Controlling a population of interconnected neurons using extrinsic stimulation is a challenging problem. The challenges are due to the inherent nonlinear neuronal dynamics, the highly complex structure of underlying neuronal networks, the underactuated nature of the control problem, and adding to these is the binary nature of the observation/feedback. To meet these challenges, adaptive, learning-based approaches using deep neural networks and reinforcement learning are potentially useful strategies. In this paper, we propose an approximation based learning framework in which a model for approximating the input-output relationship in a spiking neuron is developed. We then present a reinforcement learning scheme to approximate the solution for the Bellman equation, and to design the control sequence to achieve a desired spike pattern. The proposed strategy, by integrating the reinforcement learning and system theoretic approaches, provides a tractable framework to design a learning control network, and to select the hyper parameters in deep learning architectures. We demonstrate the feasibility of the proposed approach using numerical simulations.