  The tools of weakly coupled phase oscillator theory have had a profound
impact on the neuroscience community, providing insight into a variety of
network behaviours ranging from central pattern generation to synchronisation,
as well as predicting novel network states such as chimeras. However, there are
many instances when this theory is expected to break down, say in the presence
of strong coupling, or must be carefully interpreted, as in the presence of
stochastic forcing. There are also surprises in the dynamical complexity of the
attractors that can robustly appear - for example, heteroclinic network
attractors. In this review we present a set of mathematical tools that are
suitable for addressing the dynamics of oscillatory neural networks, broadening
from a standard phase oscillator perspective to provide a practical framework
for further successful applications of mathematics to understanding network
dynamics in neuroscience.
