: The interest in studying the brain electrical activity as a function of the development of intelligence has been spurred by the need to understand how the brain responds to environmental information. The description of sleep in mentally retarded children reveals deviant patterns of the EEG-spindles and of the eye movement activity (REM sleep) when compared to normal children. The patterns may be considered as a valuable index of mental function. According to experimental evidence, the distribution of the eye movements of sleep appears either as random or ordered. The latter are altered in the mentally handicapped in whom the appearance out of chaos, of the order which is needed for intelligence and memory to function, is altered. The sleep signs are redundant as from birth. Their pattern is also related to the psychomotor development of the infant. If their distribution remains random, or appears in long uninterrupted sequences of waves as in epilepsy, intelligence does not develop. A similar strategy appears to function in the foetus when nature organizes the structures that will lead to the development of intelligence. The eye movement patterns of sleep change in the pregnant women as a function of term and resemble those of premature babies of a similar gestational age. They also change as a function of the menstrual cycle and more generally as a function of age. The hypothesis that attention is the diurnal equivalent of REM sleep is discussed. Attempts at modelling the eye movement patterns of REM sleep as a function of near zero gravity environments have been made. 1) By means of a Montecarlo simulation using the semi Markov model during the Spacelab 1 flight. 2) With the method of the single and multiple g-phase transition analysis of the strange attractor dimension (d) during parabolic flights. The implication of the latter for the neural processes involved in learning is that the central nervous system can preserve intact, from input to output, over a period of several days, all the information it receives 3) The relation between spindles and eye movements has also been viewed by a quantum approach which is another medium between the information and the way of describing it.