An asynchronous sequential logic spiking neuron is an artificial neuron model that can exhibit various bifurcations and nonlinear responses to stimulation inputs. In this paper, a pulse-coupled system of the asynchronous sequential logic spiking neurons is presented. Numerical simulations show that the coupled system can exhibit various lockings and related nonlinear responses. Then, theoretical sufficient parameter conditions for existence of typical lockings are provided. Usefulness of the parameter conditions is validated by comparing with the numerical simulation results as well as field programmable gate array experiment results.