  Among the versatile forms of dynamical patterns of activity exhibited by the
brain, oscillations are one of the most salient and extensively studied, yet
are still far from being well understood. In this paper, we provide various
structural characterizations of the existence of oscillatory behavior in neural
networks using a classical neural mass model of mesoscale brain activity called
the linear-threshold model. Exploiting the switched-affine nature of
linear-threshold dynamics, we obtain various necessary and/or sufficient
conditions for the existence of oscillations in (i) two-dimensional
excitatory-inhibitory networks (E-I pairs), (ii) networks with one inhibitory
but arbitrary number of excitatory nodes, (iii) purely inhibitory networks with
an arbitrary number of nodes, and (iv) networks of E-I pairs. Throughout our
treatment, and given the arbitrary dimensionality of the considered dynamics,
we rely on the lack of stable equilibria as a system-based proxy for the
existence of oscillations, and provide extensive numerical results to support
its tight relationship with the more standard, signal-based definition of
oscillations in computational neuroscience.
