Why is ground water rising on these forest rings?
Nobody knows - yet. But one can say that these rings are too numerous and too regular in shape to be related to either kimberlites or (even more bizarrely) Oklo-type natural nuclear reactors, as some have suggested.

For what it is worth, here is my own hypothesis: The areas concerned are all periglacial in an area of abundant boulder clay. In colder analogues, particularly in Siberia and further north in Canada, such environments contain 'pingoes' aka 'hydroloaccoliths', which are frost heaves in the shape of ice lenses. See, for example, https://en.wikipedia.org/wiki/Pingo . These are feedback processes of circular ice formations which develop in permafrost tundra areas. The circles you refer to are exactly the right shape, diameter, distribution density, and geology to be 'fossil' (melted) pingoes. 

Consider what happens when a pingo develops. The ground is highly crazed with fractures over a disk shaped area. If there is brackish groundwater in the area then the ice freezes as pure water, thereby partitioning more saline groundwater to the perimeter - thereby leaving a high-salinity ring of brackish water which subsequently stunts tree growth (as you observe). When the ice eventually melts the pingo collapses to appear much like the surrounding boulder clay, but with a fractured and much more porous texture. Wherever there is a confined or semi-confined aquifer - which is very common because thick layers of impervious boulder clay behave as a confining layer - there will be a tendency for groundwater to rise as artesian (non-overflowing) conditions. Hence the 'bulge' of the apparent water table, which you also observe.  If this explanation is correct then the ring would be a natural exit and pressure release point for confined groundwater, and hence for entrained methane. Methane builds up under boulder clay, especially where the clay overlies peat deposits, such that oxidizing organic matter reduces the water. Moreover, this redox contrast, between surrounding oxidized near-surface/perched groundwater, and the reduced groundwater exiting from the ring, would also influence the nutrient availability, thus contributing to the ring of stressed vegetation.

As for the change in electrical potential, that is easily explained as deriving from the contrast between normal and residual more saline groundwater. Also, if the pingo was weathered during its evolution, then there would be a difference in clay content, which would also contribute to the electrical differential. Indeed, that is to be expected when a hill (the pingo) is drained by meteoric water, taking with it any dispersive clay (another common feature of boulder clay).

So no, sorry to disappoint, but the explanation is probably just normal periglacial hydrogeology - no gold deposits involved.

The test to verify this hypothesis? It needs someone to run a traverse across such a ring, measuring the hydrostatic pressure variation (horizontal and vertical), soil salinity, and soil clay content.