This paper studies spike-train dynamics of the bifurcating neuron and its pulse-coupled system. The neuron has periodic base signal that is given by applying a periodic square wave to a basic low-pass filter. As key parameters of the filter vary, the systems can exhibit various bifurcation phenomena. For example, the neuron exhibits period-doubling bifurcation through which the period of spike-train is doubling. The coupled system exhibits two kinds of (smooth and non-smooth) tangent bifurcations that can induce “chaos + chaos = order”: chaotic spike-trains of two neurons are changed into periodic spike-train by the pulse-coupling. Using the mapping procedure, the bifurcation phenomena can be analyzed precisely. Presenting simple test circuits, typical phenomena are confirmed experimentally.