Motivated by problems arising in neuroscience, we investigate properties of a class of nonlinear finite dimensional systems whose characteristic responses are pulselike traveling waves. With a view toward investigating their signal processing capabilities, we develop suitable input-output structures for these systems and show that they are, in fact, Hamiltonian relative to a suitable symplectic structure. The Hamiltonian structure provides a way to keep track of the energy flow and to study the effects of parasitic dissipation. In this framework, we develop an effective theory of interconnection, network structures, etc. This leads to a signal processing formalism in which the primary mode of communication is through pulses.