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Suppose I set the voltage value of an isolated stimulator with a floating ground. I place one electrode above the spinal cord (positive) and the other placed subcutaneously far away from the spinal cord (negative). The circuitry within the stimulator will operate in such a way that the electrical potential difference between these two electrodes is the value I specify.

The stimulator can fix the electrical potential difference between the two electrodes in multiple ways:

  1. Hold the positive electrode's potential constant while changing the negative electrode's potential.
  2. Hold the negative electrode's potential constant while changing the positive electrode's potential.
  3. Adjusting both electrode potentials using a rule.

Although each of these ways can fix the voltage between the electrodes, they do not have equal effects on the biological environment that the electrodes are placed in. Going back to my example, if the stimulator obeys 1, I would expect a larger effect to be induced local to the spinal cord, in contrast to 2, which would would have a greater effect in the subcutaneous area. This is because there is an initial heterogenous charge distribution.

How should I assume my stimulator is fixing the potential difference between my electrodes?

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The situation is quite reminiscent of a Voltage clamp experiment (Fig. 1), where one amplifier (amplifier 1 in Fig. 1) records the potential difference (V1 - V2 in Fig. 1) between two electrodes and the other opamp (2) is used to inject current (I2 in Fig. 1) to 'clamp' the system at a particular voltage using a computerized feedback system. The potential difference is thus kept constant. As OP already mentions in the question, in Fig. 1 the charge density is locally much higher at I2, because current is injected there that dissipates across the cell (or tissue in your case) the farther it gets from its source.

Having said all that and to get to the question proper...

How should I assume my stimulator is fixing the potential difference between my electrodes

...I can only say the assumption is the mother of all fu&% ups many experimental failures. Instead, I advise you to either consult the device manual, or contact the distributer or manufacturer of the device to make sure what the machine is actually doing to clamp the voltage at a certain level.

Voltage clamp
Fig. 1. Voltage clamp setup. source: wikipedia

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None of the three scenarios you describe are really different from another. Whatever process is used inside the amplifier, a voltage describes a potential difference between two points. It does not relate that to any other universal reference. If you have a ground connection, you could describe a different potential relative to ground, but that's not relevant here since you have no separate ground that gives an alternative electrical path to either of the contacts.

When you have a voltage between two points that are connected with less than infinite resistance, as you have in the biological environment you are stimulating in, you will get a current. That current will be exactly the same at each electrode. All that matters to determine how much current will flow is the voltage and the resistance/impedance between the contacts. You could get some differences in the stimulation duration/intensity around the two contacts if either acts like it has substantial capacitance, and the current density through the electrolyte solution surrounding each electrode can vary by the surface area of the electrode, but those are properties of the electrode, not how the voltages might compare to some unconnected ground reference.

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