A SINGULAR PERTURBATION MODEL OF FLUID-DYNAMICS IN THE VESTIBULAR SEMICIRCULAR CANAL AND AMPULLA

A matched asymptotic analysis is presented that describes the mechanic al response of the vestibular semicircular canals to rotation of the h ead and includes the fluid-structure interaction which takes place wit hin the enlarged ampullary region of the duct. New theoretical results detail the velocity held in a fluid boundary layer surrounding the cu pula. The governing equations were linearized for small perturbations in fluid displacement from the prescribed motion of the head and reduc ed asymptotically by exploiting the slender geometry of the duct. The results include the pressure drop around the three-dimensional endolym phatic duct and through the transitional boundary layers within the am pulla. Results implicitly include the deflected shape of the cupular p artition and provide an expression for the dynamic boundary condition acting on the two surfaces of the cupula. In this sense, the analysis reduces the three-dimensional fluid dynamics of the endolymph to a rel atively simple boundary condition acting on the surfaces of the cupula . For illustrative purposes we present specific results modelling the cupula as a simple viscoelastic membrane. New results show that the mu lti-dimensional fluid dynamics within the enlarged ampulla has a signi ficant influence on the pointwise deflection of the cupula near the cr ista. The spatially averaged displacement of the cupula is shown to ag ree with previous macromechanical descriptions of endolymph how and pr essure that ignore the fluid-structure interaction at the cupula. As a n example, the model is applied to the geometry of the horizontal semi circular canal of the toadfish, Opsanus tau, and results for the defle ction of the cupula are compared to individual semicircular canal affe rent responses previously reported by Boyle & Highstein (1990). The cu pular-shear-angle gain, defined by the angular slope of the cupula at the crista divided by the angular velocity of the head, is relatively constant at frequencies from 0.01 Hz up to 1 Hz. Over this same range, the phase of the cupular shear angle aligns with the angular velocity of the head. Near 10 Hz, the shear-angle gain increases slightly and the phase shows a lead of as much a 30 degrees. Results are sensitive to the cupular stiffness and viscosity. Comparing results to the affer ent responses represented within the VIIIth nerve provides additional theoretical evidence that the macromechanical displacement of the cupu la accounts for the behaviour of only a subset of afferent fibres.