A methodology is introduced for a neural network with local memory called a multilayered local output gamma feedback (LOGF) neural network within the paradigm of locally-recurrent globally-feedforward neural networks. It appears to be well-suited for the identification, prediction, and control tasks in highly dynamic systems; it allows for the presentation of different timescales through incorporation of a gamma memory. A learning algorithm based on the backpropagation-through-time approach is derived. The spatial and temporal weights of the network are iteratively optimized for a given problem using the derived learning algorithm. As a demonstration of the methodology, it is applied to the task of power level control of a nuclear reactor at different fuel cycle conditions. The results demonstrate that the LOGF neural network controller outperforms the classical as well as the state feedback-assisted classical controllers for reactor power level control by showing a better tracking of the demand power, improving the fuel and exit temperature responses, and by performing robustly in different fuel cycle and power level conditions.