  The nervous system displays a variety of rhythms in both waking and sleep.
These rhythms have been closely associated with different behavioral and
cognitive states, but it is still unknown how the nervous system makes use of
these rhythms to perform functionally important tasks. To address those
questions, it is first useful to understood in a mechanistic way the origin of
the rhythms, their interactions, the signals which create the transitions among
rhythms, and the ways in which rhythms filter the signals to a network of
neurons. This talk discusses how dynamical systems have been used to
investigate the origin, properties and interactions of rhythms in the nervous
system. It focuses on how the underlying physiology of the cells and synapses
of the networks shape the dynamics of the network in different contexts,
allowing the variety of dynamical behaviors to be displayed by the same
network. The work is presented using a series of related case studies on
different rhythms. These case studies are chosen to highlight mathematical
issues, and suggest further mathematical work to be done. The topics include:
different roles of excitation and inhibition in creating synchronous assemblies
of cells, different kinds of building blocks for neural oscillations, and
transitions among rhythms. The mathematical issues include reduction of large
networks to low dimensional maps, role of noise, global bifurcations, use of
probabilistic formulations.
