Hierarchical Cyclic Neural ODEs

Biological systems are inherently cyclical, operating through feedback loops where epigenetic modifications regulate transcription, driving protein synthesis and metabolic flux, which ultimately modulate the epigenome. Modeling these continuous, high-dimensional dynamics remains challenging because the underlying molecular layers are operating on disparate timescales. While standard Neural Ordinary Differential Equations (NODEs) are widely recognized for modeling continuous-time processes, they frequently experience timescale collapse and numerical stiffness when forced to map extreme multi-scale interactions. To stably learn these hierarchical oscillatory systems, the Hierarchical Cyclic Neural ODE is introduced. A Stuart-Landau limit-cycle prior is embedded into the model's latent vector field, constraining the neural network to focus primarily on the biological variations that deviate from an underlying rhythm. A cyclic small-gain analysis provides a theoretical guarantee that this architecture yields unconditionally bounded trajectories (preventing numerical instability) and exponential orbital attraction (ensuring perturbed states converge to a homeostatic orbit). Evaluated against leading continuous-time models including scNODE and Heavy Ball NODE, on a four-layer mouse liver atlas and a two-layer $\text{Neurospora crassa}$ course in constant darkness, this architecture maintains stable trajectories through the forecast window and reduces held-out error on both. In contrast, acyclic ablations suffer from temporal mode collapse and amplify perturbations over a full cycle, confirming that closed-loop structural feedback is necessary to bound system energy. Leveraging these continuous dynamics, an analysis of the system's phase response demonstrates that a perturbation confined to a single layer produces a lasting phase shift in layers reachable only around the loop, a response that vanishes identically when the single return edge is removed.