Do you absolutely need to track the rat through GPS?

You might check alternatives, i.e. those used for insects or small birds, if you are happy with active tracking from short distances (check harmonic radar; the dipole tag could just be a diminutive filament, easily attached e.g. alongside the tail), or with a limited, small track area, where you could perhaps set up an array for RFID and merely use implanted transponders. 

They won't probably work too well when well underground, though. (But neither would GPS).

That sounds helpful, Will give it a thought although the rats I am trying to follow are nocturnal

Well, then there are additional possibilities. In comparatively small and open areas, nocturnal terrestrial animals could be tracked (it's been done at least in experimental sets) with cenital (overhead) pole-mounted cameras with wide enough lenses. The animals can be tagged with small tritium light fobs (I don't know if ones small enough for your rats are available) or sprinkled with UV-phosphorescent paint, and then the area be illuminated with black light. Light dots are automatically tracked by software from the video.

But that will only work for small plots (must fit within the field of view--thence the [tall] pole and wide lens), devoid of significant vegetation cover, and certainly will not work within burrows. Your rat's activity could only be captured above ground.

Thanks a lot Arturo, from your description above, it seems a forest might not be ideal for cenital pole-mounted cameras especially when there is a thick forest understory. But that seems to be a very interesting method. Thanks again