There are numerous families of Neural Networks (NN) used in the study and development of the field of Artificial Intelligence (AI). One of the more recent NNs involves the Bursting neuron, pioneered by Rulkov. The latter has the desirable property that it can also be used to model a system (for example, the "brain") which switches between modes in which the activity is excessive ("bursty"), to the alternate case when the system is "dormant". This paper, which we believe is a of pioneering sort, derives some of the analytic properties of the Bursting neuron, and the associated NN. To be more specific, the model proposed by Rulkov [11] explains the so-called "bursting" phenomenon in the system (brain), in which a low frequency pulse output serves as an envelope of high frequency spikes. Although various models for bursting have been proposed, Rulkov's model seems to be the one that is both analytically tractable and experimentally meaningful. In this paper, we show that a "small" scale network consisting of Bursting neurons rapidly converges to a synchronized behavior implying that increasing the number of neurons does not contribute significantly to the synchronization of the individual Bursting neurons. The consequences of such a conclusion lead to a phenomenon that we call "behavioral synchronization".