This is not entirely correct. There is a lot of evidence that action potentials backpropagate into the dendrites [1]. On a local level, this allows synapses on the dendrites to "be aware" of the activity of the post-synaptic neuron. However, I do not think this accomplishes/implements the ANN backpropagation algorithm (although a few statements and a citation in [1] alludes to this being possible).
