We first present a method to mathematically build a learning rule for closed-loop neural networks. This rule is then applied to climbing fibers in the cerebellar cortex. Our analytical study is based on previous experimental non-analytical studies, which suggests that climbing fibers carry out an error signal to the brain. Thus, our goal is to find the class of functions for the activity propagated by climbing fibers, allowing the output of the Purkinje cell to converge towards a desired output. These functions must tend towards zero when the objective is reached. Our techniques are generalized to other network models.