In Figure 2 the battery is replaced with a line powered stimulator. From a biological recording standpoint there are other issues. In an instrument design context most people think of isolation from ground as it relates to electrical safety.
Stimulus isolators, as the name implies, also isolate a given stimulus from ground.
STIMULUS ARTIFACT MEANING SERIES
WPI isolators in the 300 and DLS series offer a compliance voltage of 100V with very low noise. Once this compliance voltage has been reached, further increases in tissue resistance will cause a drop in delivered current. The maximum amount of voltage that a constant current stimulator can provide is called the compliance voltage. If the resistance of the preparation becomes infinite, as might happen if one of the stimulating electrodes is removed from the tissue, the stimulator could not mount an infinite voltage to compensate. There is obviously a limit to how much voltage a constant current stimulator can provide. A constant current stimulator will "sense" resistance change and provide whatever voltage is needed to maintain delivery of current at the set rate.
Isolated Stimulation and Stimulus Isolators