Understanding the brain and creating adaptable artificial intelligence systems are both at the forefront of science and engineering. Brains have the capacity to adapt in response to novel stimuli and efficiently encode vast amounts of information for later use through learning manifested as neural plasticity. While there are a range of known plasticity mechanisms, this work focuses on brain plasticity involving neurons preferentially connecting and disconnecting with each other in a process called structural synaptic plasticity. This process is known to occur in biology, however it is difficult to study experimentally due to the long time scales involved, therefore this work explores it through simulation. Structural plasticity involves complex changes to the morphology of neural cells with the end goal of altering the connectivity of neural circuits. Morphological changes are ignored to preserve computational tractability and the size of a synapse is equated with its efficacy; this is taken to be a measure of its stability. A model of structural plasticity is implemented on the SpiNNaker neuromorphic computation platform. It operates in real time and in conjunction with spike-timing dependent plasticity, and generates higher quality topographic maps compared to experiments not involving synaptic rewiring. These networks further benefit from the rewiring mechanism creating inhibitory connections that stabilise the spiking activity of the network. Further, the same model and architecture are used in the context of handwritten digit classification, and they show that the structural plasticity model is sufficient for neurons to learn the statistics of the input in an unsupervised fashion. Finally, neurons perform unsupervised motion classification through self-organisation. Neural responses become highly tuned and resemble those observed experimentally in Visual Cortex and Superior Colliculus: a highly asymmetric response evoked by motion. For this behaviour to arise, the networks are simulated for over 5 hours, modelling the development of these circuits from the point when no synapses exist until neurons reach their full synaptic capacity. SpiNNaker's real time operation and massive parallelism ensure simulation execution is tractable, and a sufficient number can be performed to obtain statistically significant results. The work described in this thesis aims to show that structural plasticity is an important and powerful learning mechanism which can now be incorporated seamlessly in Spiking Neural Network simulations executed on SpiNNaker.