactly cancelling contributions from independent linearly mixed sources.
• The framework extends imaginary coherency and antisymmetric cross-bicoherence to arbitrary order, remains bounded by 1, and reaches its theoretical maximum under ideal cubic coupling.
• Synthetic tests and single-subject EEG show that the method resists mixing artifacts and captures cross-frequency connectivity largely independent of ImCoh, though larger studies and real-time validation remain necessary.

Motivation and problem statement

Cross-frequency coupling (CFC) is a candidate mechanism by which the brain integrates information across temporal scales, yet its identification in non-invasive electrophysiology remains compromised by two obstacles: the prevalence of same-frequency connectivity metrics (PLV, ImCoh, wPLI) that treat frequency bands in isolation, and volume conduction, which introduces spurious zero-lag couplings through linear instantaneous mixing. Existing higher-order tools address only part of this problem: the bispectrum detects quadratic (1:2) interactions, and the Antisymmetric Cross-Bicoherence (ACB) and its multivariate extension (MACB) render such estimates robust to mixing artifacts [Chella et al., 2014; Basti et al., 2024]. However, no framework existed, to the authors' knowledge, for genuine interactions among $N$ time series satisfying the harmonic constraint $f_N = \sum_{i=1}^{N-1} f_i$ while remaining intrinsically robust to linear mixing. This paper fills that gap with a general family of antisymmetric cross-polyspectral indices.

The motivation is not purely analytical. Multi-locus transcranial magnetic stimulation (mTMS) can drive several partially overlapping cortical loci at distinct rates $(f_1,\dots,f_{N-1})$, potentially inducing composite spectral drives at $\sum_i f_i$ in overlap regions. Monitoring restricted to same-frequency connectivity would miss exactly these emergent cross-frequency phenomena, so a mixing-robust estimator of high-order harmonic dependencies has direct relevance for closed-loop, personalized neuromodulation protocols.

Theoretical formulation

The framework assumes observed signals are linear mixtures of independent, zero-mean, strictly stationary sources — a model covering both sensor-level volume conduction and residual leakage after source reconstruction. For the fourth-order case, the cross-trispectrum $T_{xxxy} = \langle X(f)^3 Y(3f)^* \rangle$ retains same-source terms of the form $$\sum_i a_i^3 b_i \langle S_i(f)^3 S_i(3f)^* \rangle$$ under the null model of independent linearly mixed sources. The key construction is the antisymmetrization $T_{[x|xx|y]} = T_{xxxy} - T_{yxxx}$, obtained by swapping first and last indices. Under source independence, both terms contain identical symmetric contributions, which cancel exactly; what survives is attributable only to genuine cubic coupling between $x$ and $y$. Normalization via fourth-order amplitude norms $Q_x(f) = \langle|X(f)|^4\rangle^{1/4}$, following Hölder-based univariate normalization [Shahbazi et al., 2014], yields the Antisymmetric Cross-Tricoherence (ACT), bounded in magnitude by 1 through the triangle inequality.

A toy example establishes sharpness: when $f_N = \sum_{i=1}^{N-1} f_i$0 is exactly the cube of a cosine carried by $f_N = \sum_{i=1}^{N-1} f_i$1, ACT attains its theoretical maximum $f_N = \sum_{i=1}^{N-1} f_i$2, confirming that antisymmetrization does not attenuate true nonlinear signal.

Two generalizations complete the theory. First, the construction extends to arbitrary order $f_N = \sum_{i=1}^{N-1} f_i$3: the $f_N = \sum_{i=1}^{N-1} f_i$4-th order Antisymmetric Cross-Polycoherency (ACP) cancels symmetric mixing contributions identically and satisfies $f_N = \sum_{i=1}^{N-1} f_i$5. Second, the framework subsumes established measures as special cases: $f_N = \sum_{i=1}^{N-1} f_i$6 recovers the imaginary part of coherency [Nolte et al., 2004], and $f_N = \sum_{i=1}^{N-1} f_i$7 recovers the antisymmetric cross-bicoherence [Chella et al., 2016]. This nesting positions the new indices as the natural continuation of an existing lineage rather than an ad hoc construction. One caveat is acknowledged explicitly: degenerate frequency configurations (e.g., $f_N = \sum_{i=1}^{N-1} f_i$8) produce factorable terms that undermine robustness, but these are excluded by standard practice of analyzing positive frequencies only.

Synthetic validation

Simulations used a generative model interpolating, via a scalar parameter $f_N = \sum_{i=1}^{N-1} f_i$9, between genuine cubic coupling (a band-limited oscillator whose cube generates power at $(f_1,\dots,f_{N-1})$0 and $(f_1,\dots,f_{N-1})$1, with a lag $(f_1,\dots,f_{N-1})$2 samples at 256 Hz sampling) and artifactual instantaneous mixtures of three independent cubed noise sources. Results over 1000 runs, summarized as pointwise medians with interquartile ranges, confirm three regimes:

• Genuine coupling ($(f_1,\dots,f_{N-1})$3): ACT and CT1 ($(f_1,\dots,f_{N-1})$4) both detect the structured phase relationships imposed by cubic nonlinearity.
• Dominant mixing ($(f_1,\dots,f_{N-1})$5): both non-antisymmetrized variants (CT1, CT2) are systematically inflated by zero-lag correlations, whereas ACT remains near baseline — demonstrating immunity to spurious inflation from volume conduction.
• Intermediate mixing: CT1 exhibits a U-shaped dependence on $(f_1,\dots,f_{N-1})$6 reflecting competing influences of residual cubic structure and shared noise, while CT2 increases monotonically driven entirely by noise. Neither confound affects ACT.

The practical implication is that statistical significance testing alone cannot rescue non-antisymmetrized estimators: they can be simultaneously significant and artifactual.

Empirical EEG results

Application to roughly six minutes of resting-state EEG (61 channels, single subject) reproduces the simulation findings at sensor level. At 12 ↔ 36 Hz with an occipital seed, CT1 and CT2 produce spatially confined high-value patches around the seed and immediate neighbors — a topology consistent with volume conduction — while ACT shows no such short-range "blooms," instead revealing modest distributed frontal increases. Notably, some local CT peaks survive surrogate-based significance testing despite being artifactual, reinforcing that estimator robustness, not thresholding, is the decisive property.

A direct comparison against ImCoh quantifies the complementarity of the approach: Pearson correlations between ACT and ImCoh connectomes were $(f_1,\dots,f_{N-1})$7 (12 Hz) and $(f_1,\dots,f_{N-1})$8 (36 Hz). Although statistically significant given 3721 channel pairs, these coefficients indicate that cross-tricoherence explains less than 0.4% of ImCoh variance — the two metric families capture largely independent features of neural connectivity. Most strikingly, an ACT seed map with a parietal seed reveals a statistically robust long-range inter-hemispheric pattern involving contralateral sensorimotor cortex that no corresponding ImCoh map reproduces. This constitutes evidence that genuine higher-order cross-frequency dependencies exist in resting-state human EEG and are observable directly at sensor level without source reconstruction.

Limitations and open questions

The authors are explicit about the scope of their claims. The empirical demonstration rests on a single subject and approximately six minutes of data; whether the observed coupling patterns reflect general principles of cortical organization or highly individual characteristics remains undetermined, and validation across larger cohorts is identified as a primary objective. The frontal ACT pattern in the occipital-seed analysis was modest in magnitude and largely failed surrogate-based significance testing, so claims of long-range cubic interactions rest primarily on the parietal-seed result. The mTMS application remains prospective: no stimulation experiment was performed, and the utility of ACT as a real-time marker for adaptive multi-site protocols is untested. Achieving reliable estimates from short data segments — required for tracking rapid coupling dynamics in near real-time — demands a rigorous characterization of the trade-off among data length, SNR, and spectral estimation technique, which the paper identifies as necessary but does not provide. Finally, extension to multivariate, multidimensional scenarios (e.g., $(f_1,\dots,f_{N-1})$9 parcels each carrying vector-valued signals) is described as feasible but not developed here.

Conclusion

This paper derives a family of antisymmetric cross-polyspectral indices that quantify $\sum_i f_i$0-way harmonic interactions satisfying $\sum_i f_i$1 while cancelling, by construction, all contributions from independent linearly mixed sources. The framework generalizes ImCoh ($\sum_i f_i$2) and ACB ($\sum_i f_i$3) to arbitrary order, is provably bounded on $\sum_i f_i$4, and achieves its theoretical maximum under ideal cubic coupling. Simulations demonstrate selective sensitivity to genuine nonlinearity across the full mixing continuum, and single-subject EEG analysis shows that the resulting higher-order dependencies are largely orthogonal to linear phase-synchrony measures (<0.4% shared variance) and include statistically robust long-range patterns invisible to ImCoh. The principal open questions concern inter-subject generalizability, short-window estimation reliability, and empirical validation within closed-loop mTMS paradigms.