ABSTRACT
Networks of coupled neurons often display a rich structure of oscillatory behavior. This behavior plays an important role in several areas of the nervous system (see among others [28, 23, 14, 11, 9]). Oscillatory behavior also arises in such areas as respiration, movement, secretion, and models for scene segmentation [1, 4, 3, 11, 26, 32]. Mathematical models for these systems are typically very complicated. A single neuron is often modeled using the Hodgkin-Huxley formalism [10], and this may lead to a large system of differential equations for just the individual cells. The equations for a single cell may exhibit several different types of solutions. For example, models for electrical activity in pancreatic beta cells exhibit both bursting oscillations and continuous spiking [19, 24]. It has also been demonstrated that even simple models for bursting activity may lead to chaotic dynamics [2, 25]. Another reason why networks of neural oscillators lead to complicated dynamics is that the coupling between individual cells may be quite complicated. The coupling may involve gap junctions, chemical synapses, or ionic coupling [18]. Moreover, the coupling may be either excitatory or inhibitory, and it may include several time scales [35, 15, 27]. Since a particular physical system may involve combinations of different types of cells and different types of coupling, it is easy to understand why models for these systems lead to very challenging mathematical problems, and there has been very little rigorous analysis of them.
