  We analyze a family of clustered excitatory-inhibitory neural networks and
the underlying bifurcation structures that arise because of permutation
symmetries in the network as the global coupling strength $g$ is varied. We
primarily consider two network topologies: an all-to-all connected network
which excludes self-connections, and a network in which the excitatory cells
are broken into clusters of equal size. Although in both cases the bifurcation
structure is determined by symmetries in the system, the behavior of the two
systems is qualitatively different. In the all-to-all connected network, the
system undergoes Hopf bifurcations leading to periodic orbit solutions;
notably, for large $g$, there is a single, stable periodic orbit solution and
no stable fixed points. By contrast, in the clustered network, there are no
Hopf bifurcations, and there is a family of stable fixed points for large $g$.
