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

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 Z 2 × Z 2 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.

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“…We compute the eigenvalues of the adjacency matrix for the general model, tabulate the general form of the critical eigenvectors, and deduce the spatiotemporal symmetries of the corresponding states using symmetric Hopf bifurcation. For more detailed model-dependent analysis we adapt the methods of Stewart [50,51].…”

“…We define gain-homogeneous rate models, which have fully synchronous states without necessarily being homogeneous. We then summarize the main results of Stewart [50,51] needed for the analysis, namely: the relation between eigenvalues and eigenvectors of the adjacency matrix and those of the Jacobian of the rate model; preservation of spatiotemporal symmetry; relation between the first symmetry-breaking Hopf bifurcation and the largest eigenvalue of the adjacency matrix. (The first bifurcation from equilibrium has the possibility of creating a stable symmetry-breaking branch.…”

“…It can be justified in that case by an analysis similar to that performed for the 16-node network in Section 7. Indeed, this system of ODEs is studied in detail in Stewart [50] to model biped gaits.…”

“…We now give a rigorous analysis of some features of the rate model for Figure 10, and compare the results with the heuristic arguments of the previous section. Our method follows general results of Stewart [50,51] on rate models. We can obtain fairly detailed results because rate models (with their usual type of gain function) have special features, which are mathematically convenient.…”

“…In Stewart [50,51] it is proved that for rate equations, eigenvalues and eigenvectors of the Jacobian at a fully synchronous state can be deduced from those of the adjacency matrix A of the network. Moreover, the spatiotemporal isotropy groups of bifurcating states are preserved by this correspondence.…”