REGIONAL CEREBRAL BLOOD FLOW (RCBF RCBF): Regional cerebral blood flow monitoring may help boost the validity of the regional localization theory, in which specific parts of the brain control specific functions.. measure of the blood flow in a specific area of the brain, using positron emission tomography (PET scans) and/or single photon emission computed tomography.
If a dog goes on his back to you he is accepting you as his master. This is part of the answer but here's a little more in-depth version as reported by USA Today. In general, dogs lie on their back for three reasons.
Dogs are actually very comfortable with sleeping on a hard floor. They just want to be comfortable, and prefer the coolest spot they can find. The reason dogs can sleep anywhere is they have no sense of right or wrong, and (unlike people) they have no regrets, and can easily forget what they did five minutes ago.