https://en.wikipedia.org/wiki/Hyperventilation - This leads to hypocapnia, a reduced concentration of carbon dioxide dissolved in the blood.

https://en.wikipedia.org/wiki/Hypocapnia - Acute hypocapnia causes hypocapnic alkalosis, which causes cerebral vasoconstriction leading to cerebral hypoxia

Though not acute, you can see the effects very easily. Within couple of minutes of deep breathing we could create peripheral numbing and tingling effect. (yoga folk say you are feeling energy moving though body and that kind of stuff haha)