Phase Oscillator Model for the Central Pattern Generator in Insect Locomotion

Abstract

We investigate a coupled nonlinear 6-phase-oscillator model for the central pattern generator known to govern insect locomotion, where each oscillator corresponds to the local neural networks controlling each leg. We analyze this model and the linearization of the model about the anti-phase-locked alternating tripod gait solution, which is a gait commonly observed in rapidly-running cockroaches. The nonlinear analysis shows that coupling strengths inputted into each of the leg units must be balanced in order for the alternating tripod and tetrapod gaits to exist. We con-sider bilaterally symmetric anti-phase solutions to the nonlinear model and prove the existence of these solutions in our nonlinear analysis. We also show how coupling strengths fitted from experimental data on freely-running cockroaches generally fits the coupling strength conditions associated with different cockroach running speeds. Our study of the linearized system shows the system is asymptotically stable for the alternating tripod gait since all eigenvalues are real and in the left-half plane, with the exception of the zero eigenvalue, which corresponds to overall phase shifts. Our coupled nonlinear phase oscillator model depends on the coupling function H that arises when we reduce the 3-D bursting neuron model to a network of phase oscillators. The coupling function H is only dependent on phase differences and can be numerically calculated. In this paper, we describe how the dependence of the coupling function H on speed of the gait reflects the transition between different regimes of gaits.