We considered a clustered network of bursting neurons described by the Huber-Braun model. In the upper level of the network we used the connectivity matrix of the cat cerebral cortex network, and in the lower level each cortex area (or cluster) is modelled as a small-world network. There are two different coupling strengths related to inter- and intracluster dynamics. Each bursting cycle is composed of a quiescent period followed by a rapid chaotic sequence of spikes, and we defined a geometric phase which enables us to investigate the onset of synchronized bursting, as the state in which the neuron start bursting at the same time, whereas their spikes may remain uncorrelated. The bursting synchronization of a clustered network has been investigated using an order parameter and the average field of the network in order to identify regimes in which each cluster may display synchronized behavior, whereas the overall network does not. We introduce quantifiers to evaluate the relative contribution of each cluster in the partial synchronized behavior of the whole network. Our main finding is that we typically observe in the clustered network not a complete phase synchronized regime but instead a complex pattern of partial phase synchronization in which different cortical areas may be internally synchronized at distinct phase values, hence they are not externally synchronized, unless the coupling strengths are too large.