ABSTRACT
A class of functions is proposed, which could describe the adaptive control of the closed loop involving the cerebellar cortex. The mathematical model of the cerebellar cortex is a basic biologically constrained neural network regarding the connectivity around the Purkinje cell. The output of a neuron is the transformation, by a linear or non-linear firing function, of the summation of weighted inputs. Covariance learning rules are used for synaptic weights. It is assumed that only synapses between parallel fibers and Purkinje cell are modifiable, and that the climbing fiber transports an error signal, closing the loop between the output and the inputs of a unit. Asymptotic stability is then shown for an isolated linear unit and the same results are given with conditions on several parameters for a non-linear unit.
