We have used continuous and discrete-time versions of a neural oscillator model to analyze how various types of synaptic connections between oscillators affect synchronization and desynchronization phenomena. First, we present a synthesis of the mathematical properties of both neural oscillator versions. Then, we show that the choice of parameters leads to a relationship between the two versions. Finally, we achieve the coupling of two oscillators in order to study how synaptic connections affect the phase lag. With this in mind, we state some of the results for the continuous-time model. The second part of this paper deals with the behavior of neural networks comprising connected oscillators, which involves looking at the conditions for desynchronization of a totally synchronized oscillatory network. Such a study has been carried out both for a fully and for a sparsely connected network. This leads to the observation that some architectures enable proper desynchronization when the size of the network is large. While searching for the conditions for desynchronization, we have discovered that a macroscopic description of the network is sometimes possible. To conclude, we discuss the advantages and the limitations of this macroscopic approach.