Discriminating between physiological waves, linked to functional brain properties, and pathological waves, due to brain malfunction, is problematic, e. g., for high-frequency oscillations (HFO) as high $γ$ waves. Brain rhythms observed in EEG can also be observed in excitatory/inhibitory (E/I) NN models. Such models allow to study the relationship between waves and neuronal circuit functions, which could lead to fundamental insights to discriminating waves. To address this and other questions, we explore the emergence of high gamma rhythms in an E/I balanced neural population of integrated and fire neurons with short-term synaptic plasticity. This model generates a rich repertoire of EEG-like rhythms, including high-frequency excitatory and inhibitory oscillations. Using the integrated information decomposition framework (Phi-ID), we explore the information dynamics of the network and its relationship with excitatory and inhibitory activity and high-frequency rhythms. In regions where high gamma rhythms emerge only in the excitatory population, we see informational properties in the network more favorable for computation and processing information, corresponding then to functional regimes. However, regions where high gamma rhythms also emerge in the inhibitory population present lower mutual information in general, and the system becomes less predictable, a fact that can be linked to a less functional or ``pathological" regime. In the second case, we observe that both excitatory and inhibitory populations oscillate with the same dominant frequency, which is higher than in the first case. Higher frequency oscillations and synchronization are commonly associated with epileptic seizures so adopting an information dynamics approach could help differentiate between high-frequency oscillations related to cognitive functions from those related to neuronal disorders such as epilepsy.