It is simple and short: There is no use of this arrangement (negative "resistor" connected to a voltage source). In order to be useful, there must be a load connected in parallel. Then the voltage source will supply the load and the negative resistor will "help" it by passing an additional current through the load. Depending on the degree of the "help", various results can be obtained.
Example 1 : Load cancelling
As an illustration of this extravagant circuit technique, I will show in four steps how the resistive load of a voltage divider can be cancelled by an equivalent negative resistance. As a result, the voltage divider has the "feeling" that there is no load connected (open circuit).
1. Voltage divider unloaded. This humble circuit of two resistors in series (or a potentiometer) supplied by a perfect voltage source (with zero output resistance) behaves as a "bad" voltage source (with some output resistance). But when it is not loaded - Fig. 1, its "badness" does not manifest itself and its output voltage is exactly as it should be.
Fig. 1. Voltage divider unloaded
2. Voltage divider loaded. The problem of the high output resistance is that when a load RL is connected to the voltage divider - Fig. 2a, it "sucks" a current IL and the output voltage VL drops.
Fig. 2a. Voltage divider loaded by a resistor RL
This phenomenon can be seen everywhere in our life - if living beings are loaded, e.g. by a heavy load (Fig. 2b), they bend.
Fig. 2b. Gravitational analogy of a loading
The classic solution of this problem is to connect a voltage follower before the load to provide the load current instead of the voltage divider. This means to isolate the load from the voltage divider… to increase the load resistance "seen" from the voltage divider output.
3. Voltage divider "helped" - concept. But there is another more extravagant idea that can be borrowed from life - to "help" the voltage divider in parallel by an additional current source providing all the load current needed. This should be not the ordinary constant current source but a "proportional to voltage current source" that produces a reverse current I = -VL/RL.
We can implement this powerful idea by connecting in series a variable voltage source BH producing two times higher voltage VH= 2VL and another resistor R = RL acting as a voltage-to-current converter - Fig. 3a. This network acts as a negative resistor with resistance -RL that neutralizes the equivalent positive resistance RL and the result is infinite resistance (open circuit).
Fig. 3a. Voltage divider "helped" by an additional current source - concept
As an example of life, if someone has to raise a heavy weght - Fig. 3b, we can help it by an equivalent "anti-weght" (a powerful idea of mechanics that is widely used in lift systems, cranes etc.)
Fig. 3b. Gravitational analogy of an anti-weight
The brake booster in cars is another well-known non-electrical implementation of this idea. When the driver presses the brake pedal, the booster engages "in parallel" and assists the driver.
4. Voltage divider "helped" by INIC. In the end, all that remains is to implement this powerful idea through an op amp circuit - Fig. 4. It is simply a non-inverting amplifier (acting as the variable voltage source BH above) and a resistor R = RL in series to the op-amp output. This circuit is known as a "current-inversion negative impedance converter" (INIC).
Fig. 4. Voltage divider "helped" by an INIC
The trick of load cancelling is clever but it implies the load is a steady resistor; the input voltage and the potentiometer can vary.
Example 2 : Howland current source
Another well-known application of this powerful idea is the legendary Howland current source aka Howland current pump - Fig. 5.
Fig. 5. Howland current source - classic circuit diagram
We can see in the conceptual circuit diagram (Fig. 6), that really the idea is the same - an imperfect current source (on the left) is "helped" by an additional current source (on the right) so that the load current stays constant when the load varies.
Fig. 6. The idea behind the Howland current source
The "helping" current source acts as a negative resistor with resistance -R that neutralizes the positive resistance R of the imperfect current source; so the resulting "internal" source resistance is infinite.
The operation of the Howland current source is visualized in Fig. 7 by voltage bars and current loops.
Fig. 7. The operation of the Howland current source visualized
In contrast to the load canceller above, in the Howland circuit solution the load can vary. The negative "resistor" -R (INIC) neutralizes the steady "internal" resistance R of the imperfect current source.