In this paper, approximate optimal distributed control schemes for a class of nonlinear interconnected systems with strong interconnections are presented using continuous and event-sampled feedback information. The optimal control design is formulated as an N-player nonzero-sum game where the control policies of the subsystems act as players. An approximate Nash equilibrium solution to the game, which is the solution to the coupled Hamilton-Jacobi equation, is obtained using the approximate dynamic programming-based approach. A critic neural network (NN) at each subsystem is utilized to approximate the Nash solution and novel event-sampling conditions, that are decentralized, are designed to asynchronously orchestrate the sampling and transmission of state vector at each subsystem. To ensure the local ultimate boundedness of the closed-loop system state and NN parameter estimation errors, a hybrid-learning scheme is introduced and the stability is guaranteed using Lyapunov-based stability analysis. Finally, implementation of the proposed event-based distributed control scheme for linear interconnected systems is discussed. For completeness, Zeno-free behavior of the event-sampled system is shown analytically and a numerical example is included to support the analytical results.