A neural network model is described for how an invariant head-centered representation of 3-D target position can be autonomously learned by the brain in real time. Once learned, such a target representation may be used to control both eye and limb movements. The target representation is derived from the positions of both eyes in the head, and the locations which the target activates on the retinas of both eyes. A Vector Associative Map (VAM) learns the many-to-one transformation from multiple combinations of eye-and-retinal position to invariant 3-D target position. Eye position is derived from outflow movement signals to the eye muscles. Two successive stages of opponent processing convert these corollary discharges into a head-centered representation that closely approximates the azimuth, elevation, and vergence of the eyes' gaze position with respect to a cyclopean origin located between the eyes. VAM learning combines this cyclopean representation of present gaze position with binocular retinal information about target position into an invariant representation of 3-D target position with respect to the head. VAM learning can use a teaching vector that is externally derived from the positions of the eyes when they foveate the target. A VAM can also autonomously discover and learn the invariant representation, without an explicit teacher, by generating internal error signals from environmental fluctuations in which these invariant properties are implicit. VAM error signals are computed by Difference Vectors (DVs) that are zeroed by the VAM learning process. VAMs may be organized into VAM Cascades for learning and performing both sensory-to-spatial maps and spatial-to-motor maps. These multiple uses clarify why DV-type properties are computed by cells in the parietal, frontal, and motor cortices of many mammals. VAMs are modulated by gating signals that express different aspects of the will-to-act. These signals transform a single invariant representation into movements of different speed (GO signal) and size (GRO signal), and thereby enable VAM controllers to match a planned action sequence to variable environmental conditions.