The nervous system may be simulated as a network of model neurons as a means to understand the function of the brain. Complex as the mammalian nervous system is, such simulations of any significant scale are computationally and energetically expensive. SpiNNaker is a computer architecture designed to advance the feasible scale of neural tissue models using fifty thousand chips, each containing eighteen low-power processors, to model one billion neurons and one trillion synapses in real-time. This dissertation demonstrates the success of prototype hardware with detailed models of the rodent somatosensory cortex. Simulations are built from neuroanatomical data on a host computer using a simple declarative library of functions, and are executed on SpiNNaker atop an event-driven programming interface that neatly abstracts the intricate details of the machine. Comparisons with reference simulators show that SpiNNaker correctly reproduces established results, and power readings report that each chip draws just one watt during execution. A model of the whisker barrel, derived from the literature, exhibits key responses to simulated stimuli, and a model of the wider barrel cortex, comprising 10^5 neurons and 7*10^7 synapses, demonstrates real-time, massively parallel simulation across 360 processors on 23 chips. Ultimately, SpiNNaker is shown to be an effective architecture for the correct and efficient simulation of neural tissue.