Each oscillator has a phase \( \theta_i \), a natural frequency \( \omega_i \), and coupling coefficients \( K_{ij} \) or \( c_{ij} \). The central distinction is whether interactions depend on pairwise phase differences or on separable sensitivity--influence functions.

Key message. Kuramoto dynamics is governed by relative phase differences, whereas Winfree dynamics separates a receiving oscillator's sensitivity \(S(\theta_i)\) from neighboring oscillators' influence \(I(\theta_j)\). WONN turns this principle into a learnable neural architecture on \((S^1)^d\).

WONN is evaluated on CIFAR-10/100 and ImageNet-100/1K. The tables report accuracy and parameter count, comparing WONN with standard convolutional, transformer, and synchrony-based baselines.

^† ViT baselines use additional regularization such as CutMix and label smoothing.

On structured reasoning tasks, WONN uses oscillatory dynamics to form coherent solutions over time. We report Maze-hard pathfinding and Sudoku solving results.

Beyond final accuracy, WONN exposes interpretable dynamics: candidate paths synchronize over recurrent steps, phase distributions organize into two complementary modes, and interaction energy provides a useful diagnostic.

Synchronous Pathfinding on Maze-hard

WONN initially activates multiple diffuse path fragments. As the oscillatory dynamics evolve, compatible fragments synchronize into a coherent path while inconsistent candidates are suppressed.

Comparison with HRM on the same Maze-hard instance

For the first WONN example above, we visualize the first 12 H-block updates of HRM as evolution steps. HRM remains largely inactive during the early stages, begins to generate irregular predictions around \(T=4\), and subsequently undergoes an abrupt transition around \(T=6\) that recovers most of the correct path. The remaining steps only introduce minor refinements. This abrupt, insight-like behavior contrasts sharply with the progressive path formation exhibited by WONN.

Two-Peak Phase Distribution

Phase modes reveal complementary visual structures.

Observation

WONN exhibits a characteristic two-peak phase distribution.

Interpretation

The two synchronized phase modes capture complementary object structures.

Effect

Coarse global regions are separated from fine local details and boundaries.

Layer-wise evolution of the weighted maps associated with the two dominant phase peaks in WONN on image recognition. Left: weighted map corresponding to the first dominant phase peak. Right: weighted map corresponding to the second dominant phase peak. Here \(L\) denotes the layer index and \(T\) denotes the Winfree dynamics step within that layer. Panels are arranged according to the actual forward trajectory, from \(L1T1\) to \(L6T3\). Across layers, the phase-weighted responses are progressively refreshed and reorganized, evolving from weak local activations toward more coherent global semantic structures through the recurrent synchronization dynamics.

Interaction Energy as a Diagnostic

Energy Voting selects the lowest-energy sampled trajectory.

Energy

\(E_{\mathrm{int}}(\Theta) = -\frac{1}{2}\sum_{i,j} c_{ij}\sin\theta_i\sin\theta_j\).

Training

The final interaction energy decreases as pathfinding accuracy improves.

Test time

Energy Voting chooses the lowest-energy trajectory among sampled solutions.

BibTeX

If you find our work useful, please consider citing our paper:

@article{dai2026winfree,
  title={Winfree Oscillatory Neural Network},
  author={Jiawen Dai and Yue Song},
  journal={arXiv preprint arXiv:2605.20922},
  year={2026},
  eprint={2605.20922},
  archivePrefix={arXiv},
  primaryClass={cs.LG},
  url={https://arxiv.org/abs/2605.20922}
}