Can normal brain activity arise from no (or random) brain activity?
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.