  Neural mass models are ubiquitous in large scale brain modelling. At the node
level they are written in terms of a set of ODEs with a nonlinearity that is
typically a sigmoidal shape. Using structural data from brain atlases they may
be connected into a network to investigate the emergence of functional dynamic
states, such as synchrony. With the simple restriction of the classic sigmoidal
nonlinearity to a piecewise linear caricature we show that the famous
Wilson-Cowan neural mass model can be analysed at both the node and network
level. The construction of periodic orbits at the node level is achieved by
patching together matrix exponential solutions, and stability is determined
using Floquet theory. For networks with interactions described by circulant
matrices, we show that the stability of the synchronous state can be determined
in terms of a low-dimensional Floquet problem parameterised by the eigenvalues
of the interaction matrix. This network Floquet problem is readily solved using
linear algebra, to predict the onset of spatio-temporal network patterns
arising from a synchronous instability. We consider the case of a discontinuous
choice for the node nonlinearity, namely the replacement of the sigmoid by a
Heaviside nonlinearity. This gives rise to a continuous-time switching network.
At the node level this allows for the existence of unstable sliding periodic
orbits, which we construct. The stability of a periodic orbit is now treated
with a modification of Floquet theory to treat the evolution of small
perturbations through switching manifolds via saltation matrices. At the
network level the stability analysis of the synchronous state is considerably
more challenging. Here we report on the use of ideas originally developed for
the study of Glass networks to treat the stability of periodic network states
in neural mass models with discontinuous interactions.
