I'm not 100% sure what you're asking, so apologies if I have misunderstood.

First, there were reports (sorry, I googled and can't find the links) that there were mysterious proteins at the junction of a synapse and a neuron. It was suggested at the time that synapses were not permanent, as often thought, but regularly replaced, with these proteins telling the neuron what to do to actually rebuild a synapse. As I said, I can't find the cite or a link to this theory, so I can't offer evidence of whether it is correct or whether it was falsified. What I can say is that this may be part of the storage.

http://www.hopkinsmedicine.org/news/media/releases/neurons_constantly_rewrite_their_dna
http://www.ncbi.nlm.nih.gov/pubmed/16879837
http://www.salk.edu/news-release/study-finds-a-patchwork-of-genetic-variation-in-the-brain/
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3083085/

What has been repeatedly verified is that each neuron has unique DNA. It has been reorganized and then modified. Systematically. Different regions show different types of modification. Further, the brain has extensive mechanisms for repairing changes due to damage that do not repair these other modifications. There are far more changes (running into the thousands) than there are types of neuron, but if you multiply the number of neuron types by the maximum number of synapses, you get a number that is about right to describe what the neuron is remembering, what it is all about.

The obvious conclusion is that the DNA must be being used as storage of the state of the neuron. Unexpected changes would lead to all kinds of new synapses forming, so neurons would want to stop that.

I'm not convinced neurons don't divide, juvenile neurons must for the brain to build and there simply aren't enough stem cells to explain the structural changes observed. If neurons divide, they can preserve their state from one generation of brain cell to another. If this is the case, DNA would be the hard drive. The brain would back itself up onto an 87-billion drive striped-mode RAID array inside its own genetic code.

This could be tested, although it would take a while. If all you need to replace a brain is the DNA, then if you took a group of adaptive brains, mapped their connectome in full, then sequenced each neuron in each, you should see a perfect 1:1 relationship between each bit of connectome and the DNA in the neurons. Hard-coded brains wouldn't tell you much.

If I am correct, that this is the backup store, then things get far more lively. Tracing synapses is hard (there's an online project where you can give it a go with actual brain cells - my accuracy was horrible). Sequencing DNA is easier, reliable and faster. If all you need to define the brain is the set of changes, it's a lot less data than the full genome. (The whole human genome consists of about 3 billion pairs of nucleic acid, so a few thousand changes is about one millionth as much in terms of data volume, whether the changes are mutations or wholesale rearranging of genes.)

The rest of the DNA likely does a lot that is important and the changes may well turn other bits of the DNA on and off. But that part of the data is more-or-less fixed.

Understanding that reorganization will, I suspect, be key to the Human Brain Project and the Connectome Project. It may also be key to repairing the brain, as DNA from cells crushed into oblivion by tau protein plaques might be protected by those plaques. If you could recover it and insert it into lab-grown cells, those cells should try to reform the connections of the original cells.

http://tbistemcellstudy.ucsf.edu/
http://www.stemedica.com/info/allogeneic-adult-stem-cells/stem-cell-clinical-trials/2016-05-18-FDA-Grants-IND-Approval-to-Stemedica-for-Traumatic-Brain-Injury-Phase-IIa-Clinical-Trial.asp

Basically, similar to what these guys are actually doing in medical trials, only with preprogrammed specializations they aren't using. I wish the researchers luck, and expect them to have some success, but those changes aren't cosmetic and without them, you're missing something although nobody knows what.