EEG signals emerge from the collective behaviour of large neuronal aggregates and betrays the information processed by neocortex. This electrophysiological collective activity varies with the brain function. Thus, one can ask whether there exists any indication of that neuronal activity in the EEG record. In this work this question is considered in the particular case of sleep/awake EEG s. To this aim, the concept of lacunarity is proposed as a tool to analyse the texture of the EEG samples. From the resulting lacunarity profiles an index is defined which represents a characteristic time of each phase.The samples analysed corresponds to 30 seconds epochs from polysomnographic night records. Essential details for the computation of lacunarity patterns and the propounded index are given. The mean values of characteristic times (in seconds) are: 0.43 (phase I); 0.73 (phase II); 1.12 (phase III/IV); 0.65 (REM); 0.12 (relaxed wakefulness).With this criterium, the awake state is clearly distinguished from phase I and REM sleep, whereas REM sleep comes out to be similar to phase II. Finally, statistical analysis of results suggests the possibility to interpret this index as a complementary tool for reading polysomnographies as well as its application to different physiological or pathological situations of EEG records.