A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback.

In earlier work, we have developed an integrated model for insect locomotion that includes a central pattern generator (CPG), nonlinear muscles, hexapedal geometry and a representative proprioceptive sensory pathway. Here, we employ phase reduction and averaging theory to replace 264 ordinary differential equations (ODEs), describing bursting neurons in the CPG, their synaptic connections to motoneurons, muscle activation dynamics and sensory neurons, with 24 one-dimensional phase oscillators that describe motoneuronal activation of agonist-antagonist muscle pairs driving the jointed legs. Reflexive feedback is represented by stereotypical spike trains with rates proportional to joint torques, which change phase relationships among the motoneuronal oscillators. Restriction to the horizontal plane, neglect of leg mass and use of Hill-type muscle models yield a biomechanical body-limb system with only three degrees of freedom, and the resulting hybrid dynamical system involves 30 ODEs: reduction by an order of magnitude. We show that this reduced model captures the dynamics of unperturbed gaits and the effects of an impulsive perturbation as accurately as the original one. Moreover, the phase response and coupling functions provide an improved understanding of reflexive feedback mechanisms.