The neurons are densely interconnected and fire in concert in resonance with each other to create a single unified activation wave throughout the cerebral cortex. This unified activation wave is hypothesized to be the reason behind the unified cognitive experience that arises in us.
I don't think that it is at all necessary to find an "ever-present" neural-physical behavioral correlate to the self. The self is like the center of mass of an object...the center of mass just makes it easier for us to calculate the behavior of that mass. In a vector drawing of the object in motion an arrow will be drawn from the "center" of the object in the direction of net motion depending on the various forces applied. In the same way, our central self is a convenience for referencing this body as a relatively autonomous and somewhat free actor on the world stage at varying levels. Our brains support a range of voices more or less inter-coordinated in part based on that self-identified center.
What determines our uniqueness can be seen as pointed to without being defined as the following:
- Our personal name
- Our personal relationships
- Our cultural environment including rights and responsibilities
- Our personal experience including the sensory awareness of our bodies' position within the world we sense
In short all these things define that arrow which we continually re-interpret in ourselves and through the feedback from the environment and especially others like ourselves. We are, as Daniel Dennett has put it, the center of our own narrative gravity. And as anyone who has meditated can say we are constantly writing our own novels on a sub-conscious level.
Of course, this is an impersonal way to talk about one's self. Within the context of our evolutionary and cultural heritage we are given the privilege of referring to ourselves as equal individuals with unique and relevant experiences worth sharing. We talk about ourselves in the first person as a sort of social contract which says, "You have the right and privilege to assume you are important".