The mechanics of the growing semicircular canal.

The function of the three semicircular canals of the vertebrate labyrinth which are situated in three approximately perpendiailar planes is the detection of angular accelerations. Because of the inertia of the liquid (endolymph) in the membranous canal a fluid current is set up in the endolymph by angular acceleration. This endolymph current displaces the cupula in the ampulla. The cupula-sensory system converts the mechanical signal into a train of action potentials travelling to the brain, simultaneously with the actual deformation of the cupula. However, during small physiological accelerations the displacements of the cupula are submicroscopical. So this actual deformation can only be studied with a mathematical model, which closely approaches the real geometry of the growing semicircular canal. The model of the semicircular canal is a tube in a closed loop of non-circular shape with a narrow duct and a wide part, the cross-section being circular along the whole length of the narrow duct. The tube has rigid walls without bifurcations (the interference with the endolymphatic current by the two other canals is neglected) and contains an incompressible liquid and an elastic cupula. Since the common semicircular canal (for birds and fish) does not conform to a torus in shape, the generally accepted overcritically damped torsion pendulum model (Egmond, 1952; Groen, 1952; Jones, 1963) is not valid for a precise dimensional study of the semicircular canal (see Fig. 1). The equation of motion for the endolymph in any semicircular canal, including a non-circular region, will be used. In this overcritically damped second-order model, linear displacement, linear velocity and linear acceleration of the endolymph are related to the angular acceleration (de Vries, 1956). This equation will be solved for three different kinds of stimuli, namely stepvelocity input, step in angular acceleration and sinusoidal angular oscillations. The endolymph displacements caused by these stimuli will depend on the dimensions of the semicircular canal. A mechanical sensitivity factor can be derived and expressed in geometrical parameters of any semicircular canal. This provides the theoretical basis of a previous study on dimensions and sensitivities of semicircular canals in fish (ten Kate, 1969, 1970). The sensitivity of the canal may also inform us about the actual deformation of the cupula during small physiological accelerations. In connexion with this it should be noticed that no sufficient evidence is available for