Why don't neurons die during a stent procedure in the brain?
I am assuming you are asking because of occlusion during balloon angioplasty. If not, please clarify.

Ischemic stroke occurs when a blood vessel to the brain is occuluded by a clot or some other emboli or poor perfusion due to atherosclerosis (a type of narrowing of the vascular lumen). Given enough time, ischemia can produce tissue death, called infarction. It is possible to suffer an ischemic stroke (or ischemia to any tissue) without substantial infarction if reperfusion occurs quickly enough.

Time to tissue death

For stroke, the catchphrase "time is brain" is often used to indicate the importance of occulsion time in stroke. Cell death is not instant upon loss of oxygen, but rather occurs throughout any extended reduction in blood flow as well as during reperfusion.

The brain is more sensitive to ischemia than other tissues like the heart, but there is still a window in which little to no damage is done. By 5 minutes, however, neuron death can begin.

Balloon Angioplasty

When using balloon angioplasty (including during balloon delivery of stents), a single vessel is occluded for ~30-180 seconds. It is also possible for multiple individual inflations. Importantly, this time range is normally kept less than the ~5 minute range for damage to occur.

Embolic protection

One of the dangers during balloon angioplasty is the release of particles from the vascular wall that is being opened up. Those particles can cause distal embolisms. In other organs like the heart, this problem is fairly minor, because only small amounts of myocardial tissue are expected to be impacted which will not have a major impact on heart function. However, loss of small regions of brain could have significant impacts on patient recovery. Therefore, embolic protection strategies are sometimes employed which include devices to block distal flow or capture emboli. The use of these devices is very case-dependent, because if there is already a significant blockage it may not be possible to use particular devices for a particular case.

In any case, pharmacological protection with drugs like aspirin and clopidogrel are standard for interventional angioplasty procedures.

Balloon alternatives

Balloon-mounted stents are standard for coronary procedures, but in the periphery, including carotid and neurovascular implants, there are other stent types, such as nitinol, a shape-metal alloy of nickel and titanium. These are also known as self-expanding stents. They are deployed by removing a protective sheath and they open up on their own. Thus, there is no total occlusion during a nitinol stent deployment. However, balloons can still be used with nitinol stents to open up the original blockage, because the nitinol stents themselves don't "push" nearly as hard against the walls.

In my opinion, the major benefit to nitinol stents is not that occlusion isn't necessary for their deployment, but rather that they are not sensitive to crushing. With a regular stainless steel stent, external pressure would permanently deform the stent. In the heart that isn't a problem, because if someone is poking you in your myocardium you have bigger problems. However, the carotid arteries are fairly exposed in the neck, and a regular metal stent in the neck could be compressed and deformed. Instead, nitinol will snap back into position, like the underlying tissue.

Neurovascular circulation redundancy through collaterals

Although any ischemic tissue can eventually develop collateral circulation pathways to bypass the occlusion (assuming partial and not total occlusion, sufficient to keep the tissue alive in the meantime), the neurovascular system is unique for the high degree of redundancy present even in normal individuals with no blockages. Therefore, occlusion of a single vessel even permanently may not cause substantial symptoms. Of course this will vary from patient to patient and depends on the patency of collaterals, overall vascular health, and specific anatomy at the site of treatment.

Caveats

Of course there are risks in any surgical procedure. It is unlikely that there is zero neuronal death during any neurovascular procedure. Rather, these procedures are used when there is already significant risk or imminent damage without intervention; the intent is to create as little damage as possible while either solving an acute problem that is much worse, or reducing the likelihood of future infarction events.

References

Kulik, T., Kusano, Y., Aronhime, S., Sandler, A. L., &amp; Winn, H. R. (2008). Regulation of cerebral vasculature in normal and ischemic brain. Neuropharmacology, 55(3), 281-288.

Lee, J. M., Grabb, M. C., Zipfel, G. J., &amp; Choi, D. W. (2000). Brain tissue responses to ischemia. The Journal of clinical investigation, 106(6), 723-731.

Mousa, A. Y., Campbell, J. E., AbuRahma, A. F., &amp; Bates, M. C. (2012). Current update of cerebral embolic protection devices. Journal of vascular surgery, 56(5), 1429-1437.

Pan, Jie, et al. "Reperfusion injury following cerebral ischemia: pathophysiology, MR imaging, and potential therapies." Neuroradiology 49.2 (2007): 93-102.

Saver, J. L. (2006). Time is brain—quantified. Stroke, 37(1), 263-266.

(disclaimer: I worked for a major manufacturer of intravascular catheters and stents for about 4 years. I was last employed by that company ~8 years ago and no longer have any financial relationship with that company. Everything in this post represents my own personal views. I am not a medical doctor; this information should not be taken as medical advice. Only your doctors/interventionalist can give appropriate medical advice according to your specific case)