For energy-efficient computation in specialized neuromorphic hardware, we present the Spiking Neural Coding Network, an instantiation of a family of artificial neural models strongly motivated by the theory of predictive coding. The model, in essence, works by operating in a never-ending process of "guess-and-check", where neurons predict the activity values of one another and then immediately adjust their own activities to make better future predictions. The interactive, iterative nature of our neural system fits well into the continuous time formulation of data sensory stream prediction and, as we show, the model's structure yields a simple, local synaptic update rule, which could be used to complement or replace online spike-timing dependent plasticity. In this article, we experiment with an instantiation of our model that consists of leaky integrate-and-fire units. However, the general framework within which our model is situated can naturally incorporate more complex, formal neurons such as the Hodgkin-Huxley model. Our experimental results in pattern recognition demonstrate the potential of the proposed model when binary spike trains are the primary paradigm for inter-neuron communication. Notably, our model is competitive in terms of classification performance, can conduct online semi-supervised learning, naturally experiences less forgetting when learning from a sequence of tasks, and is more computationally economical and biologically-plausible than popular artificial neural networks.