Understanding the operations of neural networks in the brain requires an understanding of whether interactions among neurons can be described by a pairwise interaction model, or whether a higher order interaction model is needed. In this article we consider the rate of synchronous discharge of a local population of neurons, a macroscopic index of the activation of the neural network that can be measured experimentally. We analyse a model based on physics' maximum entropy principle that evaluates whether the probability of synchronous discharge can be described by interactions up to any given order. When compared with real neural population activity obtained from the rat somatosensory cortex, the model shows that interactions of at least order three or four are necessary to explain the data. We use Shannon information to compute the impact of high-order correlations on the amount of somatosensory information transmitted by the rate of synchronous discharge, and we find that correlations of higher order progressively decrease the information available through the neural population. These results are compatible with the hypothesis that high-order interactions play a role in shaping the dynamics of neural networks, and that they should be taken into account when computing the representational capacity of neural populations.