The network of neurons with linear threshold (LT) transfer functions is a prominent model to emulate the behavior of cortical neurons. The analysis of dynamic properties for LT networks has attracted growing interest, such as multistability and boundedness. However, not much is known about how the connection strength and external inputs are related to oscillatory behaviors. Periodic oscillation is an important characteristic that relates to nondivergence, which shows that the network is still bounded although unstable modes exist. By concentrating on a general parameterized two-cell network, theoretical results for geometrical properties and existence of periodic orbits are presented. Although it is restricted to two-dimensional systems, the analysis can provide a useful contribution to analyze cyclic dynamics of some specific LT networks of high dimension. As an application, it is extended to an important class of biologically motivated networks of large scale: the winner-take-all model using local excitation and global inhibition.