| The neural aspects of locomotion of animals have been an active topic of research in neurophysiology for the past 30 years. From these studies, central pattern generators (CPG), networks of neurons producing the rythmic output to the limbs, were found. Central pattern generators for primitive animals, like the lamprey, are located and mapped. A complete study in mammals is obstructed by the complexity of the nervous system. Recently, mathematical models have been used to conjecture the possible architecture of mammalian CPGs.;Gaits are periodic patterns of leg movements that animals use for locomotion. In particular, gaits possess symmetry. In recent years, mathematical models of the gaits of quadrupeds incorporate symmetry as a modeling assumption. In particular, Collins and Stewart model CPGs for quadruped locomotion with the assumption that the symmetries of gaits is a mirror of the symmetries of the CPG, and they use four symmetrically coupled cells in their models. However, Collins and Stewart do not find a single system that produces all of the standard quadruped gaits without conjugating trot and pace, and this conjugacy does not agree with observations.;In this thesis, we show that four cell networks can never produce all standard gaits without unwanted conjugacies. We then present a model for the CPG of quadrupeds that overcomes the conjugacy problem. In Chapters 2 and 3 we show the existence and stability of the primary gait patterns that are produced by Hopf bifurcation. In Chapter 4, we study bifurcations from periodic solutions with abelian symmetry, and in Chapter 5 apply these results to find secondary gait patterns that bifurcate from the primary gaits. Numerical simulations are performed in Chapter 6. |
