'A low-carbohydrate diet based on animal sources was associated with higher all-cause mortality in both men and women, whereas a vegetable-based low-carbohydrate diet was associated with lower all-cause and cardiovascular disease mortality rates.'
https://pmc.ncbi.nlm.nih.gov/articles/PMC2989112/
It is of course rather difficult if not impossible, to have a vegetable diet completely devoid of carbohydrates.
Alas, it doesn't have much bearing on the part of the comment I wanted to focus on 'You have to have carbs for your brain to run.' Because the diets in question here still have some carbs. (And the changes in mortality rates are fairly modest, too.)
What I wanted to say is that even if you eat no carbs at all, you brain will still get its energy. You won't just keel over after a while, like if you didn't eat anything at all henceforth.
Of course, there might be second order health effects, like the study you linked suggest.