Oscillations and other patterns of neuronal activity arise throughout
the central nervous system. This activity has been observed in sensory
processing, motor activities, and learning, and has been implicated in
the generation of sleep rhythms, epilepsy, and parkinsonian tremor.
Mathematical models for neuronal activity often display an incredibly
rich structure of dynamic behavior. In this lecture, I describe how the
neuronal systems can be modeled, various types of activity patterns that
arise in these models, and mechanisms for how the activity patterns are
generated. In particular, I demonstrate how methods from geometric
singular perturbation theory have been used to analyze a recent model
for activity patterns in an insect's antennal lobe.

The set of multiplicative arithmetic functions over a ring R
(commutative with identity) can be given a unique functorial ring
structure for which (1) the operation of addition is Dirichlet
convolution and (2) multiplication of completely multiplicative
functions coincides with point-wise multiplication. This existence of
this ring structure can be derived from the existence of the ring of
``big'' Witt vectors, and it yields a ring structure on the set of
formal Dirichlet series that are expressible as an Euler product. The
group of additive arithmetic functions over R also has a naturally
defined ring structure, and there is a functorial ring homomorphism
from the ring of multiplicative functions to the ring of additive
functions that is an isomorphism if R is a Q-algebra. An application
is given to zeta functions of schemes of finite type over the ring
of integers.