In this work gravicommunication (GC) is introduced, as a new form of communication (different from the gravitational waves), which involves gravitons (elementary particles of gravitation). This research is based on quantum modification of the general relativity. The modification includes effects of production /absorption of gravitons, which turn out to have small, but finite mass and electric dipole moment. It is shown, that such gravitons form the dipole Bose-Einstein condensate, even for high temperature. The theory (without fitting parameters) is in good quantitative agreement with cosmological observations. In this theory we got an interface between gravitons and ordinary matter, which very likely exist not only in cosmos, but everywhere, including our body and, especially, our brain. Subjective experiences are considered as a manifestation of that interface. A model of such interface is presented and some new experimentally verifiable aspects of natural neural systems are considered. According to the model, GC can be superluminal, which will solve the problem of quantum entanglement. Probable applications of these ideas include health (brain stimulation), new forms of communication, computational capabilities, energy resources and weapons. Potential social consequences of these developments can be comparable with the effects of discovery and applications of electricity. Some developed civilizations in the universe may already master gravicommunication (with various applications) and so should we.