Biol. Cybern. 63, 161-167 (1990) Biological Cybemetics Springer-Verlag 1990 Distributed Parallel Processing in the Vertical Vestibulo-Ocular Reflex: Learning Networks Compared to Tensor Theory T. J. Anastasio t and D. A. Robinson 2 1 Department of Otolaryngology - Head and Neck Surgery, University of Southern California, Los Angeles, CA 90033, USA 2 Department of Ophthalmology and Biomedical Engineering, The Johns Hopkins School of Medicine, Baltimore, MD 21205, USA Received November 21, 1989/Accepted in revised form February 26, 1990 Abstract. The vestibulo-ocular reflex (VOR) is capable of producing compensatory eye movements in three dimensions. It utilizes the head rotational velocity sig- nals from the semicircular canals to control the contrac- tions of the extraocular muscles. Since canal and muscle coordinate frames are not orthogonal and differ from one another, a sensorimotor transformation must be produced by the VOR neural network. Tensor theory has been used to construct a linear transformation that can model the three-dimensional behavior of the VOR. But tensor theory does not take the distributed, redun- dant nature of the VOR neural network into account. It suggests that the neurons subserving the VOR, such as vestibular nucleus neurons, should have specific sensi- tivity-vectors. Actual data, however, are not in accord. Data from the cat show that the sensitivity-vectors of vestibular nucleus neurons, rather than aligning with any specific vectors, are dispersed widely. As an alterna- tive to tensor theory, we modeled the vertical VOR as a three-layered neural network programmed using the back-propagation learning algorithm. Units in mature networks had divergent sensitivity-vectors which resem- bled those of actual vestibular nucleus neurons in the cat. This similarity suggests that the VOR sensorimotor transformation may be represented redundantly rather than uniquely. The results demonstrate how vestibular nucleus neurons can encode the VOR sensorimotor transformation in a distributed manner. Introduction The primary function (Fig. 1) of many vestibular nu- cleus (VN) neurons is to relay head-velocity signals from semicircular canal primary afferents to extraocular muscle motoneurons. The purpose of this relay, com- monly known as the three-neuron-arc of the vestibulo- ocular reflex (VOR), is to stabilize the retinal image by producing eye movements that compensate for head movements (Wilson and Melvill Jones 1979). The VOR operates in three dimensions. In frontal-eyed mammals, the VOR produces almost compensatory eye rotations for yaw and pitch head rotations, but eye movements in response to roll are less than compensatory (ibid.). To simplify the development to follow, however, it will be assumed that the VOR produces perfectly compensa- tory eye movements in all three rotational dimensions. In Fig. 1, the geometrical arrangement of the six canals and the extraocular muscles of the left eye in the cat are shown; geometrical data are from Ezure and Graf (1984). Because the coordinate frames defined by the geometry of the semicircular canals and extraocular muscles are not orthogonal and differ from one an- other, a sensorimotor transformation must occur within the relay. The tensor theory of the VOR (Pellionisz and Llmas 1980; Pellionisz and Graf 1987) utilizes certain mathematical forms to develop a linear, matrix trans- formation that can describe the three-dimensional be- havior of this reflex. Tensor theory specifies unique connectivity patterns for the synaptic projections from the canal afferents to the VN neurons and from the VN neurons to the motoneurons. By seeking a unique solu- tion, it describes the connections from the six semicircu- lar canals to the six extraocular muscles of one eye (Pellionisz and Graf 1987) as though they were medi- ated by only six VN neurons. Given that there are many hundreds of such interneurons, this mathematical description indicates little about how individual VN neurons actually behave. The three-dimensional behavior of any vestibulo- ocular neuron can be described by its sensitivity-vector, defined as the axis of head (or eye) rotation for which its activity is a maximum. The sensitivity-vectors of the canal afferents are specified by canal geometry just as those of the motoneurons are specified by the pulling directions of the muscles (Robinson 1982). Since tensor theory describes a unique pattern of connectivity from the canal afferents to the VN neurons, it also specifies a unique set of sensitivity-vectors for these neurons. If such a unique connectivity pattern existed, then the sensitivity-vectors of real VN neurons should line-up along these specified directions. Instead, single-unit recordings in the cat have shown that the sensitivity-