Thank you for asking. I will try.
To understand the Advaita worldview, I will request that you allow a possibility that the deep sleep state is not lack of consciousness but is consciousness of lack (lack of subject-object division and of space-time-objects). Furthermore, it helps to allow the possibility that consciousness is like an unlimited ocean on which individual selves are like individual surface waves. In this worldview, in deep sleep, the individual wave goes down into the body of the ocean and loses individuality temporarily.
Deep sleep is characterised by lack of desire and thus lack of subject and a corresponding world. The state is devoid of space-time-objects. Being devoid of a body-mind, it is a state of pure bliss (although we remember of the bliss only upon waking). According to Vedanta, deep sleep is not a state of unconsciousness but is a state of ignorance. It is a state of the pure non-dual ground of consciousness.
In deep sleep, we are pure potential awareness — nothing is known in deep sleep since the realm is non-dual; it being devoid of any sort of contrast of sound, touch, taste, sight, or smell. It is pure dense potential awareness. From the pure potential awareness of the deep sleep arises the subtle dream body to which the mind attaches and experiences a subtle world. This manifestation is entirely mental, devoid of input from senses. But, while in a dream, our experiences of pain or joy are not unreal. Only on waking, we come to know the unreality.
Again when dream transitions to waking, senses open up and in association with mind, it shows us a gross world and a gross body that is used for experiencing inputs from five senses — sound, smell, sight, taste, and touch. The touch function is killing, it delineates a boundary as if — necessary for sexual enjoyment etc. — and this leads to the idea “I am this body”.
There are the physical expressions of the states of waking, dreaming, and sleep of a person's brain that a third-party observer can record (as the brain waves) employing EEG. In waking and dreaming, the beta waves dominate. In deep sleep, the delta waves dominate (remember that these are the waking state observations of a third person and not the actual first-person subjective consciousnesses of waking, dreaming, and sleeping states).
What are Brainwaves ? Types of Brain waves | EEG sensor and brain wave – UK
We can influence waking state (and possibly dream state to some extent). But we cannot do anything in a deep sleep state since the ego-intellect-mind is absorbed and is practically absent. But from the waking state, we can follow prescribed methods of mediation: yoga-TM, Zen, Mindfulness etc. and attain blissful waking sleep state, also known as Samadhi. The ultimate attainment is loss of subject-object distinction with full consciousness present. Under this condition, the reality will be known as 'unbounded ocean' like rather than 'interacting billiard balls' like.
Different types of mediations generate different kinds of brain waves: gamma, alpha1, theta, or delta. All these expressions are different from the common everyday waking-dreaming beta brain wave type. In deep sleep and in the rare ultimate state of samadhi meditation (characterised by total loss of subject-object distinction), delta waves predominate. In other intermediate meditative states, the brain emits theta, alpha 1 or gamma waves.
It is now well established, thanks to advances in neuroscience, that we control the brain and not the opposite. I link below two informative videos of well-respected scientists.
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