How do neural networks learn to represent information? Here, we address this question by assuming that neural networks seek to generate an optimal population representation for a fixed linear decoder. We define a loss function for the quality of the population read-out and derive the dynamical equations for both neurons and synapses from the requirement to minimize this loss. The dynamical equations yield a network of integrate-and-fire neurons undergoing Hebbian plasticity. We show that, through learning, initially regular and highly correlated spike trains evolve towards Poisson-distributed and independent spike trains with much lower firing rates. The learning rule drives the network into an asynchronous, balanced regime where all inputs to the network are represented optimally for the given decoder. We show that the network dynamics and synaptic plasticity jointly balance the excitation and inhibition received by each unit as tightly as possible and, in doing so, minimize the prediction error between the inputs and the decoded outputs. In turn, spikes are only signalled whenever this prediction error exceeds a certain value, thereby implementing a predictive coding scheme. Our work suggests that several of the features reported in cortical networks, such as the high trial-to-trial variability, the balance between excitation and inhibition, and spike-timing dependent plasticity, are simply signatures of an efficient, spike-based code.