The claim isn't that the cells are constantly replaced, but that the atoms are; see e.g. this blog post as an example of the claim being made. [EDITED to add:] More specifically, an example where the claim is made and "new cells" and "new atoms" are explicitly distinguished.
I have found this reddit discussion:
https://www.reddit.com/r/askscience/comments/1q16hw/im_32_is_there_a_single_cell_or_even_molecule/
and there are some pro and contra comments in there about atoms and molecules of the nerve cells. Probably I need a lot more reading on this subject , txs for the guidance.
BR
http://www.technologyreview.com/view/541311/the-false-science-of-cryonics/
Michael Hendricks is a neuroscientist and assistant professor of biology at McGill University.