Imagine we want to read impulses within single neurons in the brain. Can we do that now for a single region within the brain? If not, what is the smallest region that can be monitored for an activity in real time? Is there a more advanced method than electroencephalography for scaning brain activity in real time?
Electroencephalography has a good time resolution (milliseconds) but poor spatial resolution (several centimers). The usual estimated figure is that at least 50000 neurons need to fire simultaneously so that the activity can picked up by EEG.
The answer provided by @Jeremy Kemball is not very accurate. The reason why the spatial resolution of EEG is poor is that the signal from even a point-like source on the cortical surface spreads about to every sensor along the scalp due to the conductivity properties of the brain, CSF, skull, and scalp.
This means that in order to get that few centimeter accuracy I mentioned above, one must solve an inverse problem from sensors -> cortex. Inverse problems are mathematically ill-posed, and have infinite number of solutions unless some constraints to the solution are added.
I'll give a practical example. Say that a subject is presented some sounds. You record the EEG. Now, if you look at the sensor-level data, you can find brain activity related to the sounds from almost any sensor, from both sides of the brain (each ear projects to both hemispheres). If you want to know where the signal actually came from, you need to do the source localization using some inverse solver. Some commonly used are for example dSPM, sLORETA, or MNE.
Finally, I want to emphasize that
Brain regions deeper in the brain produce a 'blurred' signal that shows up on several electrodes at once.
that this happens to all activity recorded by EEG. Actually, EEG does not even pick subcortical activity too well.
Is there a more advanced method than electroencephalography for scaning brain activity in real time?
Yes, that is called magnetoencephalography (MEG). MEG records the magnetic fields related to the electric activity. It reaches sub-centimeter localization accuracy, while still having millisecond time resolution. However, this is now strictly only for the cortical activity; MEG does not see subcortical activity almost at all. Google for Elekta Neuromag or CTF to see some MEG systems.
If you would be interested in studying some deep structure, like amygdala, your best choice is then fMRI, which has about ~2 second time-resolution and 1mm^3 spatial resolution.
The smallest area that the EEG can measure is related to the electrode density. Even with research setups with 100s of electrodes, the smallest measurable region is on the order of a square centimeter or so on the skull surface. Brain regions deeper in the brain produce a 'blurred' signal that shows up on several electrodes at once.
fMRI, on the other hand, can measure smaller areas in three dimensions, but not as often.