Oscillatory neural networks and oscillatory neurons are common in the nervous system. Yet the function of neural networks that contain OsCillatory elements is largely unexplored. The proposed work aims to develop a body of technique relevant to addressing this relationship, and to use this technique to further the understanding of some particular neural networks. The work will focus on one vertebrate and one invertebrate neural network, both central pattern generators (CPGS) engaged in producing rhythmic motor output. In each case, the work will be collaborative with neurophysiologists whose data will constrain the modelling activities. The vertebrate CPG to be studied is that for undulatory locomotion; the main experimental animal is the lamprey. This work continues a collaborative effort to formulate and investigate a general and flexible mathematical framework within which conclusions may be drawn from experimental data on the lamprey about how the network functions. Work in the immediate future will focus on the properties of long range coupling among the ii local oscillatory elements, the implications of the design of local networks for the global emergent behavior, and the interaction between the neural behavior and the mechanical activity that it regulates. Relevant experiments will be designed and carried out. The invertebrate CPG is the crustacean stomatogastric ganglion (STG). This work is part of a larger project involving modelling and experiments. Our part of the project is to develop mathematics to help address such questions as: (i) How can the behavior of the network be explained on the basis of the properties of the component neurons? (ii) How do pacemaker and emergent modes of oscillation, that are both known to exist in the STG, interact and cooperate to produce stable and flexible system behavior?