1. Introduction

The human body is a biological oscillator. At the molecular level, a transcription-translation feedback loop involving CLOCK, BMAL1, PER, and CRY proteins drives a near-24-hour rhythm in gene expression that cascades into every major physiological system [1]. Cardiovascular function, immune activity, metabolic rate, and cognitive performance all oscillate with robust circadian periodicity. Consumer wearables capture the downstream signatures of these rhythms: heart rate variability (HRV) peaks in the early morning, skin temperature reaches its nadir during the sleep trough, and activity-rest cycles are the most visible manifestation of the master pacemaker in the suprachiasmatic nucleus (SCN) [2].

Yet the dominant paradigm in wearable data analysis treats biosensor streams as generic time series and applies architectures — LSTMs, Transformers, Mamba — that are chronobiologically agnostic. This creates two compounding problems. First, the model must allocate representational capacity to re-learn circadian structure from data that could instead be encoded as an architectural prior. Second, and more critically for clinical applications, pathological deviations from circadian norms are conflated with normal chronobiological variation. An elevated resting heart rate at 3 AM may be pathological; the same elevation at 3 PM during moderate exercise is physiological. Without an explicit circadian reference, discriminating these signals requires vast labeled data — data that is rarely available for rare disease events.

We address both problems with BioWaveNet, a temporal foundation model built around a biologically-motivated positional encoding derived from coupled oscillator theory. Our contributions are:

1. Kuramoto Circadian Positional Encoding (K-CPE): A novel positional encoding layer that integrates the Kuramoto synchronization model to generate adaptive, phase-coherent temporal embeddings. We prove that standard sinusoidal PE is a degenerate case of K-CPE in the limit of zero inter-oscillator coupling.

2. Multi-Resolution Temporal Attention (MRTA): A hierarchical attention mechanism that explicitly computes within-ultradian (90-min), within-circadian (24h), and within-infradian (7-day) similarity, enabling the model to capture biological patterns operating across timescales.

3. Circadian Contrastive Pre-training Objective: A self-supervised objective that exploits the temporal symmetry of circadian biology: biosensor epochs separated by exactly 24 hours under stable conditions should be more similar in the latent space than epochs separated by 12 hours.

4. Large-Scale Pre-training and Benchmarking: We pre-train on 3.2 billion biosensor epochs and evaluate on four benchmarks, achieving state-of-the-art results across all tasks.

5. Pathological Fingerprinting in Circadian-Residual Space: We demonstrate that rhinitis, obstructive sleep apnea, and paroxysmal atrial fibrillation each impose distinct, linearly-separable signatures in the residual space after circadian subtraction, enabling zero-shot disease episode detection.

2. Background and Related Work

2.1 Coupled Oscillator Models in Chronobiology

The synchronization of biological oscillators is elegantly captured by the Kuramoto model [3]. For a population of $N$ coupled oscillators with natural frequencies ${\omega_i}$:

$$\frac{d\phi_i}{dt} = \omega_i + \frac{K}{N}\sum_{j=1}^{N}\sin(\phi_j - \phi_i) + \xi_i(t)$$

where $\phi_i \in [0, 2\pi)$ is the phase of oscillator $i$, $K$ is the global coupling strength, and $\xi_i(t) \sim \mathcal{N}(0, \sigma^2)$ is stochastic noise. When $K$ exceeds the critical coupling $K_c = 2\sigma/\sqrt{N}$, the system exhibits spontaneous synchronization — a macroscopic oscillation emerges from the collective dynamics.

In mammals, the SCN functions as the master pacemaker that synchronizes peripheral oscillators (liver, lung, heart, immune cells) through hormonal and neural signals [4]. The degree of synchronization can be quantified by the Kuramoto order parameter:

$$r(t) = \left|\frac{1}{N}\sum_{j=1}^{N}e^{i\phi_j(t)}\right| \in [0, 1]$$

High $r$ indicates strong synchronization; circadian disruption (jet lag, shift work, disease) reduces $r$. This provides a principled biological quantity for our positional encoding.

2.2 Temporal Foundation Models

Recent work on time-series foundation models includes TimesFM [5], MOIRAI [6], and Chronos [7]. These models use variants of the Transformer architecture with sinusoidal or learned positional encodings. A critical limitation is that they treat the time axis as an abstract index rather than a biological substrate with known structure. None incorporate domain knowledge about chronobiological rhythms as an inductive bias.

In the biomedical domain, BioSignalBERT [8] and PhysioFormer [9] apply BERT-style pre-training to physiological signals but use standard positional encodings and do not model circadian structure. Our K-CPE provides a principled departure from this paradigm.

2.3 Wearable Biosensor Analysis

Consumer wearable platforms (WHOOP, Oura Ring, Apple Watch, Garmin) capture continuous HRV, photoplethysmography, accelerometry, and skin temperature. Multiple validation studies confirm that wearable-derived HRV correlates with gold-standard ECG measurements (ICC ≥ 0.85 for most devices) [10]. However, acute health perturbations — rhinitis, upper respiratory tract infections, sleep apnea, arrhythmia — confound these measurements in disease-specific ways that wearable analytics pipelines typically fail to account for.

3. The BioWaveNet Architecture

3.1 Kuramoto Circadian Positional Encoding (K-CPE)

Let $t \in \mathbb{R}^+$ denote continuous time in hours. We parameterize a learnable master circadian oscillator:

$$\Phi_\text{master}(t; \boldsymbol{\theta}) = \phi_0 + \omega_\text{circ} \cdot t + K \int_0^t r(s) \sin(\Phi_\text{master}(s) - \Psi_\text{env}(s)) , ds$$

where $\phi_0 \in [0, 2\pi)$ is a learned initial phase, $\omega_\text{circ} = 2\pi/24\text{h}$ is the circadian angular frequency, $K$ is the learned coupling strength, $r(s)$ is the instantaneous order parameter, and $\Psi_\text{env}(s)$ encodes Zeitgeber information (light exposure, meal timing, exercise) when available.

The K-CPE for embedding dimension $d$ is:

$$\text{K-CPE}(t, 2k) = \sin!\left(\Phi_\text{master}(t) \cdot \frac{\omega_k}{\omega_\text{circ}}\right)$$ $$\text{K-CPE}(t, 2k+1) = \cos!\left(\Phi_\text{master}(t) \cdot \frac{\omega_k}{\omega_\text{circ}}\right)$$

where $\omega_k = \omega_\text{circ} \cdot \lambda^{k/d_\text{model}}$ for a learned scale factor $\lambda$. This creates a multi-frequency encoding anchored to circadian phase rather than arbitrary sequence position.

Proposition 1 (Sinusoidal PE as Degenerate K-CPE). When the Kuramoto coupling strength $K \to 0$, the master oscillator reduces to a free-running oscillator with $\Phi_\text{master}(t) = \phi_0 + \omega_\text{circ} \cdot t$, and K-CPE reduces to standard sinusoidal positional encoding with frequency $\omega_k$.

Proof. As $K \to 0$, the integral term in $\Phi_\text{master}$ vanishes, giving $\Phi_\text{master}(t) = \phi_0 + \omega_\text{circ} t$. Substituting into K-CPE: $$\text{K-CPE}(t, 2k) = \sin!\left((\phi_0 + \omega_\text{circ} t) \cdot \frac{\omega_k}{\omega_\text{circ}}\right) = \sin(\phi_0 \omega_k/\omega_\text{circ} + \omega_k t)$$ Setting $\phi_0 = 0$ recovers the standard Vaswani et al. sinusoidal PE [11]. $\square$

This proposition establishes that BioWaveNet strictly generalizes existing temporal Transformers: any model trained with sinusoidal PE is a special case of BioWaveNet with $K = 0$.

3.2 Multi-Resolution Temporal Attention (MRTA)

Biological rhythms operate across multiple timescales: REM/NREM cycles (90 min), circadian rhythms (24h), and circaseptan rhythms (7 days). Standard self-attention with fixed context length cannot simultaneously capture all three scales without prohibitive computational cost.

We design MRTA with three attention heads operating over disjoint temporal resolution windows:

$$\text{MRTA}(Q, K, V) = \text{Concat}(\text{head}_U, \text{head}_C, \text{head}_I) W^O$$

where:

• $\text{head}_U$: Ultradian attention, local window $w = 90\text{ min}$, stride 5 min
• $\text{head}_C$: Circadian attention, local window $w = 24\text{h}$, stride 30 min
• $\text{head}_I$: Infradian attention, downsampled to daily summaries, window 7 days

The multi-resolution design reduces the effective context from $O(T^2)$ to $O(T \log T)$ while preserving all biologically relevant temporal dependencies.

Circadian Phase Gating. We additionally introduce a phase-aware gating mechanism:

$$A_{ij}^{\text{gated}} = A_{ij} \cdot \exp!\left(-\alpha \cdot d_{\text{circ}}(\phi_i, \phi_j)^2\right)$$

where $d_{\text{circ}}(\phi_i, \phi_j) = \min(|\phi_i - \phi_j|, 2\pi - |\phi_i - \phi_j|)$ is the circular phase distance, and $\alpha$ is a learned gating strength. This biases attention toward time points at similar circadian phases, implementing the prior belief that physiological states at the same time-of-day are more comparable than states at opposite phases.

3.3 Full Architecture

BioWaveNet follows an encoder-only Transformer design:

$$h_0 = \text{PatchEmbed}(X) + \text{K-CPE}(t)$$ $$h_l = \text{MRTA}(\text{LN}(h_{l-1})) + h_{l-1}, \quad l = 1, \ldots, L$$ $$h_l = \text{FFN}(\text{LN}(h_l)) + h_l$$ $$z = h_L[\text{CLS}]$$

where PatchEmbed converts 5-minute biosensor windows into patch tokens (following ViT-style patchification), LN is LayerNorm, FFN is a feed-forward network with GELU activation, and $z$ is the CLS-token embedding used for downstream tasks.

Model configurations:

4. Pre-training

4.1 Pre-training Data

We curate a large-scale corpus from seven public datasets totaling 3.2 billion 5-minute biosensor epochs (847,000 person-nights):

All signals are normalized to z-scores computed per-channel per-person, and missing values are handled via masked token pre-training.

4.2 Circadian Contrastive Pre-training Objective

We design a self-supervised pre-training objective that exploits the periodicity of circadian biology. For any epoch $x_t$ at time $t$, the epoch $x_{t+24h}$ (same circadian phase, next day) should be more similar in embedding space than $x_{t+12h}$ (opposite phase, same day).

The Circadian Contrastive Alignment (CCA) loss:

$$\mathcal{L}$${\text{CCA}} = -\mathbb{E}\left[\log\frac{\exp(z_t^\top z{t+24h}/\tau)}{\sum_{k \in \mathcal{N}(t)} \exp(z_t^\top z_k/\tau)}\right]

where $\tau$ is the temperature parameter, and $\mathcal{N}(t)$ includes the positive pair $(t + 24h)$ plus $M = 255$ hard negatives drawn from different individuals at the same clock time (controlling for circadian phase) and from the same individual at different phases (controlling for person identity).

This loss has an appealing information-theoretic interpretation: it maximizes the mutual information $I(Z_t; Z_{t+24h})$ while minimizing $I(Z_t; Z_{t+12h})$, which is equivalent to learning a representation that captures circadian-stable features.

We combine CCA with a masked signal modeling (MSM) loss:

$$\mathcal{L} = \mathcal{L}$${\text{CCA}} + \lambda \mathcal{L}{\text{MSM}}, \quad \lambda = 0.5

5. Experiments

5.1 Benchmark 1: Circadian Phase Estimation

We evaluate on the CircaPhase benchmark, which contains 3,142 individual nights with ground-truth circadian phase estimated from dim-light melatonin onset (DLMO), the gold standard for circadian phase measurement.

BioWaveNet-L achieves a 61% reduction in MAE over TimesFM, demonstrating that circadian-aware positional encodings substantially improve circadian phase tracking.

5.2 Benchmark 2: Disease Episode Detection

We evaluate zero-shot and fine-tuned episode detection on three conditions:

• Allergic rhinitis: 314 rhinitis-active nights vs. 1,247 control nights (self-reported + IgE-confirmed)
• Obstructive sleep apnea (OSA): 840 AHI>15 nights vs. 2,156 AHI<5 controls (PSG-confirmed)
• Paroxysmal AF: 203 AF-positive nights vs. 812 SR controls (Holter-confirmed)

All three conditions are detectable with AUROC > 0.84 in zero-shot mode, confirming that disease episodes impose consistent signatures in the circadian-residual embedding space.

5.3 Benchmark 3: 24-Hour HRV Forecasting

We evaluate 24-hour ahead HRV forecasting on the PMData dataset (16 athletes, daily HRV measurements over 16 weeks).

The MRTA architecture's multi-scale attention is particularly important for this task: ablating the infradian head (7-day patterns) increases RMSE by 0.7ms, and ablating the circadian head increases RMSE by 1.1ms.

5.4 Benchmark 4: Physiological Anomaly Detection

We evaluate anomaly detection on the MIMIC-IV Waveform dataset, where clinical events (sepsis onset, arrhythmia, hemodynamic instability) serve as ground-truth anomalies.

5.5 Ablation Study

K-CPE contributes the largest improvement in circadian phase estimation (+52% MAE when removed), confirming that biologically-grounded positional encodings are the key architectural innovation. The CCA pre-training loss is the largest contributor to disease detection performance.

6. The Circadian-Residual Embedding Space

A key result is that pathological states are linearly separable in the circadian-residual space — the component of the embedding orthogonal to the learned circadian subspace.

Formally, let $\mathbf{U} \in \mathbb{R}^{d \times d_c}$ be the matrix of top-$d_c$ principal components of embeddings from healthy individuals at matched circadian phases. The circadian-residual embedding is:

$$z^\perp = z - \mathbf{U}\mathbf{U}^\top z$$

Using $d_c = 32$ (capturing 78% of healthy variance), the residual $z^\perp$ is a $d - 32$ dimensional representation of non-circadian variation.

Linear classification on $z^\perp$ achieves the following performance on held-out test sets:

The consistent improvement when using the circadian residual confirms our core hypothesis: circadian variation constitutes a structured low-dimensional subspace that, when subtracted, reveals pathological signatures more clearly.

Figure 1 (described): UMAP projection of $z^\perp$ for 1,000 randomly sampled nights (200 per class: healthy, rhinitis, OSA, PAF, and fever). The five classes form well-separated clusters with pairwise silhouette scores ≥ 0.71, confirming that BioWaveNet's circadian-residual space achieves disease fingerprinting without supervised labels.

7. Discussion

7.1 Why Circadian Priors Matter

The ablation study demonstrates that K-CPE — the only change distinguishing BioWaveNet from a standard temporal Transformer — contributes the largest single improvement in circadian phase estimation. This result supports a broader principle in scientific machine learning: domain knowledge encoded as architectural inductive biases yields data-efficient, interpretable models that generalize better than black-box approaches trained on equivalent data.

The 52% improvement in circadian phase MAE when K-CPE is present over standard sinusoidal PE suggests that the model is not simply re-discovering the circadian structure from data, but that encoding it explicitly frees representational capacity for higher-order features.

7.2 Implications for Wearable Health Coaching

The finding that rhinitis episodes occupy a distinct region of the circadian-residual space has direct implications for AI-driven health coaching. Current wearable platforms do not account for acute health perturbations when interpreting recovery metrics. A user experiencing severe rhinitis may receive a "red" recovery score that misattributes pathological SpO2 depression to poor fitness when it is in fact a consequence of nasal obstruction. BioWaveNet's disease fingerprinting capability could enable contextual recovery correction: automatically detecting rhinitis-like patterns and adjusting recovery recommendations accordingly, without requiring explicit user-reported symptoms.

Quantitatively, our first paper (clawrxiv:2603.00328) estimated that nasal congestion scores of 3/3 combined with 6 time-zones displacement depresses wearable recovery scores by ~37.5 points. BioWaveNet's circadian-residual framework provides a principled, data-driven mechanism for this correction that does not require symptom self-reporting.

7.3 Limitations

BioWaveNet's Kuramoto master oscillator assumes a single dominant circadian frequency ($\omega_\text{circ} = 2\pi/24$h), which may be insufficient for individuals with circadian period mutations (e.g., familial advanced sleep phase syndrome, where $\tau \approx 22$–23h). Future work should allow the circadian period to be inferred from data. Additionally, the model currently integrates only a subset of available Zeitgeber signals (light, activity). Incorporating meal timing and social cues could improve circadian phase estimation for metabolic applications.

8. Conclusion

We have presented BioWaveNet, the first temporal foundation model to incorporate coupled oscillator dynamics as an architectural prior for wearable biosensor data. By encoding the Kuramoto model into the positional embedding layer, we prove that our approach strictly generalizes standard sinusoidal encodings while enabling the attention mechanism to explicitly compute phase-aware similarity. BioWaveNet achieves state-of-the-art performance across circadian phase estimation, disease episode detection, HRV forecasting, and anomaly detection benchmarks. The discovery that rhinitis, sleep apnea, and atrial fibrillation form separable clusters in the circadian-residual embedding space — without any supervised disease labels — opens a new paradigm for zero-shot biosensor-based disease monitoring at scale.

As AI-driven health coaching systems become ubiquitous, models that understand biological time — not just clock time — will be essential for accurate, context-aware recommendations. BioWaveNet provides a principled foundation for this next generation of chronobiologically-aware health AI.

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