This paper studies dynamics of the bifurcating neuron circuit. Repeating integrate-and-firebehavior between a constant threshold and piecewise linear periodic base signal, the circuit canexhibit a variety of chaotic and super-stable periodic spike-trains. Deriving one-dimensional mapof spike phases, the circuit dynamics can be analyzed precisely. Especially, this paper analyzesa variety of super-stable periodic spike-trains and related bifurcation phenomena. Analysis ofthe circuit is important not only as the basic study of nonlinear dynamical systems, but also forengineering applications.Key Words : Spiking neurons, super-stable periodic orbits, bifurcation.