The primary learning mechanism of artificial neural networks (ANN) is back-propagation, which is not biologically plausible .
Trevor Berkolay created an alternative to this learning with the Neurological Engineering Framework (NEF) and Nengo called hPES (Homeostatic Prescribed Error Sensitivity) . But how does it's learning capabilities compare to the standard supervised and unsupervised learning of ANNs in terms of computational power required and speed of learning?
 See also, "How to Build a Brain" by Chris Eliasmith chapter 6.4
 Further details around this claim can be found in the question "Is back-prop biologically plausible?"
Note: This question portrayed ANNs and the NEF as adversarial, which really isn't the case. Spaun the brain model, which is kind of the poster child of the NEF, uses ANNs (Convolutional Neural Networks specifically) converted into spiking neurons in it's vision system.