(A,B) Simulation of constant velocity rotation lasting 60 s, followed by 60 s where the head doesn’t rotate (top row, , blue). The semicircular canal signal (top row, V, magenta) decreases exponentially (time constant = 4 s) during the rotation ( = –; represented by cyan lines). The deceleration at t = 60 s induces a canal after-effect. During active rotation (A), the final estimates of rotation velocity ( lower row, blue) and canal dynamics (, lower row, cyan) are identical to their real respective values (A, top panel). In contrast, during passive rotation (B), the final estimate of rotation differs from the stimulus: decreases exponentially toward zero, although the decrease is slower (time constant = 16.5 s) than that of the canals. This prolonged estimate arises from the contribution of the internal state variable , which rises at the beginning of rotation as in (A) but reaches a maximum and decreases towards zero. (C) Simulation of optokinetic stimulation, i.e., a visual stimulus rotating at constant velocity (, top row, green) while the head is immobile. The rotation estimate rises immediately to 70% of stimulus velocity at the beginning of the stimulation, and then increases exponentially to 96% of stimulus velocity. rises exponentially to the stimulation velocity but doesn’t exhibit an immediate increase at the beginning of the stimulation. Both and persist at the end of the rotation. The simulated results in B and C match the perceptual (Bertolini et al., 2011) and reflex (Raphan et al., 1979) responses to these stimuli. Furthermore, the rotation estimate is equal to the Kalman feedback signal during passive rotation (B) and visual surround rotation (C). As expected, VN neurons responses follow the same dynamics than in these conditions (Waespe and Henn, 1977; Yakushin et al., 2017). Previous work (Raphan et al., 1979; Laurens and Angelaki, 2011) used an intermediate variable called ‘velocity storage’ to model low-frequency responses to vestibular and visual stimulation. The state variable in the Kalman filter model is identical to the velocity storage. An optimal model based on particle filtering also produced identical results (Laurens and Droulez, 2007; Laurens and Droulez, 2008). The Kalman filter also predicts that rotation perception should last indefinitely during active constant-velocity rotation (A), unlike passive rotation (B), in agreement with experimental findings (Solomon and Cohen, 1992; Guedry and Benson, 1983; Howard et al., 1998; Jürgens et al., 1999). Note, however, that sustained rotation perception during active rotation saturates at about 10°/s in humans (Guedry and Benson, 1983; Howard et al., 1998), possibly related to the saturation of the velocity storage () at velocities of ~20–30°/s in humans (Laurens et al., 2011). Therefore, actively rotating at substantially higher velocities may lead to disorientation and vertigo (as commonly experienced by children and waltz dancers). Note that all Kalman filter simulations do not include this saturation. Sustained rotation perception does not saturate in macaques at velocities up to 200°/s (Solomon and Cohen, 1992).