They are not interacting to each other in both cases.
Connect electrical wire to rubber mat. Nothing happens.
However, in some cases when both objects are for example metals.... this happens:
1) positive - neutral case
if you connect positively charged (object P, experiencing short of electrons) to neutral object (object N), after some time they both will become positive. This means that object P would gain some missing electrons from the object N. The object P becomes a bit less positively charged than before. The object N becomes from neutral to positively charged as it loses some of it's electrons.
2) same story with negatively charged and neutral object.
If you connect them, after some period of time negatively charged (object A) will exchange electrons with neutral object B. After some period of time they both will become negatively charged. Object A a bit less negatively charged compare to initial stage
