Here's a long answer.
Actually, pO2/pCO2 levels have little to do with respiration rate! It has to do with [H+]....for a quick review revisit the chapter on cellular respiration, pulmonary respiration, and the medulla oblongata (even brain stem overall will do). Usually when you're number-crunching a huge math problem, you are not likely sprinting through the jungle (usually!). Therefore, your plasma H+ levels are relatively lower (much lower) than during intense physical exercise. This means reduced medullary upregulation of breathing rate via the vagus nerve. Therefore, the lower breathing rate. There is also no evidence (that I found on Google Scholar or PubMed) in support of a mechanism of deeper/slower breaths versus shallower/rapid breaths.
Then again, if you're concentrating hard to beat a clock on the MCAT, your heart rate's higher, but your breathing rate stays constant (maybe even slows down further!). This again makes sense if you're sitting down and there is little physical activity besides neuronal activity. And there is always enough oxygen in your blood to meet the brain's demand for short intense bouts of concentration while resting. Over longer periods, you just wear out and start getting distracted. This also makes sense, because O2 levels have little to do to up regulate breathing. Taking a quick break to do some pushups or other physical exercise really boosts concentration/mental performance because these can actually raise respiratory rates by increasing cellular respiration which in turn boosts H+ concentrations.
Hope this helps.
