The atman in Hindu philosophy is the univeral Self; in Buddhism the concept of anatta refers to the non-Self. These two terms, by literal translation and conceptual understanding, are opposites (Self vs. non-Self). Yet experientially and philosophically I've come the conclusion they are the same. As is usually the case, this observation isn't unique to me.
When one meditates, the point is to silence the aspect of the self (you can call it the ego) that is the the chattering monkey, the part that perceives, senses, feels, and thinks. Behind this thinking, perceiving self is the real Self. Eckhart Tolle refers to it as a "space" in between or the spatial consciousness (as opposed to the doing consciousness). When you meditate, you instinctively experience this real Self when your thinking, perceiving self is quietened, which leads to your awakening. This non-Self, the space between the self, is the same as the atman, i.e., the true universal Self. While we don't know for sure, all those who meditate talk about the same experience. Each self is different from another. One self may experience pain for pleasure and the other may wince at it, but the Self experienced by everyone is identical.
The space in-between is the univeral substrate for reality. In this manner, the connection between the metaphysical (or "spiritual") dimension and the physical worlds are manifest, where there is a biological rationale for the evolution of the observing mind-made self. (If you want to label it, you can call it the ego.)
A final point I want to make is that unlike the biological basis of the self which is a hypothesis that can be falsified, this atman-anatta distinction is about experience, not about reading/watching someone like me discuss in intellectual terms or "proving" their equivalence. As I write elsewhere, meta-thinking or thinking about thinking isn't experiential. When you meditate, you will have this experience: The space that is the passive observer, which is not your Self, is the atman.