  Oscillations arise in many real-world systems and are associated with both
functional and dysfunctional states. Whether a network can oscillate can be
estimated if we know the strength of interaction between nodes. But in
real-world networks (in particular in biological networks) it is usually not
possible to know the exact connection weights. Therefore, it is important to
determine the structural properties of a network necessary to generate
oscillations. Here, we provide a proof that uses dynamical system theory to
prove that an odd number of inhibitory nodes and strong enough connections are
necessary to generate oscillations in a single cycle threshold-linear network.
We illustrate these analytical results in a biologically plausible network with
either firing-rate based or spiking neurons. Our work provides structural
properties necessary to generate oscillations in a network. We use this
knowledge to reconcile recent experimental findings about oscillations in basal
ganglia with classical findings.
