  For coupled oscillator networks with Laplacian coupling the master stability
function (MSF) has proven a particularly powerful tool for assessing the
stability of the synchronous state. Using tools from group theory this approach
has recently been extended to treat more general cluster states. However, the
MSF and its generalisations require the determination of a set of Floquet
multipliers from variational equations obtained by linearisation around a
periodic orbit. Since closed form solutions for periodic orbits are invariably
hard to come by the framework is often explored using numerical techniques.
Here we show that further insight into network dynamics can be obtained by
focusing on piecewise linear (PWL) oscillator models. Not only do these allow
for the explicit construction of periodic orbits, their variational analysis
can also be explicitly performed. The price for adopting such nonsmooth systems
is that many of the notions from smooth dynamical systems, and in particular
linear stability, need to be modified to take into account possible jumps in
the components of Jacobians. This is naturally accommodated with the use of
\textit{saltation} matrices. By augmenting the variational approach for
studying smooth dynamical systems with such matrices we show that, for a wide
variety of networks that have been used as models of biological systems,
cluster states can be explicitly investigated. By way of illustration we
analyse an integrate-and-fire network model with event-driven synaptic coupling
as well as a diffusively coupled network built from planar PWL nodes, including
a reduction of the popular Morris--Lecar neuron model. We use these examples to
emphasise that the stability of network cluster states can depend as much on
the choice of single node dynamics as it does on the form of network structural
connectivity.
