Feelings of spiritual transcendence and oneness with God and/or the universe can be attributed to a purely physiological phenomenon. There is a region of the brain that neuroscientists call the orientation association area (OAA) that orients us in physical space. It is this area that controls our perception of ourselves in relation to the objects around us and differentiates between "me" and "not me." People with damage to this part of the brain have trouble navigating through even familiar spaces, and will frequently bump into things that a typical person will easily avoid.
It has been found that meditation and some drugs can cause activity in the OAA to decrease. As its function of differentiating between a person and the rest of the world decreases, the person feels a sense of connection to things outside himself. Damage to the OAA, in addition to bumping into things, can cause people to spontaneously experience feelings of spiritual transcendence.
This is a physiological explanation for the feelings people may experience during davening. Davening is a form of meditation, and can at times cause the OAA to go into hibernation. The spiritual feelings people experience are likely not the result of a connection with the divine, but are the result of a malfunction in the brain that blurs the boundaries between "me" and "not me."