Humans often dream of random events that occurred during the day. Could the reason for this be that our brains are backpropagating errors while we sleep, and we see the result of these "backpropagations" as dreams?
This is a very interesting question and also an important one for AI. All the current Deep Learning successes are built on the effectiveness of backprop, so what if it doesn't play a role in the only examples of intelligence currently around?
It makes a lot of sense for the brain to employ some form of backprop, because that would allow it to create low-level features in a way most conductive for the high-level features that finally give rise to intelligent behavior. And if the brain employs backprop, dreams are a logical candidate.
Of course the problem here is that the brain doesn't calculate and propagate gradients, so the question is whether the backpropagation algorithm can somehow be implemented with the biological constraints in mind, such as the locality of neural computation and the non-symmetrical feedback connections.
It turns out this is actually possible, as explained in these talks by Hinton and Bengio. Symmetrical feedback connections (and in fact continually updated feedback connections) aren't necessary for backpropagation, however unintuitive this may seem. And backpropagation can be emulated with a form of temporal difference learning, that is plausible within biological constraints.
Of course this doesn't conclusively answer the question, but it certainly makes the possibility much more plausible than a priori assumed.
There has been a lot of research in cognitive science on the relationship of sleep/dreaming and memory/learning.
I don't know enough about the subject to say if it resembles backprop in spirit, by as the BlindKungFuMaster points out, that may be corollary.
Here's a paper from 2004, Memory Consolidation in Sleep: Dream or Reality?, which concludes that "there is no compelling evidence to support a relationship between sleep and memory consolidation."
However, a more recent article from the Harvard Medical School, Sleep Helps Learning, Memory (2015), comes to a different conclusion, citing a 2010 study:
A 2010 Harvard study suggested that dreaming may reactivate and reorganize recently learned material, which would help improve memory and boost performance.
The general idea is that sleep/dreaming is a process where neural connections in the brain are reinforced or suppressed. Following are several papers that touch on that topic. The first one, in particular may be of interest, and references the work of Hinton, although be aware it's over 20 years old, before recent NN breakthroughs:
Hinton et al.6 have provided an elegant new theoretical framework for creating efficient memory representations in hierarchical neural network models. In this model (Figure 1), the feedback connections generate patterns on the input layers of the network that correspond to the representations at the higher level, when the external inputs to the cortex and feedforward processing have been suppressed. During this generative sleep stage, the strengths of the feedforward synaptic strengths are altered. Conversely, during the awake stage, the feedback connections are suppressed and the sensory inputs drive the feedforward system, during which the weights on the feedback connections can be altered.
These observations suggest that sleep-dependent memory consolidation in humans is facilitated by the offline reactivation of recently formed memories, and furthermore, that dream experiences reflect this memory processing. That similar effects were not seen during wakefulness suggests that these mnemonic processes are specific to the sleep state.
Learning while you sleep: Dream or reality? (Harvard Medical School, 2012)
Neuroscientists Reveal How The Brain Can Enhance Connections (MIT Tech Review, 2015)
Sleep and the Price of Plasticity: From Synaptic and Cellular Homeostasis to Memory Consolidation and Integration (National Institutes of Health, 2014)
The brain uses REM sleep to cut unneeded connections (Ars Technica, 2017)
The Brain’s Connections Shrink During Sleep (The Atlantic, 2017)
And, just for fun: