Does Swimming Kill Brain Cells?

In an effort to keep my majorly injury-prone body injury-free in 2012 (with the hopeful outcome of a completed marathon AND a marathon PR!), I've semi-adopted the Run Less, Run Faster approach.  This new plan involves only 3 runs per week (but focusing on quality runs over quantity, each run has a purpose and involves working hard, which isn't my favorite thing to do!) and adding in more cross-training on the other days.  The authors of the plan don't really like the elliptical (my go-to cross-training exercise!) as a cross-training activity (nor pool running for that matter, which I found odd) since it too closely mimics running and using those muscles.  So extremely long story short, I've recently taken up swimming since swimming and cycling are the "endorsed" activities.  This is unfortunate because not only do I not really like swimming all that much, but swimwear is far from my favorite apparel.

Let's get one thing straight up-front.  I am NOT a good swimmer.  I cannot adequately perform any strokes besides the doggie paddle and the crawl.

Cooper is a much better swimmer than I. 

Although in typical Al Bundy "glory days" behavior I might claim to fame that I went to state in swimming my senior year of high school, the truth is I went to a small school with 3 good swimmers and the next decent swimmer was a 7th Day Adventist who couldn't swim on Saturdays.  Enter the mediocre me, who was therefore needed for the 400m relay (Districts was on a Saturday).  Despite this minor bit of glory (though it's far from glorious to be the worst on the relay team by far!), I still don't really know how to swim.  

                      That's not me.  I don't even know how to attempt to do the butterfly.  source

So the last few weeks I've been swimming 1-2 times per week.  On Tuesday I built up to swimming a whole half mile without stopping (18 laps!) and was pretty proud of myself.  On Friday I only made it 15 laps without stopping, but completed a full mile (36 laps!), which took me about 40 minutes. 

But here's my dilemma.  Although at first it was my shoulders that killed me when swimming (because I'm a weak little baby), the problem remaining is my breathing gasping.  The reason I have to stop (after 18 laps or 15 laps or whatever) is what I believe to be MY BRAIN CELLS DYING.  You see, my heart rate is high and I am a little short of breath, but the main issue stopping me is the massive headache that always, without fail, ensues.  My theory about this massive headache (which goes away quickly upon resting a minute or two) is that it is caused by a shortage of oxygen, which in turns KILLS MY PRECIOUS FEW BRAIN CELLS!

    This is a picture of a brain cell.  You're welcome.      source

So what should I do?  Keep swimming and risk my precious brain cells?  Do you swim?  How long did it take you to be good at it?  Does it kill your brain cells?