Will the person live if his/her brain disappeared only for microseconds?
Disappearing only for microseconds means absent for microseconds & it reappears again in its original state.
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The brain is an electro-chemical device; meaning it also depends on electrical fields. If the matter of the brain disappears, presumably the unattached electrical fields do not; they will dissipate in microseconds.
Remember a microsecond is 1000 nanoseconds; light, at 186,282 miles per second, can travel 983.6 feet in one microsecond, or 0.19 miles. Although electrical fields in materials tend to travel around 2/3 the speed of light, a "few microseconds" is more than enough time for them to dissipate.
However, microseconds will not be enough time for blood and other fluids to move very much at all; and I will presume the return position is exact to the molecular level; so no severing of nerves occurs. I will also presume the magical disappearance and reappearance do not create an absolute vacuum that causes the head to implode; although I doubt even that could be completed in microseconds (it is just air pressure that causes the implosion) we will handwave away the fact that it might start. In order to reappear in the skull as the OP obviously intends, the skull cannot be deformed by the beginning of a head implosion.
That said, when the brain returns in its "original" state, it is likely to be devoid of any electrical signals in progress. The person will most likely be unconscious; and may not recover (this might kill them). Their brain would need an electrical reboot; that might occur through natural action of blood moving through the brain because of the beating heart, but might not. In fact the lack of electrical signal in the part of the brain controlling the heart beat might cause a heart attack; the cycle of charges in the brain would have been disrupted by the brain vanishing, and have a cascade effect, even though the vanishing was momentary.
As an analogy, imagine a baseball game in which the first baseman vanishes for a tiny moment, just enough to fail to catch a ball that would otherwise have ended the World Series. The first baseman may be fine, but history is changed, tens of millions of dollars in bets goes to what would have been losers, and is taken from what would have been winners. The point of this analogy is this: The vanishing of the baseball player causes a domino effect of major changes.
The same can be true for the momentary vanishing of the brain; it could easily lead to death by interrupting (and perhaps halting) various cycles of electro-chemical charges that are constantly moving through the brain. In fact we call them brainwaves, detectable purely by their electrical activity, as detected by Electroencephalography.