What are the neurological mechanisms behind risking one's own life for another? Every so often, we hear of people who go that extra mile and put their life on the line (and sometimes, give their life) for another person - whether it is a loved one or a complete stranger (sometimes, an enemy).  

This has happened to me and several people that I know - and there many cases floating around in the media, a couple of examples are Jordan Rice and Joan North, there are millions of cases such as this, where ordinary people, young and old have 'stepped up' and did all they could to help.  Nearly all of them reject the title 'hero'.

Having experienced this myself, I can honestly say, it was not really a 'choice', but a 'compulsion' - seemingly, the reflexes seem to become more acute, the reactions faster and fear is either suppressed or gone altogether.

So, my question is, what are the neurological mechanisms that compel a person to act to help another, often without regard to their own safety?
This book What Makes a Hero?: The Surprising Science of Selflessness by Elizabeth Svoboda, covers the answer to this question in considerable detail.
The author explores the genetic predisposition to be heroic, whether it can be developed and learnt, and the idea of group selection; which proposes this is an instinctual reaction, the sacrifice of one member can be in the aide of the greater good of the species.
This discussion on Moral maze  examines whether human altruism really exists, or whether it is an evolutionary process. Examples of mutual reciprocity within various species, demonstrates that self-sacrifice can contribute to an individual's overall well-being. It is a cynical look at human nature, making the basic assumption that the survival instinct within human beings (or the preservation of an individual's gene pool)  is the greatest drive and that any seeming detour from this, ultimately serves this.

&quot;Even in human societies we can still trace many behaviours seemingly
carried out for the benefit of the group back to benefits for the
individual,&quot; Stanford explains.
&quot;Someone who jumps into a river to save a drowning man will become a
hero. And heroes accrue all sorts of benefits. For example, they may
become more attractive to women,&quot; he adds.

My personal take on the two examples you provided:
In the case of Jordan Rice he has sacrificed himself for his younger brother. An act that conforms with the notion of group selection and maintains the integrity of the individual's aim to preserve the own gene pool.
In the case of Amanda (aka Joan) North is quoted as saying:

I felt like there was a reason I was there. I felt a  compulsion to go over to you.

Amanda North is in her 50s and has two children; past her child bearing and raising years (from a survival viewpoint).
Erika Brannock (the woman Amanda North rescued) is a 29 year old woman. A member of the species in the prime of her child bearing years, hence better able to contribute to the ongoing survival of our species. An added, an unrelated observation, is that she also happens to be a preschool teacher - demonstrating the, more likely odds, that she is a member of the species in the role of raising the next generation.
A note  about the state of mind of a person in such a crisis, it is survival mode at it's most primitive. The body's fight and flight response, with is managed by the sympathetic nervous system. This optimizes the individual's for self preservation and protection of others.
The body's reaction to fight and flight:


heart rate and blood pressure increase

pupils dilate to take in as much light as possible

blood-glucose level increases

muscles tense up, energized by adrenaline and glucose (responsible    for goose bumps -- when tiny muscles attached to each hair on surface
of skin tense up, the hairs are forced upright, pulling skin with
them)

smooth muscle relaxes in order to allow more oxygen into the lungs      nonessential systems (like digestion and immune system) shut down to
allow more energy for emergency functions

trouble focusing on small tasks (brain is directed to focus only on    big picture in order to determine where threat is coming from)



Why Do People Risk Their Lives To Help Others During Emergencies Such as the Boston Marathon Bombing?
note - if you like I can investigate studies which show the demographics of likely heros, which is a little out of the scope of this question.