One of the advantages of oscillatory neural networks is their dynamic links among related features; the links are based on input-dependent synchronized oscillations. This paper investigates the relations between synchronous/asynchronous oscillations and the connection architectures of an oscillatory neural network with two excitatory-inhibitory pair oscillators. Through numerical analysis, we show synchronous and asynchronous connection types over a wide parameter space for two different inputs and one connection parameter. The results are not only consistent with the classification of synchronous/asynchronous connection types in König's model (1991), but also offer a useful guideline on how to construct a network with local connections for a segmentation task.