  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
linear-threshold dynamics. Exploiting the switched-affine nature of this
dynamics, we obtain various necessary and/or sufficient conditions on the
network structure and its external input 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.
