This work studies rhythmic activity and stability in a neural network built by coupling three Wilson–Cowan oscillatory nodes, each with interacting excitatory and inhibitory populations. We compare two connectivity architectures (bus and ring) using a six-dimensional delay differential equation model with discrete transmission delays, constant external inputs, and synaptic weight matrices capturing both local feedback and inter-node coupling. Through linearization, we study one synchronous equilibrium and two asynchronous ones that redistribute activity across nodes. This structure reduces the full linearized system to three independent two-dimensional delayed subsystems, enabling stability analysis. For each equilibrium, characteristic equations are derived and examined with and without delay. In the zero-delay case, stability criteria yield explicit links between equilibrium stability, coupling strength, and topology.