The dynamic vestibular response behaves like a second degree differential equation with a very short time constant (~1/100 sec) and a long time constant (20-30 sec).
The stimulus is almost certainly rotational head movements, which cause angular acceleration.
The short-term response mimics the angular velocity, which is taken to indicate a mechanical integration.
The long-term response is characterized by an adaptive return to baseline in the presence of a sustained constant angular velocity.
The model is generally characterized by a driving function that is a torque, therefore proportional to the second derivative of an angular displacement. The semicircular duct responds with an inertial drag from the fluid within the duct, resistance to flow of the fluid because of viscosity (proportional to first derivative of angular displacement), and distortion of the cupular membrane as it resists the relative displacement of the fluid (proportional to angular displacement). To this one might add the distortion of the duct itself as it tries to accommodate the back flow, which is proportional to fluid displacement or angular displacement.
The semicircular duct is quite narrow, which results in a substantial resistance to flow along its axis, and it is flattened, which means that has a cross-sectional area that is less than maximal for its circumference. If it is necessary for fluid to flow through the duct, there will be a resistance that will be partially offset by outwardly directed pressure on the duct walls, which will cause the duct to become more circular in cross-section. This conformational change will act as a capacitive or spring element.
A pressure difference across the cupula will cause the cupula to billow like a sail in a breeze. Since the cupula is elastic, it will provide a restorative force that resists fluid flow and returns the fluid to status quo if it is displaced. Most head movements are apt to be of short duration and moderate speed. Under those conditions there is probably very little movement of the cupula and it acts as a linear capacitive element.
As the cupula billows it produces a transient pressure changes in the ampulla and the proximal duct. These are partially absorbed by fluid flow through the duct and partially by changing the duct’s cross-sectional shape to accommodate more or less fluid. The resistance to fluid flow through the duct is proportional to the velocity of flow. The distension/compression of the duct is proportional to the amount of fluid flow.
Because of the much greater resistance of the duct to fluid flow, one can open the circuit and to the first approximation model the cupular-duct dynamics as two chambers with an elastic diaphragm between. On one side there is low resistance to flow and a large reservoir of fluid. On the other side is a high resistance to flow and a small volume of fluid. The driving force is the pressure at the utriculo-ampullary junction, Fu. The magnitude of the force is proportional to the rate of angular displacement in a manner that depends on the axis and direction of rotation, the location, orientation, and cross-section of the junction. The amount of fluid flow will be proportional to the driving force, V = Fu * R. The cupular membrane is distorted by the fluid displacement, V, and it generates a restoring force proportional to the amount of distortion. In addition, the compression or rarefaction of the fluid on the duct side of the cupula will cause conformational changes in the duct and will cause fluid flow through the duct. The conformational changes will be small and therefore linearly related to the volume displaced. The flow through the duct will be laminar, therefore proportional to the rate of change of the volume displaced.
The forces on the cupula will be balanced so we can write a number of expressions