Self organizing maps (SOM's) are an extremely interesting and powerful tool in the field of artificial neural networks. My question is: Does something similar exist in the brain? Or are these systems completely "man made"? Also, if there are such things in the brain, how do they work?
These are great questions. The SOM was absolutely biologically inspired. Sure, there are many examples.
Examples of self-organizing topographical feature maps in the human brain include the the sensory homunculus, retinotopic maps, and the entorhinal cortex. There are even maps such as these in the midbrain auditory nucleus of the barn owl as well as the auditory cortex of the mustache bat.
As to how they work and organize in nature, that is still an active area of research, and it seems like there are no experimental conclusions that suggest the existence of universal rules for self-organization of sensory maps in the brain. As far as I know, however, Kohonen's theory that these maps organize via a combination of lateral inhibition and Hebbian learning has not yet been displaced by a more modern idea.