# Gait transitions in a phase oscillator model of an insect central pattern generator

## TL;DR

This study models insect gait transitions using a phase oscillator framework, revealing how increased stepping frequency causes bifurcations from tetrapod to tripod gaits, aligning with observed insect behavior.

## Contribution

It introduces a reduced phase oscillator model for insect gait transitions, providing a mathematical explanation for gait changes with speed.

## Key findings

- Bifurcations from tetrapod to tripod gaits occur as speed increases.
- The model predicts stable gait patterns consistent with insect observations.
- Gait transitions are explained through bifurcation analysis of coupled oscillators.

## Abstract

Legged locomotion involves various gaits. It has been observed that fast running insects (cockroaches) employ a tripod gait with three legs lifted off the ground simultaneously in swing, while slow walking insects (stick insects) use a tetrapod gait with two legs lifted off the ground simultaneously. Fruit flies use both gaits and exhibit a transition from tetrapod to tripod at intermediate speeds. Here we study the effect of stepping frequency on gait transition in an ion-channel bursting neuron model in which each cell represents a hemi-segmental thoracic circuit of the central pattern generator. Employing phase reduction, we collapse the network of bursting neurons represented by 24 ordinary differential equations to 6 coupled nonlinear phase oscillators, each corresponding to a sub-network of neurons controlling one leg. Assuming that the left and right legs maintain constant phase differences (contralateral symmetry), we reduce from 6 equations to 3, allowing analysis of a dynamical system with 2 phase differences defined on a torus. We show that bifurcations occur from multiple stable tetrapod gaits to a unique stable tripod gait as speed increases. Finally, we consider gait transitions in two sets of data fitted to freely walking fruit flies.