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Symmetry 2014, 6(1), 23-66; doi:10.3390/sym6010023
Article

Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator

Received: 28 October 2013 / Revised: 13 December 2013 / Accepted: 13 December 2013 / Published: 3 January 2014
(This article belongs to the Special Issue Symmetry Breaking)

Abstract

The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.
Keywords: biped gait; symmetry-breaking; rate model; network; Hopf bifurcation; tetrahedral group biped gait; symmetry-breaking; rate model; network; Hopf bifurcation; tetrahedral group
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Stewart, I. Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator. Symmetry 2014, 6, 23-66.

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