A really interesting question. It would seem far fetched at first that backpropagation could be used by the brain, because it's unclear what the mechanism for transmitting the error at every synapse backwards, to the synapses in the previous layer, would look like physically.

This is a really interesting lecture given by Geoffrey Hinton (https://www.youtube.com/watch?v=VIRCybGgHts) where he discusses the various issue commonly raised with "biological backpropagation" and proposes a solution based on Spike Timing Dependent Plasticity (STDP). Basically he argues that you can interpret the STDP learning rule as a derivative filter on a firing rate and get backpropagation in this way. This is just on wild idea though and has not been shown to work experimentally or through simulations.

There are also a couple of interesting pointers in this Stackexchange thread: https://cogsci.stackexchange.com/questions/16269/is-back-pro...

I once attended some talk or other and that question came up - the answer given was flippant, but I always thought it was oddly perceptive...... "Dreaming"