Nature uses something like STDP (spike timing dependent plasticity), where the change in a synapse strength is proportional to the sign and delta time of the last spike at each end of the synapse. You can approximate something like backprop by using symmetric STDP (discarding the sign).

"Towards a Biologically Plausible Backprop" from Benjamin Scellier and Yoshua Bengio (2016) would be a recent paper on the topic.