Models of central pattern generators for quadruped locomotion - I. Primarygaits

In this paper we continue the analysis of a network of symmetrically couple d cells modeling central pattern generators for quadruped locomotion propos ed by Golubitsky. Stewart, Buono, and Collins. By a cell we mean a system o f ordinary differential equations and by a coupled cell system we mean a ne twork of identical cells with coupling terms. We have three main results in this paper. First, we show that the proposed network is the simplest one m odeling the common quadruped gaits of walk, trot. and pace. In doing so we prove a general theorem classifying spatio-temporal symmetries of periodic solutions: to equivariant systems of differential equations. We also specia lize this theorem to coupled cell systems. Second, this paper focuses on pr imary gaits: that is. gaits that are modeled by output signals from the cen tral pattern generator where each cell emits the same waveform along with t ract phase shifts between cells. Our previous work showed that the network is capable of producing six primary gaits. Here, we show that under mild as sumptions on the cells and the coupling of the network, primary gaits can b e produced from Hopf bifurcation by varying only coupling strengths of the network. Third, we discuss the stability of primary gaits and exhibit these solutions by performing numerical simulations using the dimensionless Morr is-Lecar equations for the cell dynamics. .