Chaotic bursting is a fundamental behavior of neurons. In this paper, global and local burst synchronization is studied in a heterogeneous small-world neuronal network ofnon-identical Hindmarsh–Rose(HR) neurons with noise. It is found that the network can achieve global burst synchronization much more easily than phase synchronization and nearly complete synchronization. Moreover, local burst synchronized clusters have already formed before global burst synchronization happens. We study the effect of the shortcut-adding probability and the heterogeneity coefficient on local and global burst synchronization of the network and find that the introduction of shortcuts facilitates burst synchronization while the heterogeneity has little effect. Moreover, we study the spatio temporal pattern of the network and find that there is an optimal coupling strength range in which the periodicity of the network is very apparent.