A feed forward network is just matrix multiplication with a nonlinearity. It could, at best, be described as biologically inspired.
To put things in perspective, we've fully mapped the connectome (map of neuron connections) of the simplest animal, C. Elegans, which has only about 300 neurons, yet we still can't simulate this organism's behavior computationally.
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...
E.g. http://journal.frontiersin.org/article/10.3389/fnins.2016.00...