Does stimulus-independent ERP make sense/ exist? Is anyone aware of any concept of stimulus independent ERP (an apparent oxymororn, an event related potential that is neither sensory, cognitive nor motor elicited) ...or a procedure that looks specifically at noise waveform during ERP trials.....

Basically I am trying to think of an analogue of resting-state fMRI for EEG. My logic is that we can find a stimulus-independent ERP signature by
a) either taking EEG values every say 1 sec without any repetitive stimulus and averaging them just like in traditional ERP to get a stimulus independent ERP waveform
b) we can have a repetitive sound stimulus say every 1 sec and average the square of the waveform over the trials and subtract the square of the ERP calculated in usual manner. the resulting waveform will be reflective of noise.

The idea is that a waveform that is stimulus-independent / noise driven may be equivalent in value to resting-state fMRI procedures and the components of such waveforms may be of some help in delineating one person from another based around neural speed etc.

I want to find out if the concept already exists or makes sense?
An ERP is an Event Related Potential, making it inherently connected to an event. Having stimulus/response independent ERPs is thus impossible. There do exist potentials that seem to be stimulus independent, or at least vary in timing, but these are often the result of cognitive processes. These potentials are the result of Event Related (De-)Synchronization (Pfurtscheller, 1999), for which you need to look into the time-frequency domain, instead of the time domain (in which the ERP is visible). 

By use of a Fourier transform you can subdivide brain activity into sinussoids with different frequencies and amplitudes. These sinussoids are grouped based on their frequencies in the groups commonly referred to as the "frequency bands". These include the delta- (0-4 Hz), theta- (4-8 Hz) alpha- (8-12 Hz), beta- (12-±30 Hz) and gamma frequency band (30+ Hz). The exact frequencies and groupings tend to vary between researches however (wiki). 

It is the time-frequency domain that you can study independent of stimuli. The power in the frequency bands can be computed in real time by use of a time-frequency analysis. The steady state potentials are often used to compare brain activity between doing (nearly) nothing, and performing a particular task. This allows you to investigate what frequency bands show more (or less) activity (i.e. a higher or lower power) in particular brain areas. See Method/Unit for measuring activity in specific frequency band for EEG and Are brain waves electromagnetic waves? for some more specific info on EEG and analyzing data in the (time-)frequency domain. 



Pfurtscheller, G., &amp; Da Silva, F. L. (1999). Event-related EEG/MEG synchronization and desynchronization: basic principles. Clinical neurophysiology, 110(11), 1842-1857.