The neocortex is the most recently evolved part of the mammalian brain and enables the intelligent, adaptable behaviour that has allowed mammals to conquer much of planet earth. The human neocortex consists of a thin sheet of neural tissue containing approximately 20*10^9 neurons. These neurons are connected by a dense network of highly plastic synapses whose efficacy and structure constantly change in response to internal and external stimuli. Understanding exactly how we perceive the world, plan our actions and use language, using this computational substrate, is one of the grand challenges of computing research. One of the ways to address this challenge is to build and simulate neural systems, an approach neuromorphic systems such as SpiNNaker are designed to enable.The basic computational unit of a SpiNNaker system is a general-purpose ARM processor, which allows it to be programmed to simulate a wide variety of neuron and synapse models. This flexibility is particularly valuable in the study of synaptic plasticity, which has been described using a plethora of models. In this thesis I present a new SpiNNaker synaptic plasticity implementation and, using this, develop a neocortically-inspired model of temporal sequence learning consisting of 2*10^4 neurons and 5.1*10^7 plastic synapses: the largest plastic neural network ever to be simulated on neuromorphic hardware. I then identify several problems that occur when using existing approaches to simulate such models on SpiNNaker before presenting a new, more flexible approach. This new approach not only solves many of these problems but also suggests directions for architectural improvements in future neuromorphic systems.