Neurons practically never die, so the DNA itself will not directly reflect what a decades old braincell stores, though some mutations might influence how or if at all a certain neuron stores something. But since these mutations are random they are closer to defects (the article draws a connection to psychiatric diseases) than features.
But DNA can also be methylated, which is something that also happens in neurons and might be involved with actual memory.
And if you ever want to digitize a brain, then you might have to grab the DNA of 100 billion neurons as well, and there straightforward mutations are easier than just a somewhat transient methyl group here and there.
I read a research paper a couple years ago making an argument that neurons were using DNA as a kind of Turing tape for information storage. Very interesting but I can't seem to find it now.
