Movement learning results from synaptic plasticities in various sites of the brain. Three sites háve been particularly studied: the cortico-cortical synapses in the cerebral cortex, the parallel fiber-Purkinje cell synapses in the cerebellar cortex and the cerebello-thalamo-cortical pathway at the level of the thalamocortical synapses. We intended to understand how these three adaptive processes cooperate for optimal performance during the arm reaching movement, and how the cerebellar learning is supervised. A neural network model was developed on the basis of two main prerequisites: the columnar organization of the cerebral cortex and the Marr-Albus-Ito theory of cerebellar learning. The synaptic plasticities observed on these sites were incorporated in the model as differential equations. The analytical resolution of the set of rules showed two main results. First, the adaptive processes taking place in different sites do not interfere but complement each other during the learning of the arm reaching movement. Secondly, any linear combination of the cerebral motor commands may generate olivary signals able to supervise the cerebellar learning process.