Do neurons fire at a faster rate during dreaming? When I dream it seems that the (subjective) time is slower than the objectively passed time as seen on my clock radio. Is this because neurones fire collectively at a faster rate, so you can "put an hour of dream time in a minute of objective time"? If they fire at a faster rate, then this does not mean that you perceive the events in your dream as going faster, as you adapt to your experience of time in the real world.
In REM sleep, the EEG is remarkably similar to that of the awake state (Purves et al., 2001). Although the EEG represents the synchronized activity of many neurons in the cortex, it does give us a clue whether they are firing faster or not.

Wakefulness is mainly dominated by  beta and gamma waves (source: Scholarpedia), i.e. 12 - 100 Hz. REM sleep is characterized by low-amplitude mixed-frequency brain waves, quite similar to those experienced during the waking state - theta waves, alpha waves and even the high frequency beta waves more typical of high-level active concentration and thinking, i.e. 4-30 Hz (table 1) (source: Sleep).


Table 1. EEG bands. source: Neurosky

So if anything, I would say REM sleep, being largely devoid of the gamma band, is associated with slightly lower-frequency oscillations and hence lower neural activity overall. 

References
- Purves et al. (eds). Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates (2001)