  We review the theory of weakly coupled oscillators for smooth systems. We
then examine situations where application of the standard theory falls short
and illustrate how it can be extended. Specific examples are given to
non-smooth systems with applications to the Izhikevich neuron. We then
introduce the idea of isostable reduction to explore behaviors that the weak
coupling paradigm cannot explain. In an additional example, we show how
bifurcations that change the stability of phase locked solutions in a pair of
identical coupled neurons can be understood using the notion of isostable
reduction.
