A Bidirectionally Coupled Hopf–Kuramoto Model of Hair Cell–Neuron Dynamics: Saddle-Node-Mediated Phase-Locking/Quasi-Periodic Transitions and Coherence Resonance

This paper presents a minimal dynamical model of bidirectional coupling between a hair cell and a neuron, combining a Hopf oscillator (representing the self-sustained mechanical oscillations of the hair bundle) with a Kuramoto phase oscillator (modeling neuronal spiking). This framework captures the essential reciprocity of sensory transduction: mechanical deflection drives neural activity, while efferent feedback modulates hair bundle mechanics. Through bifurcation analysis, we identify conditions for stable phase locking, where both oscillators synchronize at a common frequency with a fixed phase difference. Stable entrainment arises most robustly under moderate nonisochronicity (amplitude-dependent frequency shifts) and balanced coupling strengths. In contrast, weak or excessively strong coupling, or large mismatches in intrinsic frequencies, typically lead to quasiperiodic or desynchronized dynamics. Phase response curves reveal type-II characteristics, with perturbations causing either phase advances or delays depending on oscillation phase. Notably, perturbations to the hair cell induce significantly larger phase shifts than those applied to the neuron, highlighting the mechanical oscillator’s dominant influence on system timing. Finally, we assess the impact of stochastic noise using the Kuramoto order parameter as a synchrony metric. Noise can either disrupt coherence or enhance synchronization via coherence resonance under specific coupling regimes, particularly when the hair cell strongly drives the neuron. This suggests that stochasticity may not degrade signal fidelity but can, in certain configurations, promote functional synchrony in sensory processing. The model thus provides a tractable platform for exploring how mechanical–neural interactions, parameter tuning, and noise jointly shape the dynamic behavior in auditory and vestibular systems.