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A novel electrical circuit spiking neuron model that has a piece-wise-constant vector field with a state-dependent reset is presented. It is shown that the model exhibits six kinds of border-collision bifurcations, where their bifurcation sets are derived and are summarized into two parameter diagrams. Then, using the diagrams, systematic synthesis procedures of the presented model so that it can reproduce four kinds of bifurcation scenarios that are typically observed in standard neuron models are presented. It is shown that the model can reproduce the bifurcation scenarios as well as corresponding nonlinear response characteristics observed in model and biological neurons. Occurrences of typical neuron-like bifurcations are confirmed by experimental measurements.