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This paper presents the algorithm and software developed for parallel simulation of spiking neural networks on multiple SpiNNaker universal neuromorphic chips. It not only describes approaches to simulating neural network models, such as dynamics, neural representations, and synaptic delays, but also presents the software design of loading a neural application and initial a simulation on the multi-chip SpiNNaker system. A series of sub-issues are also investigated, such as neuron-processor allocation, synapses distribution, and route planning. The platform is verified by running spiking neural applications on both the SoC Designer model and the physical SpiNNaker Test Chip. This work sums the problems we have solved and highlights those requiring further investigations, and therefore it forms the foundation of the software design on SpiNNaker, leading the future development towards a universal platform for real-time simulations of extreme large-scale neural systems.