Traditionally, EEG is understood as originating from the synchronous activation of neuronal populations that generate rhythmic oscillations in specific frequency bands. Recently, new neuronal dynamics regimes have been identified (e.g. neuronal avalanches) characterized by irregular or arrhythmic activity. In addition, it is starting to be acknowledged that broadband properties of EEG spectrum (following a 1/f law) are tightly linked to brain function. Nevertheless, there is still no theoretical framework accommodating the coexistence of these two EEG phenomenologies: rhythmic/narrowband and arrhythmic/broadband. To address this problem, we present a new framework for EEG analysis based on the relation between the Gaussianity and the envelope of a given signal. EEG Gaussianity is a relevant assessment because if EEG emerges from the superposition of uncorrelated sources, it should exhibit properties of a Gaussian process, otherwise, as in the case of neural synchronization, deviations from Gaussianity should be observed. We use analytical results demonstrating that the coefficient of variation of the envelope (CVE) of Gaussian noise (or any of its filtered sub-bands) is the constant root 4/pi - 1 approximate to 0.523, thus enabling CVE to be a useful metric to assess EEG Gaussianity. Furthermore, a new and highly informative analysis space (envelope characterization space) is generated by combining the CVE and the envelope average amplitude. We use this space to analyze rat EEG recordings during sleep-wake cycles. Our results show that delta, theta and sigma bands approach Gaussianity at the lowest EEG amplitudes while exhibiting significant deviations at high EEG amplitudes. Deviations to low-CVE appeared prominently during REM sleep, associated with theta rhythm, a regime consistent with the dynamics shown by the synchronization of weakly coupled oscillators. On the other hand, deviations to high-CVE, appearing mostly during NREM sleep associated with EEG phasic activity and high-amplitude Gaussian waves, can be interpreted as the arrhythmic superposition of transient neural synchronization events. These two different manifestations of neural synchrony (low-CVE/high-CVE) explain the well-known spectral differences between REM and NREM sleep, while also illuminating the origin of the EEG 1/f spectrum.