Modeling brain dynamics to better understand and control complex behaviors underlying various cognitive brain functions are of interests to engineers, mathematicians, and physicists from the last several decades. With a motivation of developing computationally efficient models of brain dynamics to use in designing control-theoretic neurostimulation strategies, we have developed a novel data-driven approach in a long short-term memory (LSTM) neural network architecture to predict the temporal dynamics of complex systems over an extended long time-horizon in future. In contrast to recent LSTM-based dynamical modeling approaches that make use of multi-layer perceptrons or linear combination layers as output layers, our architecture uses a single fully connected output layer and reversed-order sequence-to-sequence mapping to improve short time-horizon prediction accuracy and to make multi-timestep predictions of dynamical behaviors. We demonstrate the efficacy of our approach in reconstructing the regular spiking to bursting dynamics exhibited by an experimentally-validated 9-dimensional Hodgkin-Huxley model of hippocampal CA1 pyramidal neurons. Through simulations, we show that our LSTM neural network can predict the multi-time scale temporal dynamics underlying various spiking patterns with reasonable accuracy. Moreover, our results show that the predictions improve with increasing predictive time-horizon in the multi-timestep deep LSTM neural network.