Based on computational models of resting brain activity, it can be assumed that the brain is a multistable dynamical systems that operates on the edge of a "critical" bifurcation point seperating a low firing spontaneous state (which can be seen as more or less random low firing activity) from a state of high frequency activity.
see for example the work from Deco et al. or Hansen et al.:
- Deco, G., & Jirsa, V. K. (2012). Ongoing cortical activity at rest:
criticality, multistability, and ghost attractors. The Journal of
Neuroscience : The Official Journal of the Society for Neuroscience,
32(10), 3366–75. doi:10.1523/JNEUROSCI.2523-11.2012
- Deco, G., Ponce-Alvarez, A., Mantini, D., Romani, G. L., Hagmann, P.,
& Corbetta, M. (2013). Resting-state functional connectivity emerges
from structurally and dynamically shaped slow linear fluctuations.
The Journal of Neuroscience : The Official Journal of the Society for
Neuroscience, 33(27), 11239–52. doi:10.1523/JNEUROSCI.1091-13.2013
- Deco, G., Jirsa, V. K., & McIntosh, A. R. (2013). Resting brains
never rest: computational insights into potential cognitive
architectures. Trends in Neurosciences, 36(5), 268–74.
- Hansen, E. C. a., Battaglia, D., Spiegler, A., Deco, G., & Jirsa, V.
K. (2015). Functional connectivity dynamics: Modeling the switching
behavior of the resting state. NeuroImage, 105, 525–535.
This state is the most economic state since it is on the edge of stability, which means that small inputs will lead to a bifurcation where the spontaneous state becomes unstable and the high firing "task-activated" state becomes stable. This provides high flexibility to brain operations. The control parameter in most of the models is the coupling strength between neurons. As Philippe mentioned, there is always some degree of spontaneous activity in normal functioning (healthy) brains, and I'm only aware of models which simulate neurons by means of chaotic systems or realistic neuronal spiking networks which have some low degree of connectivity/coupling to simulate a low firing "random" state, but there may be other models, which I do not know about.
But let's assume you simulate a model just with random spontaneous activity. Then there would be no information exchange between neurons when there is a low coupling, thus there would not emerge any form of resting state network, which can be seen as a form of "ghost attractors" where brain areas exchange information by spike synchronization.
So to answer your question, I'd say that the normal healthy brain is wired in a way that it can respond to environmental stimuli very flexibly and is able to switch from a resting state (for example when closing eyes and relax without any form of stimulation) to a task-activated or stimulus activated state very quickly.
But this can all be different when there is some form of physical damage to central hubs of brain function (think of the brain as small world network), which can't be compensated or substance induced changes like LSD or Psilocybin. Or think about mental disorder like schizophrenia or psychosis with visual and auditory hallucinations asf. There are probably many ways of altering the optimal operation point of the brain.