How can I link bones from the same armature? Head to Tail and Vice-versa? Hey guys.

So I'm trying to rig the lamp in the picture(s) but I'm running into a technical difficulty with the armature. I've already read up on the other questions posted in here but they don't really apply to my problem as far as I understood them. 

The bone nodes that have been circled are part of the same armature. What I want to do is close the loop of the armature and make one close loop armature as a whole. So I was hoping I could just merge the nodes that have been circled or even link them so that when I move one set of the bones the others shift automatically (I have Auto IK ON) and not disconnect like it's happening in the second picture.

Any suggestions if I can do what I want? Thanks.
IK is indeed the solution. It looks similar in principle to rigging a piston, which is just a type of four-bar linkage: https://vimeo.com/25357097

I think the main insight to make your rig work is the use of IK "Target" bones. For example, where you want the spring to connect, create a small bone parented to the link's bone. Give the spring bone an IK modifier with chain length of one and allow stretching. That should be all it takes.