The Buddhist body-mind model is similar to the hardware-software model of how a computer or robot works.
Hardware and software (of a computer or robot) are both machines or automatons on their own right and also depend on each other to continue functioning. It is similar in the case of the mind and the body. Even if the exact mechanism is not the same, the general principle is the same.
Just as how a robot or computer (hardware-software) does not require a soul or Self to function, similarly the body-mind also does not require a soul or Self to function. The self is just an idea of the mind.
Please note from DN 2:
“When his mind is thus concentrated, pure and bright, unblemished,
free from defects, malleable, wieldy, steady and attained to
imperturbability, he directs and inclines it to knowledge and vision.
He understands thus: ‘This is my body, having material form, composed
of the four primary elements, originating from father and mother,
built up out of rice and gruel, impermanent, subject to rubbing and
pressing, to dissolution and dispersion. And this is my consciousness,
supported by it and bound up with it.’
The last statement shows the dependence of mind on body to continue functioning ("supported by it") and also the inter-dependence of mind and body with each other ("bound up with it").
From here, I quote the Buddhist teacher Stephen Batchelor who describes this operation as a fluidic stream of events:
I would argue that there is really no present moment. The present
moment is one of these things that Buddhists have become terribly
attached to. If you think about it, you try to find the present
moment, you will never find anything. The present moment is actually
just a concept; it can be a very useful, strategic concept … I will
often say when instructing in meditation, “Stay in the present
moment.” But I don’t mean by that, try to find this elusive thing
called the present moment and stay in it. It’s basically a way of
saying, Don’t get caught up in the unknown future. Don’t get caught up
in reminiscing about the past. But confront the situation at hand. And
the situation at hand is always unfolding. It’s fluid. It’s like
water, it’s like a stream. Things are constantly impacting your
senses, constantly bubbling up in your thoughts, constantly emerging
as emotions and feelings; it’s always in motion, it’s moving. And it’s
such a mobile experience that notion of ‘The present moment’ really
has no place there. There’s no point really. So, every situation that
occurs, and you could call it ‘at the present moment’ if you wish, but
it’s basically an unfolding of events that is calling forth an
appropriate response. And in this sense, it’s always in time.
A river looks like it flows uninterruptedly, just like the mind. But the river is composed of moving water molecules - each time you look at a flowing stream, you are looking at a different set of water molecules.
Similarly, the movement of the mind involves arising and ceasing cittas - so fast, that they appear uninterrupted.
Now scientific studies are also validating this - please see this answer for details.