The vestibular semicircular canals are internal sensors that signal the magnitude, direction, and temporal properties of angular head motion. Fluid mechanics within the 3-canal labyrinth code the direction of movement and integrate angular acceleration stimuli over time. Directional coding is accomplished by decomposition of complex angular accelerations into 3 biomechanical components-one component exciting each of the 3 ampullary organs and associated afferent nerve bundles separately. For low-frequency angular motion stimuli, fluid displacement within each canal is proportional to angular acceleration. At higher frequencies, above the lower corner frequency, real-time integration is accomplished by viscous forces arising from the movement of fluid within the slender lumen of each canal. This results in angular velocity sensitive fluid displacements. Reflecting this, a subset of afferent fibers indeed report angular acceleration to the brain for low frequencies of head movement and report angular velocity for higher frequencies. However, a substantial number of afferent fibers also report angular acceleration, or a signal between acceleration and velocity, even at frequencies where the endolymph displacement is known to follow angular head velocity. These non-velocity-sensitive afferent signals cannot be attributed to canal biomechanics alone. The responses of non-velocity-sensitive cells include a mathematical differentiation (first-order or fractional) imparted by hair-cell and/or afferent complexes. This mathematical differentiation from velocity to acceleration cannot be attributed to hair cell ionic currents, but occurs as a result of the dynamics of synaptic transmission between hair cells and their primary afferent fibers. The evidence for this conclusion is reviewed below.