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Entrained Oscillatory State Space Models for Phase Aware Speech Enhancement

Abstract

Cortical oscillations track speech not as fixed resonators but through neural entrainment, their phase and frequency are continuously modulated by the acoustic input. We ask whether this distinction matters computationally for monaural speech enhancement, where reconstructing the phase of the complex spectrogram is the dominant bottleneck. We study a family of diagonal state-space models (SSMs) that vary one biologically-motivated axis: from passive oscillatory resonance (LinOSS, a linear time-invariant harmonic-oscillator bank) to entrainment (a selective oscillatory SSM whose per-step eigenvalue magnitude and phase are input-dependent). Because naïve cross-architecture comparisons confound the transition geometry with state layout, parameterization, and capacity, we isolate the mechanism within a single architecture — a Mamba-S6 backbone whose real, decay-only transition is replaced by a complex, scaled-rotation (oscillatory) one — and toggle the rotation between off, static, and input-dependent. In a TF-domain enhancement model on EARS-WHAM and VoiceBank-DEMAND, we test whether entrained oscillation specifically improves phase-sensitive metrics (phase loss, LSD, SI-SDR, DistillMOS), and whether the learned modulation frequencies align with speech-relevant cortical bands. The result characterizes what an oscillatory inductive bias contributes to speech enhancement and when input-driven modulation — not mere resonance — is what delivers it.

Introduction

• Phase reconstruction is the limiting factor in TF-domain speech enhancement; magnitude denoising saturates.
• Neuroscience motivation: cortical oscillations entrain to speech (delta/theta → phrasal/syllabic; gamma → phonemic). Entrainment = input-modulated oscillation, not passive resonance.
• Hypothesis: an oscillatory state, and specifically an entrained one, is a useful inductive bias for phase.
• Contribution: a controlled, confound-free isolation of the oscillation/entrainment axis, plus a neuro-aligned analysis.

Background

• Diagonal SSMs as sequence mixers; real-decay (Mamba/S6) vs complex-rotation (S4D, LRU) vs second-order oscillator (LinOSS).
• Speech enhancement framework (SEMamba-style TF blocks, magnitude+phase losses).
• Neural entrainment and phase coding in auditory cortex.

Methodology

Design Space and Confounds

Model	Selective	State Layout	Stability	Oscillation	Order
S4D	-	SISO	\(Re\{A\} < 0\)	static	1st
LinOSS	-	MIMO	IMEX constraint	static	2nd
LRU	-	SISO	\(\|\lambda\|<1\)	static	1st
Mamba	\(\Delta t\), B, C	SISO	\(A < 0\)	-	1st
Selective LRU	\(\Delta \nu\), \(\Delta \theta\), B, C	SISO	\(\|\lambda\|<1\)	selective	1st

• Factorization table: order, eigenvalue geometry, rotation (none/static/selective), selectivity & its location (B/C/Δ vs eigenvalues), SISO vs MIMO, effective state, normality.
• Why cross-model comparisons (e.g. Mamba-1 vs MambOSS) are uninterpretable — multiple axes move at once (real↔complex, SISO↔MIMO, ~256× state, selectivity location).
• Design principle: vary one axis inside one architecture.

Method: Entrained Oscillatory S6

• Base: Mamba-S6 (SISO per-channel, shared selective B/C, low-rank Δ).
• Modification: replace real decay λ=exp(ΔA) with complex scaled rotation \(λ_k = ν_k e^{iθ_k}\); structural stability via \(|λ|=ν<1\).
• The entrainment knob: ν (damping) and θ (frequency) as static-per-mode (resonance) vs input-dependent (entrainment).

Experimental Design

• The factorial inside the architecture: rotation {off, static, selective} × damping {static, selective}; θ≡0 recovers a real selective SSM in the identical surround.
• Reference anchors (not the factorial): Mamba-1 (real), LinOSS (LTI 2nd-order), S4D (static complex).
• Controls: iso-parameter / iso-compute, ≥3 seeds with variance, conv parity, fixed loss/schedule.
• Datasets: EARS-WHAM + VoiceBank-DEMAND. Metrics: phase loss, LSD, SI-SDR, PESQ/STOI, DistillMOS.

Analysis (mechanism + neuro-alignment)

• Phase-localized gains: where in the metric set and where in frequency the oscillatory bias helps.
• Pole/eigenvalue plots; normality.
• Band analysis: map learned time-axis θ (rad/frame, at the 160 Hz frame rate) to Hz; test clustering near phrasal/syllabic bands.
• Entrainment probe: does selective θ phase-lock to the clean-speech envelope / syllabic rate?
• Passive-vs-entrained ablation as the causal test.

Discussion

• What the oscillatory prior buys, and the honest boundary: resonance alone vs entrainment; oscillation vs selectivity for overall quality.
• Limits of the cortical analogy (no nonlinearity / cross-frequency coupling); analogy motivates, evidence carries.

Conclusion

• Entrained oscillation as a neuro-grounded, interpretable mechanism for phase-aware enhancement.