Can normal brain activity arise from no (or random) brain activity? I wonder how stable the brain is as a dynamical system. In other words, how important the state (current activation) of the brain is for its further functioning. Would the brain recover from a state of zero or random activity given sensible inputs from the environment, e.g. visual and auditory input?

By zero activity I mean equilibrium potential everywhere and no spikes. By random activity I mean independent random potentials and thus an initially random spike pattern.

Of course, normal brain activity is needed to keep people alive (e.g. control breathing) but maybe there were cases where machines ensured vital control for brain-dead subjects or computer simulations of neuronal brain models. An answer from either view would be interesting.
There is no such thing as "zero" activity in regards to brain circuits. Even when you record neurons from a brain slice in vitro, you have spontaneous network activity. There is a few studies that suggest that brain circuits are chaotic, where infinitesimal difference in the initial state will lead to exponentially diverging patterns of activity: 


Banerjee, A., Seriès, P., &amp; Pouget, A. (2008). Dynamical constraints
on using precise spike timing to compute in recurrent cortical
networks. Neural Computation, 20(4), 974–993.
London, M., Roth, A., Beeren, L., Häusser, M., &amp; Latham, P. E.
(2010). Sensitivity to perturbations in vivo implies high noise and
suggests rate coding in cortex. Nature, 466(7302), 123–127.


To answer your question, the interaction between the internal state of a system and external stimuli is an important topic in neuroscience, and it is not clear how internally and externally generated dynamics interact. Here is a paper that discuss it under the term of "state-dependent computing":


Buonomano, D. V., &amp; Maass, W. (2009). State-dependent computations:
spatiotemporal processing in cortical networks. Nature Reviews
Neuroscience, 10(2), 113–125.


And there are multiple studies investigating spontaneous brain activity: 


Foster, B. L., He, B. J., Honey, C. J., Jerbi, K., Maier, A., &amp;
Saalmann, Y. B. (2016). Spontaneous Neural Dynamics and Multi-scale
Network Organization. Frontiers in Systems Neuroscience, 10.


Hope it helps answer your question.