P.L. Buono and M. Golubitsky
Models of Central Pattern Generators For Quadruped Locomotion:
I. Primary Gaits
J. Math. Biol. 42 No. 4 (2001) 291-326.
In this paper we continue the analysis of a
network of symmetrically coupled cells modeling
central pattern generators for quadruped locomotion
proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean
a system of ordinary differential equations and by a coupled cell
system we mean a network 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 modeling 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 specialize this theorem to coupled cell systems.
Second, this paper focuses on primary gaits;
that is, gaits that are modeled by output signals
from the central pattern generator where each cell emits
the same waveform along with exact phase shifts between cells.
Our previous work showed that the network is capable
of producing six primary gaits.
Here, we show that under mild assumptions on
the cells and the coupling of the network, primary
gaits can be 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 Morris-Lecar equations for the cell dynamics.