This paper studies a spiking neuron circuit with a periodic triangular base signal. The circuit can output rich pulse-trains and the dynamics can be analyzed exploiting a piecewise linear one-dimensional pulse position map. Using two neurons we construct a pulse-coupled system whose dynamics can be integrated into the composite map of the pulse position maps of two neurons. The composite map is piecewise linear and we can analyze rich phenomena precisely. For example, periodic behavior of each neuron is changed into chaotic behavior and chaotic behavior of each neuron is changed into periodic behavior. These results provide basic information to construct flexible pulse-coupled neural networks.