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A novel rotate-and-fire digital spiking neuron is presented. The digital neuron is a wired system of shift registers and thus it is suited to on-chip learning unlike many other analog spiking neuron models. By adjusting the wiring pattern among the registers, the digital neuron can generate spike trains with various spike patterns and can exhibit related bifurcations. A discrete-continuous hybrid map, which describes the neuron dynamics without any approximation, is derived analytically. Using the hybrid map, it is shown that the digital spiking neuron can mimic typical bifurcation phenomena and various nonlinear responses of biological neurons.