Optimization of the mechanical performance of a two-duct semicircular duct system--part 2: excitation of endolymph movements.

The endolymph flow inside the semicircular ducts is analytically investigated by considering a system of two hydrodynamically interconnected ducts. Rotation of this system adds an amount of motion (momentum) to parts of it. This results in an endolymph flow in generally all vestibular parts. The "external impulses" are the impulses which emerge by rotation of exclusively a particular vestibular part. The real impulses can be calculated from a set of equations which contain the external impulses. Analytical expressions are derived for the initial velocities in the ducts and for the maximum endolymph displacements. These formulae contain the external impulses and the ratios of: (1) the radii of crus commune and ducts (gamma), (2) the lengths of crus commune and ducts (lambda). It was proven that an interconnected system composed of two ducts, and also a system composed of two such semicircular duct systems, behaves as a pure rotation transducer (like a single duct does), also when it is rotated excentrically. Duct systems with polygonal and circular geometries were used to evaluate whether an optimal value of lambda would exist (gamma was already considered elsewhere). Optimum values of lambda in a range of about 0.10-0.52 were found. This rather wide range of values agrees with values from measurements. Optimization of an interconnected duct system appeared to be equal to optimization of a system composed of separate ducts.

Optimization of the mechanical performance of a two-duct semicircular duct system--part 1: dynamics and duct dimensions.

The classical representation of the semicircular duct system consists of three separate duct circuits. The ducts are, however, in reality, hydrodynamically interconnected. Muller & Verhagen (1988a,b) derived equations for the mechanical behaviour of an interconnected system with three ducts (anterior, posterior and horizontal). An analytical solution of these equations would, however, be too complex to provide surveyable formulae. A system of two interconnected ducts avoids this complexity whilst keeping the essentials of the coupling of ducts intact. The solution of the equation of motion leads to expressions for time constants and maximal endolymph excursions which are functions of morphological parameters, viz. the ratios of radii (gamma) and lengths (lambda) of the common vestibular part (crus commune or utriculus) and the ducts. The system possesses two short time constants which are shown to have similar values. The maximum endolymph displacements in the two ducts after a steplike stimulus are the products of the respective initial velocities and combinations of time constants. The initial velocities depend strongly on the position of the labyrinth with respect to the excitating rotation vector. Measured data of gamma and lambda are compared with the theoretical results. For gamma, excellent agreement was found. lambda is treated elsewhere.

Optimization of the mechanical performance of a two-duct semicircular duct system--part 3: the positioning of the ducts in the head.

In the majority of vertebrates, the horizontal duct of the vestibular system lies approximately in the yawing plane of the head. The positioning of the vertical ducts, however, is not in the pitch- and roll planes but the vertical ducts generally lie under an angle of about 30-45 degrees relative to the medial plane. Using the equations for a hydrodynamically interconnected two-duct system, optimal positions of the vertical and horizontal ducts in different vertebrate groups can be derived. It was stated that the mean response of the vertical ducts should be optimized. This leads to a symmetrical positioning of the vertical ducts with respect to the medial plane. In all observed vertebrate groups, a solution of mu =(pi-alpha)/2 is found (mu is the angle of the vertical ducts relative to the medial plane, alpha is the angle between the vertical duct planes). For alpha=90 degrees, this provides an equal sensitivity for pitch- and roll- movements. For alpha>90 degrees, a larger sensitivity for pitch movements is obtained, at the expense of a lower sensitivity for roll movements. It is argued that the angle alpha between the vertical ducts may vary from 90 to 120 degrees. In most vertebrates, the centre of mass is stabilized by e.g. fins, tri- or quadrupedal stability, a crawling body or upside-down resting positions (e.g. bats). Birds are generally biped, so in walking they are also rather sensitive to roll. These features are related to labyrinth positioning in the head.