I know that many new studies have been done recently regarding the relationship between the human eye and the brain and how we are able to create a seamless experience with them, but I am curious as to whether a study has ever been done to capture a moment of what the human brain receives from the eye. Specifically, I am curious as to whether anyone has ever captured, digitally or otherwise, what we see when our eyes are closed.

No. This is because all we recieve are electro-chemical impulses and it is not possible to capture, digitally or otherwise, the ultimate perception. People have managed to capture the image that the lens projects to the retina though. It's surprisingly poor (and upside down), which is an indicator of the amount of work the brain does to produce the finished percept.

 

As for the relationship between the eye and the brain, the eye (or the retina, to be precise) is actually part of the brain.

Where did you see these pictures taken through the retina?

There are no pictures taken through the retina to my knowledge. The images were taken through the lens & cornea by a camera that had been placed where the retina would be, had the eye still been in the body. I saw them in a book, although I can't remember which one now. It was years ago.