# The role of phase shifts of sensory inputs in walking revealed by means of phase reduction

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## Abstract

Detailed neural network models of animal locomotion are important means to understand the underlying mechanisms that control the coordinated movement of individual limbs. Daun-Gruhn and Tóth, Journal of Computational Neuroscience *31*(2), 43–60 (2011) constructed an inter-segmental network model of stick insect locomotion consisting of three interconnected central pattern generators (CPGs) that are associated with the protraction-retraction movements of the front, middle and hind leg. This model could reproduce the basic locomotion coordination patterns, such as tri- and tetrapod, and the transitions between them. However, the analysis of such a system is a formidable task because of its large number of variables and parameters. In this study, we employed phase reduction and averaging theory to this large network model in order to reduce it to a system of coupled phase oscillators. This enabled us to analyze the complex behavior of the system in a reduced parameter space. In this paper, we show that the reduced model reproduces the results of the original model. By analyzing the interaction of just two coupled phase oscillators, we found that the neighboring CPGs could operate within distinct regimes, depending on the phase shift between the sensory inputs from the extremities and the phases of the individual CPGs. We demonstrate that this dependence is essential to produce different coordination patterns and the transition between them. Additionally, applying averaging theory to the system of all three phase oscillators, we calculate the stable fixed points - they correspond to stable tripod or tetrapod coordination patterns and identify two ways of transition between them.

## Keywords

Central pattern generators Inter-segmental coordination Phase oscillator model Stepping patterns Transition Speed control 6-legged locomotion## Notes

### Acknowledgements

We would like to thank Philip Holmes for useful discussions in the course of the work. This research was supported by Deutsche Forschungsgemeinschaft (DFG) grants GR3690/2-1, GR3690/4-1, DA1953/5-2, and by BMBF-NSF grant 01GQ1412.

## Compliance with Ethical Standards

## Conflict of interests

The authors declare that they have no conflict of interest

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