Oscillations arise throughout the central nervous system. The thalamus, for example, is centrally important in the generation of sleep rhythms. It has also been implicated in the generation of Parkinson tremor and epilepsy. Models for these oscillations can exhibit an incredibly rich structure of dynamic behavior. The rhythms are often the result of interactions between the intrinsic properties of the individual neurons within the network and the synaptic properties of the coupling between the neurons. Both the intrinsic and synaptic dynamics depend on numerous parameters and may involve multiple time scales; each of these may have a profound influence on the emergent network behavior. In this chapter, I will present recent models for some of these rhythms, review the sorts of oscillatory behavior which can arise from the models and then discuss how geometric dynamical systems methods can be used to analyze the emergent oscillatory behavior.
The dynamics of even a single cell may be quite complicated. It may, for example, fire bursts of action potentials that are followed by a silent phase of near quiescent behavior. There are, in fact, several different types of such bursting solutions. The different types can be classified by the geometric properties of how solutions evolve in phase space. I will describe each of the known types of bursting oscillations and then review mathematical analysis related to these oscillations.
I will then discuss the dynamics of small networks of neurons. The models will include synaptic coupling since this is the primary form of coupling within the brain. The coupling may, however, be excitatory or inhibitory and it may involve multiple time scales. I will be primarily concerned with how the intrinsic and synaptic properties of the cells interact to either synchronize or desynchronize the population rhythms. This has been a very active area of research; I will review and extend recent results.
Finally, I will discuss larger networks of neuronal oscillators. In particular, I will demonstrate how dynamical systems methods can be used to analyze specific applications including models for sleep rhythms and models for scene segmentation. I will conclude with a discussion of recent results related to wave propagation in one-dimensional arrays of synaptically coupled neuronal networks.
Jan 28 1998