Population-wide synchronized rhythmic bursts of electrical activity are present in a variety of neural circuits. The proposed general mechanisms for rhythmogenesis are often attributed to intrinsic and synaptic properties. For example, the recurrent excitation through excitatory synaptic connections determines burst initiation, and the slower kinetics of ionic currents or synaptic depression results in burst termination. In such theories, a slow recovery process is essential for the slow dynamics associated with bursting. This thesis presents a new hypothesis that depends on the connectivity pattern among neurons rather than a slow kinetic process to achieve the network-wide bursting. The thesis begins with an introduction of bursts of electrical activity in a purely excitatory neural network and existing theories explaining this phenomenon. It then covers the small-world approach, which is applied to modify the network structure in the simulation, and the Morris-Lecar (ML) neuron model, which is used as the component cells in the network. Simulation results of the dependence of bursting activity on network connectivity, as well as the inherent network properties explaining this dependence are described. This work shows that the network-wide bursting activity emerges in the small-world network regime but not in the regular or random networks, and this small-world bursting primarily results from the uniform random distribution of long-range connections in the network, as well as the unique dynamics in the ML model. Both attributes foster progressive synchronization in firing activity throughout the network during a burst, and this synchronization may terminate a burst in the absence of an obvious slow recovery process. The thesis concludes with possible future work.