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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.

Cross-reference: SSM Domain Evaluation compares largely the same SSM families (S4D, LinOSS, Mamba, Selective LinOSS) but along the input-domain axis (time/STFT/filterbank/token) rather than this frequency-vs-time/entrainment axis — the two design-space tables overlap in scope but answer different questions; don't conflate the two projects.

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.

Results

June 2026 — LinOSS vs. S4D vs. Selective LRU in SEMamba

This project is about testing LinOSS (Linear Oscillatory State Spaces) for speech enhancement. LinOSS is a State Space Model based on a second-order ODE, meaning its hidden state evolves with the dynamics of forced harmonic oscillators. Practically, this means LinOSS carries a bank of oscillators as a hidden state, where the input drives the oscillations. This is inspired by the cortical oscillations found in human brains. LinOSS is particularly well suited for long-sequence predictions that are periodic in nature. For discretization, LinOSS offers two options: A decaying implicit discretization (IM) and a symplectic implicit-explicit discretization (IMEX). Dampened LinOSS (d-LinOSS) introduces dampening into the IMEX-discretized state dynamics, so the model can learn symplectic and decaying oscillations, combining the IMEX and IM discretization. In the following work, LinOSS will always refer to d-LinOSS.

Speech is similarly composed of glottal harmonic oscillations, filtered by the shape of the vocal tract. Furthermore, we observe that speech modulates brain waves to concentrate on the theta band, which locks to the slow moving magnitude envelope of the time-domain waveform of speech, corresponding to syllables in speech. Thus, the LinOSS oscillator bank might lock on to these oscillatory dynamics in speech, capturing the underlying system better than a first-order model.

Experiments

The first experimental setup integrates LinOSS in SEMamba, replacing the selective SSM backbone with an oscillatory SSM. SEMamba acts on spectro-temporal STFT features, which contain the periodic structure of voice harmonics along the frequency axis and a quasi-periodic structure over time.

Time-Frequency Dynamics

SEMamba uses a DenseNet to encode the STFT magnitude and phase into a 3-dimensional latent speech representation, with the shape (T, F // 2, C). Each time-frequency bin is thus embedded as a vector of size C. The SSM is then run bi-directionally across the time-axis (observing frames of shape (F//2, C)) and across the frequency axis (observing frames of shape (T, C)).

Replacing Mamba with LinOSS has slightly different effects for the time and frequency axis: - On the time-axis, LinOSS almost matches Mambas performance. PESQ slightly reduces, but MOS scores stay similar. - On the frequency-axis, LinOSS has more detrimental effects, worsening PESQ even more, and also reducing MOS scores and Phase losses. - Adding LinOSS on both axes accumulates the negative impact.

Comparison to other SSMs

In a second experiment, other SSMs are implemented and matched for parameters with Mamba/LinOSS.

First-Order & Non-Selective: S4D (SISO)

The SSM S4D is very similar to LinOSS except that its hidden state evolves with first-order dynamics. In comparison to S4D, LinOSS slightly outperforms S4D in terms of perceptual scores and SI-SDR. LinOSS significantly outperforms S4D in terms of matching the phase of the clean waveform. This suggests an advantage of oscillatory dynamics for modeling phase information in speech.

First-order Complex & Selective: Selective LRU (similar to GateLoop & Mamba 3)

Important Note: These results were gathered from MIMO selective LRU

The SSM selective Linear Recurrent Unit is a first-order complex selective SSM. It uses scaled rotations for the transition matrix, where scale and rotation are each independently selective to the input (\(h_{k}=\nu_{k}e^{i\theta_{k}}h_{k-1}+b_{k}u_{k}\)). The complex nature allows the model to learn rotational dynamics, and thus better conforms to oscillatory signals as well. However, in comparison to LinOSS, it lacks the inductive bias of physical harmonic oscillators.

This model strongly outperforms LinOSS and performs within ~0.1 PESQ below Mamba and maintains phase slightly better. Along other metrics, both models perform about equal. Again, this strengthens the hypothesis that oscillatory dynamics help modeling phase information in speech.

Mamba 3 actually incorporates rotational dynamics as well, and GateLoop basically the same SSM as this model, so the theoretical novelty is rather limited. Executing this model in parallel SISO style, like Mamba, the performance is slightly worse, which matches the findings in Mamba 3.

Summary

This experiment suggests that complex state dynamics in SSMs help in phase modeling, which is a logical connection. While the complex selective SSMs can model oscillations as well, they lack the inductive bias of an actual bank of harmonic oscillators, which is the bio-physical foundation of LinOSS. Unfortunately, an input-selective LinOSS-style oscillatory bank is difficult to construct due to system instability: The IMEX discretization in LinOSS has a closed-form solution for the dampened second-order LTI dynamics, with eigenvalues within the unit-circle. But once time-varying dynamics are introduced, a Lyapunov stability proof is required, which I didn't solve yet.

[!note]- Superseded The Lyapunov stability proof for selective (input-varying) LinOSS was later worked out in LinOSS Stability: IM-discretized selective LinOSS is unconditionally stable, and IMEX is stable above a damping floor \(G_n \ge \tfrac{\Delta t_n}{2}A\). S-LinOSS Proposal builds on this to propose a normal, scaled-rotation transition as a structurally stable alternative that keeps the full unit-disk spectrum.

Conclusion and Future Ideas

LinOSS has not proven superior to other SSMs for speech processing. It shows improvements in phase-modeling, but this is likely reproduced by a complex state-space as well. The inductive bias of harmonic oscillators in particular have not shown any advantage.

Potential explanations - For frequency modeling, the SSM must be able to adapt to a time-varying pitch, which LinOSS struggles most with. Also, formants and plosives contain little oscillatory structure for LinOSS to adapt to. - For time modeling, LinOSS is better suited due to its long-range modeling capabilities. However, it's still outperformed by Mamba, and a selective SSM with similar long-range modeling capabilities (e.g. Mamba 3 with RoPE) likely outperforms both. - For raw waveform modeling, LinOSS has trouble adapting its oscillator banks to the complex waveform dynamics. This might be caused by lack of hyperparameter tuning, or might be a general shortcoming of the model.

Future ideas: - Incorporating LinOSS for specialized tasks: - Use in combination with Mamba blocks in SEMamba to improve tracking of periodic structure (along frequency-axis) and quasi-periodic structures (along time-axis) - Explore parallel LinOSS-Mamba blocks, where input is passed through both SSMs and their output is combined.