Your link to Wikipedia's cryptobiosis article mentions that Tardigrades are capable of reducing their metabolic activity to 0.01% of normal levels, and their water content to 1%, a state which they can maintain for up to 10 years. Possessing a considerable nervous system, it is almost certain that Tardigrade neurons are not active at all during that period, especially since neurons are some of the most energy-demanding cells. Some nematodes, which apparently are also cryptobiotic, would be a similar case.
Another relevant context would be hypoxic insults to brain tissue. Hypoxia-induced neurological damage results partly from a cascade of glutamate-release that is caused by diminished ATP production in neurons. According to a study by Bickler and Donohoe (2002), some animal's neurons, in order to preserve ATP levels, are equipped to block action potentials at the onset hypoxic incidents. See the section "Why does the lack of O2 kill typical neurons rapidly?"--they even cite a study that performed EEGs (as you suggested) on anoxic turtles.
Whether it is anything-but-rare for spike trains to be completely inhibited during the course of regular nervous system function is another question, and I wish I knew the answer.
Bickler, P. E., & Donohoe, P. H. (2002). Adaptive responses of vertebrate neurons to hypoxia. Journal of experimental biology, 205(23), 3579-3586. [PDF]