"tl;dr Mutations are happening in your neurons every day! We looked at individual neurons to find out how many.
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Not saying this is an absolute correct way of going about this (no one really knows), but it shows they did at least thought about this issue.
Not enough to scan brain connectivity info, neuron types and per-person genome, you will now also need per-neuron genomes to truly capture and reproduce a brain.
Not that we were anywhere close to any sorts of brain uploading, but this news just made it orders of magnitude more difficult.
Is the cup half-full or half-empty? From one perspective, if the known fixed functionality of the brain is implemented in part using DNA mutations, this is good long-term news - after all, if there is one thing you can expect to be kept intact and be recoverable in a vitrified brain, it's the DNA. That makes decoding more difficult, but also makes information-theoretic death less likely.
If they aren't random and there is some purpose or information encoded in them, then you have a good point. I think that's unlikely though.
The brain would work in a very similar way. If a neuron has a mutation that effects it's function, surrounding neurons would adapt to that.
It is likely that some mutations affect the efficiency of signal propagation, and encoding might adapt to this, so that equivalent inputs would create more or fewer or "stronger" or "weaker" synapses to generate the same output from the second neuron. But this could be detected functionally, if you could observe the signals and responses of neuron pairs.
It probably also is related to the fact that neurons use a lot of energy and have a lot of mitochondria. Energy requires oxidative phosphorylation in the mitochondria, which produces a lot of free radicals, which damage DNA.
Yet, 1700 (the number they reported is 1700, not 1000) is about typical. Many cancer cells have been sequenced; typical findings are that about 100 genes in cancer cells have acquired mutations, about 10 of which contribute to the cancer. Genes comprise about 1% of DNA; this suggests that the typical somatic cell has 1000 to 10,000 acquired mutations. (1000 assuming that all 90 mutations past the 10 that caused the cancer happened after the tissue went cancerous; 10,000 assuming they all happened earlier. Though that linear interpolation is a bad estimate, because genes and intergenic DNA mutate at different rates, owing to transcription and chromatin.)
However, if they sequenced just the exome (the expressed genes) rather than the entire genome, then 1000 is 10x typical. I can't tell, since their paper is paywall-protected.
We've got lots of cells that are turning over continually, but plenty of long lived structures as well.
Also, if it's related to cell turnover, you would expect cells which replicate through our life to have higher numbers of mutations compared with our initial zygotic state due to the potential for mutation at each point
If they were claiming that this were an unusually high mutation rate, I'd wonder why everyone doesn't die of brain cancer. If other tissue had N mutations per genome, and cancer typically takes 10 mutations to create, in the absence of compensatory mechanisms, the ratio of brain cancer to other cancers should be about (1700 choose 10) / 50 * (N choose 10). If N were, say, 100, that number would be so large that not one person would ever had developed any cancer other than brain cancer.