If you take a Jungian point of view, it neither starts nor ends anywhere. :-) While it remains for us all (or - nearly all, at least!) to find those infinite connections, it sounds like something worth exploring. Getting funding for it appears hopeless, though.
This is a good question with many possible answers.
I agree with @Costas Drossos: the mind starts at -\infty and ends at +\infty!
This answer is an example of cutting edge transcendental philosophy. From that point of view, the mind intersects with the infinite. It is not bound physically. It transcends the physical world rather easily either imaginatively or mathematically. The beauty of the mind is its ability to conceptualize distant frontiers in the past and in the future. It belongs to an open interval with -\infy as lower bound and +\infy as an upper bound. And this is outside a Cantorian framework. In other words, the mind (its scope, its life and its extent) is boundless.
The mind begins within us but with an infinite expanse and immense power in which its effect is limited by our imagination to project and think in higher orders. Our mind ends when we biologically stop to live.
The mind is the dark matter of our cognition power, that fuels and replenishes our capacity of abstraction and thinking. I will say the beginning of the mind as a big bang and its end as a big collapse.
Mind begins and ends with consciousness. But, this consciousness is not limited to an individual. The skills that we have evolved through millions of years, are all consolidated and embedded in the mind of the unborn fetus, that come into play when it comes out into the world and consciously interacts with the environment.
A part of the mind can also certainly belong to groups and societies. We don't exist in isolation and much of our mental processes are implicitly determined by external influences. In more pronounced case, people can fall prey to groupthink and mob psychology.
A mind has no tangible beginning or end. It is largely defined through our consciousness of its existence.
Interesting question, with indeed many possible different answers.
1- If we have free will, then mind begins, as per the Free Will Theorem, at the elementary particle level. Any individual mind is then embodied within some evolving wave function, and your statement that a mind is wholly circumscribed inside a physical body becomes then untenable. Its spacelike boundaries are those of the associated wave function: infinity.
2- If we do not have free will, and our impression of free will is an illusion, then it becomes harder to ascertain where mind begins. In that case however the very concept of mind becomes moot - we're reduced, however unwittingly, to mindless robots. Owing to the limitations imposed by the Bekenstein bound, the mind then cannot possibly be infinite at any instant of time.
If we take the reasonable view of wave function realism, we almost certainly end up in case 1 above. It's not entirely clear, to me at least, how we could have both a time-dependent wave function attached to a mind and no free will.
The mind begins & ends in its owner. No one can borrow a mind of another one & add it to his mind in a physical sense. There is a difference between the mind & the perception of the mind. The perception is limited by the powers of senses which send signals, about objects, to a mind which has previous limited information upon which judgment is made. Since what man knows is far less than what is unknown in dimensions & spectrum, then there will be limitations whatever imaginations are there.
Mind begins with a preset code as like computer ROM based on certain set of beliefs that has been nurtured to it. This could be a preset for the new born from the divine creator which changes based on our learning and what the mind acquires and it ends where we want our mind to wonder based on our desires, wants, understanding, reactions... The beginning and ending of the mind is a continuous process and the answer to it is the answer to infinity (as we define, another limitation of human mind) Its like the hub of the Brain...
A great mind do never ends, it is a continuous process of heritage for others. From several centuries many philosophers, scientists, researchers etc... do have let to us their singular minds from ideas and discoveries; The mind begins with all this past heritage of human knowledge added to some particularities of cultural or sociological behavior that by time the smart mind could merge them or not as part of useful or intelligent data. The mind should be exercised to more logic.
1)Greek Fire. 2) Roman Cement. 3) Silphium (Roman wonder herbal drug for birth control). 4) Damascus steel (very strong metal which no one can manufacture today). 5) Antikythera Mechanism (Greek bronze machine which no one can duplicate today). 6) Nepenthe (Greek primitive anti-depressant drug). 7) The art of embalming as was done in ancient Egypt. 8) The lions of Alhambra which work without pumping machines . No one knows how kinetic energy transforms into potential energy & vice versa in these lions. 9) Building pyramids with huge stones using no lifting machines in old Egypt. 10) Craft of digging stones to build cities with specific colors & with durability for centuries (such as Petra in Jordan). These examples give clear evidence that the mind begins & ends. Had there been coherent accumulation of knowledge through centuries " with continuous mind flow" , then we would have been now with more progress than the old Egyptians or Romans or Greeks or Syrians or Nabataeans.
Both our genetic heritage and our cultural heritage give transgenerational continuity to the human mind. The mind, in a general sense, emerged as our species evolved and will continue to evolve, or become extinct, depending on the success or failure of our collective strategies for survival. Our individual minds are part of this continuum, of course; they emerge as our nervous systems develop in our mothers' wombs and fade when our bodies cease to function.
Nizar's examples point out the danger of taking too literally the idea of an extracorporeal collective mind. Our individual minds, nevertheless, reflect millions of years of evolution, while assimilating and building on culturally transmitted knowledge, values, and ways of thinking.
As @Nizar mentioned, some lost knowledge and technologies, are generally the consequences of some human stupidity, cupidity, ego, terror...etc.... Many invasions and wars destroyed several libraries and records of human mind history... and many stupid and egoistic political power ships decreased the intelligence ability of their citizens to slavery, loosing in the same time their power and the mind creative history and knowledge of their people ...... However the mind usually evolves from its genetics and social heritage to new or improved knowledge. From the stone age...to nowadays numerical, laser, nano....etc ...age, human have constructed civilizations of knowledge (savoir et savoir faire). When the stupidity is fight, human intelligence is able to innovate and create knowledge, but when the stupidity becomes the maîtresse even the history of human mind could be lost