Is "tonic activity" common for neurons firing in the brain?
Most neurons that have been studied have spontaneous activity. Tonic firing is common in sensory neurons as well as cortical neurons. However, this could be very biased since it is difficult to study neurons that are silent in vivo. In vivo studies are blind, meaning that we usually do not see which neurons we are recording from. This is known as the "dark matter" problem (ref 1).

Also, it is known that there are homeostatic mechanism for maintaining average firing rate. Basically, if a neuron doesn't fire much, then the input sensitivity of the neuron increases to produce more activity (ref 2). But we don't know how widely this can be applied.

References:


Shy Shoham, Daniel O’Connor, Ronen Segev. How silent is the brain: is there a "dark matter" problem in neuroscience? Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioral Physiology, Vol. 192, No. 8. (1 August 2006), pp. 777-784  doi:10.1007/s00359-006-0117-6
Gina G. Turrigiano. Homeostatic plasticity in neuronal networks: the more things change, the more they stay the same. Trends in Neurosciences, Vol. 22, No. 5. (1 May 1999), pp. 221-227, doi:10.1016/S0166-2236 (98)01341-1