In this thesis we study aspects of periodic activity in model mutually-coupled oscillators inspired by the nervous system. We define and use maps describing the timing of activity on successive cycles. The central theme here is to examine emergent behavior in networks through the properties of the individual oscillators.In the first chapter, we describe Phase Response Curves (PRCs), which map the changes in theperiod of an oscillator to perturbations at dierent phases along the cycle. We consider various networks of oscillators, pulse-coupled through their PRCs: rings, chains, arrays, and global coupling.We study conditions under which stable patterns, such as synchrony and waves, may be found.In the second and third chapters, we model beta (12-30 Hz) and gamma (30-80 Hz) rhythmsin the nervous system in reduced networks of excitatory and inhibitory neurons. We look at theintriguing results of experiments that show increases in beta band activity in human MEGs upon taking the sedative Diapam. We show that the model network is able to mimic the experimental data. The model then clarifies the inhibitory action of the drug in tissue.We look at another experiment that finds disruption of long-range synchrony of gamma oscillations in transgenic mice with altered excitatory kinetics. We study this behavior in a reduced network that encodes for conduction delays across spatially distal sites. The model provides an explanation of this phenomenon in terms of the properties of the cells involved in generating the rhythm.In our analyses, we use maps to study stability of the patterns of activity.
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