Neuromorphic computing systems simulate spiking neural networks that are used for research into how biological neural networks function, as well as for applied engineering such as robotics, pattern recognition, and machine learning. In this paper, we present a neuromorphic system based on an asynchronous event-based hardware platform. We represent three algorithms for implementing spiking networks on our asynchronous hardware platform. We also discuss different trade-offs between synchronisation and messaging costs. A reinforcement learning method known as Rewardmodulated STDP is presented as an online learning algorithm in the network. We evaluate the system performance in a single box of our designed architecture using 6000 concurrent hardware threads and demonstrate scaling to networks with up to 2 million neurons and 400 million synapses. The performance of our architecture is also compared to existing neuromorphic platforms, showing a 20 times speed-up over the Brian simulator on an x86 machine, and a 16 times speed-up over a 48-chip SpiNNaker node.