Although it is true that maintaining the ionic concentration gradient across the neuronal membrane requires energy, I doubt that non-maintenance of ionic concentration is the main reason for quick brain damage when one is deprived of oxygen.
If the sodium potassium pump which maintains the gradient is poisoned, I think one only gets a 6-10 mV decrease in the magnitude of the potential difference across the membrane (out of a starting potential difference of 70 mV). If one poisons only the sodium potassium pump, it usually takes hours (not minutes) for the potential difference to run down to an unhealthy level.
I've extracted from the paper below the most common hypothesis about why neurons die (glutamate poisoning) after oxygen deprivation. A depolarization from -70 mV to -60 mV will cause neurons to spike more, and release more glutamate at synapses. However, it is unclear whether glutamate release from synapses is what kills neurons. It may be glutamate released from extrasynaptic locations that is much more important.
https://www.ncbi.nlm.nih.gov/pubmed/20670828
Neuron. 2010 Jul 29;67(2):181-98. doi: 10.1016/j.neuron.2010.07.002.
The science of stroke: mechanisms in search of treatments.
Moskowitz MA, Lo EH, Iadecola C.
"Excitotoxicity and calcium overload are major factors contributing to the early stages of ischemic cell death. The canonical pathway asserts that glutamate, the most abundant neurotransmitter, accumulates into the extracellular space as a consequence of energy and ion pump failure, as well as failure of reuptake mechanisms (Choi and Rothman, 1990). The glutamate overload leads to prolonged stimulation of AMPA and NMDA ionotropic receptor subtypes to dramatically enhance the influx of calcium, sodium, and water into neurons. Massive calcium influx activates catabolic processes mediated by proteases, lipases, and nucleases (Ankarcrona et al., 1995). In addition, activation of nNOS, PLA2, and other Ca2+-dependent enzymes leads to production of NO, arachidonic acid metabolites, and superoxide, which act as additional triggers of cell death (Dirnagl et al., 1999, Lo et al., 2003). For these and other reasons, oxidative phosphorylation becomes uncoupled, leading to further ATP depletion, ROS production, and release of stored Ca2+ from
mitochondria, further accelerating a series of catastrophic events that lead to acute cell death."
https://www.ncbi.nlm.nih.gov/pubmed/10411944
Proc Natl Acad Sci U S A. 1999 Jul 20;96(15):8733-8.
Inhibition of uptake unmasks rapid extracellular turnover of glutamate of nonvesicular origin.
Jabaudon D, Shimamoto K, Yasuda-Kamatani Y, Scanziani M, Gähwiler BH, Gerber U.
"These results show that under basal conditions, the activity of glutamate transporters compensates for the continuous, nonvesicular release of glutamate from the intracellular compartment. As a consequence, acute disruption of transporter activity immediately results in significant accumulation of extracellular glutamate."
https://www.ncbi.nlm.nih.gov/pubmed/10805815
Proc Natl Acad Sci U S A. 2000 May 9;97(10):5610-5.
Acute decrease in net glutamate uptake during energy deprivation.
Jabaudon D, Scanziani M, Gähwiler BH, Gerber U.
"The extracellular glutamate concentration ([glu](o)) rises during cerebral ischemia, reaching levels capable of inducing delayed neuronal death. The mechanisms underlying this glutamate accumulation remain controversial. ... we demonstrate that energy deprivation decreases net glutamate uptake within 2-3 min and later promotes reverse glutamate transport. This process accounts for up to 50% of the glutamate accumulation during energy deprivation. Enhanced action potential-independent vesicular release also contributes to the increase in [glu](o), by approximately 50%, but only once glutamate uptake is inhibited. These results indicate that a significant rise in [glu](o) already occurs during the first minutes of energy deprivation and is the consequence of reduced uptake and increased vesicular and nonvesicular release of glutamate."