Brain's information flow also isn't just through direct neuron-to-neuron connections. Firing also releases neuromodulators into the extracellular space which affects how other neurons operate. Furhermore neuron connection architecture is very different between brain and feedforward ANNs, with the former exhibiting a lot of recurrent connections.
Deep learning doesn't try to mimic all the intricacies of biological processes. It tries to approximate the end result (information processing).