Auditory and Vestibular Systems NeuroReport 0959-4965 # Lippincott Williams & Wilkins Simulation of pathological ocular counter-roll and skew-torsion by a 3-D mathematical model Stefan Glasauer, CA Marianne Dieterich and Thomas Brandt Department of Neurology, Klinikum Grosshadern, Ludwig-Maximilians-University, Marchioninistr. 15, D-81366 Mu È nchen, Germany CA Corresponding Author A basic version of a 3-D mathematical model for simulation of otolithic control of binocular static eye position was extended by introducting excitatory com- missural ®bers between the vestibular nuclei, and physiological non-linearities: the force±response rela- tionship of utricular neurons and a quadratic relation- ship between eye muscle innervation and force. These modi®cations appeared to be necessary in order to simulate the gain asymmetry of ocular counter-roll to lateral head tilt in patients with unilateral utricular loss. The current model can adequately simulate skew- torsion in patients with unilateral utricular loss, lesions of the vestibular nuclei, and central graviceptive path- way lesions. The direction of simulated skew-torsion corresponds satisfactorily to data from normals and patients with acute vestibular loss. The relatively low values of predicted eye deviations for peripheral vestib- ular lesions suggest that part of the effects seen in patients is caused by affection of the semicircular canals. NeuroReport 10:1843±1848 # 1999 Lippincott Williams & Wilkins. Key words: Mathematical model; Ocular counter-roll; Ocular torsion; Otolith; Skew deviation; Vestibulo-ocular re¯ex Introduction In a previous study [1], we reported our ®rst attempt to model the static part of vestibulo-ocular responses: roll of the head in the frontal plane causes static eye torsion equivalent to a shift of Listing's plane, while pitch of the head in the sagittal causes a pitch of Listing's plane [2]. Other models of vesti- bulo-ocular function presented so far have concen- trated on the dynamic aspects, e.g. nystagmus, which is caused by semicircular canal input (angular vestibulo-ocular re¯ex) and dynamic otolith input (linear vestibulo-ocular re¯ex). Our original basic model assumed that graviceptive modulation of static eye position is caused by the utricles, which transmit information to the extraocular muscles via anatomically known vestibulo-ocular pathways. To be able to simulate peripheral and central lesions of the otolith±ocular pathways, the model included a detailed matrix description of the utricles, the ves- tibular nuclei (VN), the otolith±ocular pathways, and the extraocular eye muscles. The aim of the current study was to further develop the basic model in order to adequately simulate asymmetrical ocular counter-roll (OCR) in response to head roll in the frontal plane in patients with unilateral vestibular lesions [3,4]. This task required modi®cations of the model: a more realistic force±response relationship of the utricular periph- eral neurons based on electrophysiological ®ndings in the squirrel monkey [5] was incorporated and the non-linear innervation-force characteristics of the extraocular eye muscles [6±8] were introduced. In addition excitatory commissural connections, demonstrated electrophysiologically [9], were intro- duced to approximate the predicted values of skew deviation, which surpassed the smaller values re- ported in clinical data [10]. In the following, we compare model simulation of OCR and skew-torsion to human data available from studies on normal subjects and neurological patients. Materials and Methods The structure and the algorithms of the basic version of the complete bilateral 3-D mathematical model of otolith±ocular control of static eye position version is described elsewhere in detail [1]. A short account of the main aspects together with a description of the additional features (see Fig. 1) necessary for simulating peripheral and central lesions is given below. The model contains linear static elements, such as gain and transformation matrices and summations, and non-linear static transformations. All pathways NeuroReport 10, 1843±1848 (1999) Vol 10 No 9 23 June 1999 1843