## Bars : The bars below are the support graph of the gait….

v f/I.~.’11 I / I / 1 1 f . ~ • VI///[//A I IT[/I/////Zl/J/////Y///L//AI/I//I//IIIII///•]///////#II//I////IIII// Fig. 1.

Slow rack-like walk of giraffe is left-right reflected by a phase shift of half a period.

The bars below are the support graph of the gait, and show when each foot is in contact with the ground. [From P.

Gambaryan (1974), How Mammals Run: Anatomical Adaptations.

Distributed by John Wiley & Sons, Inc., New York.] Coupled Nonlinear Oscillators and the Symmetries of Animal Gaits 351 Our aim in this paper is to draw attention to some remarkable parallels between the generalities of coupled nonlinear oscillators and the observed range of gait patterns in animal locomotion.

It is widely held that locomotion might be controlled by a central pattern generator (CPG), which is a network of neurons capable of producing rhythmic output; see Sect. 3.

One explanation of the observed parallel is that a locomotor CPG may have a degree of rectangular or square symmetry.

If so, then our results may be interpreted as a symmetry classification of the general architecture of CPGs, describing which symmetry types of gaits can occur for each.

The traditional approach to CPG architecture has been to hypothesize specific neural circuitry and analyze its dynamics, either analytically for linear models or numerically for nonlinear ones.

We approach this question from a different perspective, namely, that of nonlinear dynamics and local bifurcation theory.

This approach makes the rigorous mathematical analysis of nonlinear systems more tractable, reveals general universal patterns and recognizable phenomena, and may be relevant to the neurophysiology of animals.

These general results form a useful starting point for more detailed model-dependent analysis, and separate the questions to be answered into two types: a.

What are the general phenomena to be expected in symmetrically coupled systems of nonlinear neuronal oscillators? b.

What specific phenomena among these actually occur, and what does that imply about model-dependent features of the network architecture? Here we answer some questions of type a.

In particular we show that for each of a number of symmetry types of CPG, there is a natural universal hierarchy of symmetry-breaking oscillation patterns, many of which correspond to actual gaits.

The patterns depend strongly on the architecture of the CPG network, but have a certain amount in common.

Among quadrupedal gaits (Figures 2–4), the pace, trot, and bound (and also the rarer pronk) are highly symmetric, with relative phase lags of zero or half a period, and are very robust.

The walk involves phase lags of a quarter of a period and also has a natural interpretation in terms of symmetry-breaking.

The rotary and transverse gallops have less symmetry (but despite often being termed asymmetric, they retain s o m e symmetry), are less robust, and involve somewhat arbitrary phase lags.

They appear to correspond to networks of oscillators having an odd internal symmetry, a characteristic shared by pendulums and van der Pol oscillators.

The canter is a more curious gait, rather fragile, with very little symmetry, and it remains mysterious.

It is worth observing that it is often a trained gait.

Moreover, transitions between these gaits strongly resemble the typical types of symmetry-breaking bifurcation that can occur in the corresponding nonlinear dynamical systems.

Varying parameters in a CPG, such as the coupling strengths between the component neuronal oscillators, may thus permit the s a m e CPG to control a variety of distinct gaits, and to cause transitions from one to another.

The results of this paper complement those of Schrner e t at. (1990), which approach the same problem from the point of view of synergetics rather than equivariant bifurcation theory.

They describe the main observed gait symmetries group-theoretically, analyze the corresponding phase dynamics, and obtain gait transitions as phase transitions in model dynamical systems.

In our approach all possible 352 J.

J.

Collins and I.

N.

Stewart symmetries of gaits are derived as a natural consequence of generic symmetry-breaking in CPG dynamics, and gait transitions are viewed as generic symmetry-breaking bifurcations.

Moreover, our main results (Sects. 7, 9 and Appendices 2, 3) are modelindependent.

Thus, Sch6ner et al. (t990) take symmetries of gaits as their starting point, and discuss their consequences for animal behavior; we concentrate on how such symmetries might arise from the dynamics of symmetric CPG networks.

The analogy between the general phenomenology of periodic oscillations of symmetric dynamical systems, and the various more-or-less symmetric gaits of animal locomotion, is thus quite striking.

Below we investigate how far this analogy can be carried.

We describe symmetric gaits and discuss how they can be classified.

We present experimental evidence for nonlinear effects and symmetry-breaking in animal gaits and discuss the conclusions which can be drawn about the general nature of locomotion, without considering specific model equations for the dynamics. 2.

Animal Gaits

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