A simple state-determined model reproduces entrainment and phase-locking of human walking

Theoretical studies and robotic experiments have shown that asymptotically stable periodic walking may emerge from nonlinear limit-cycle oscillators in the neuro-mechanical periphery. We recently reported entrainment of human gait to periodic mechanical perturbations with two essential features: 1) entrainment occurred only when the perturbation period was close to the original (preferred) walking period, and 2) entrainment was always accompanied by phase locking so that the perturbation occurred at the end of the double-stance phase. In this study, we show that a highly-simplified state-determined walking model can reproduce several salient nonlinear limit-cycle behaviors of human walking: 1) periodic gait that is 2) asymptotically stable; 3) entrainment to periodic mechanical perturbations only when the perturbation period is close to the model's unperturbed period; and 4) phase-locking to locate the perturbation at the end of double stance. Importantly, this model requires neither supra-spinal control nor an intrinsic self-sustaining neural oscillator such as a rhythmic central pattern generator. Our results suggest that several prominent limit-cycle features of human walking may stem from simple afferent feedback processes without significant involvement of supra-spinal control or a self-sustaining oscillatory neural network.