A central problem in learning theory is how the vertebrate brain processes reinforcing stimuli in order to master complex sensorimotor tasks. This problem belongs to the domain of supervised learning, in which errors in the response of a neural network serve as the basis for modification of synaptic connectivity in the network and thereby train it on a computational task. The model presented here shows how a reinforcing feedback can modify synapses in a neuronal network according to the principles of Hebbian learning. The reinforcing feedback steers synapses towards long-term potentiation or depression by critically influencing the rise in postsynaptic calcium, in accordance with findings on synaptic plasticity in mammalian brain. An important feature of the model is the dependence of modification thresholds on the previous history of reinforcing feedback processed by the network. The learning algorithm trained networks successfully on a task in which a population vector in the motor output was required to match a sensory stimulus vector presented shortly before. In another task, networks were trained to compute coordinate transformations by combining different visual inputs. The model continued to behave well when simplified units were replaced by single-compartment neurons equipped with several conductances and operating in continuous time. This novel form of reinforcement learning incorporates essential properties of Hebbian synaptic plasticity and thereby shows that supervised learning can be accomplished by a learning rule similar to those used in physiologically plausible models of unsupervised learning. The model can be crudely correlated to the anatomy and electrophysiology of the amygdala, prefrontal and cingulate cortex and has predictive implications for further experiments on synaptic plasticity and learning processes mediated by these areas.