Computation of linear acceleration through an internal model in the macaque cerebellum

Abstract

A combination of theory and behavioral findings support a role for internal models in the resolution of sensory ambiguities and sensorimotor processing. Although the cerebellum has been proposed as a candidate for implementation of internal models, concrete evidence from neural responses is lacking. Using unnatural motion stimuli, which induce incorrect self-motion perception and eye movements, we explored the neural correlates of an internal model that has been proposed to compensate for Einstein's equivalence principle and generate neural estimates of linear acceleration and gravity. We found that caudal cerebellar vermis Purkinje cells and cerebellar nuclei neurons selective for actual linear acceleration also encoded erroneous linear acceleration, as would be expected from the internal model hypothesis, even when no actual linear acceleration occurred. These findings provide strong evidence that the cerebellum might be involved in the implementation of internal models that mimic physical principles to interpret sensory signals, as previously hypothesized.

Figure 1. Schematic illustrating the Tilt-While-Rotating (TWR) stimulus

(a) Constant velocity earth-vertical axis rotation (EVAR, red arrow), superimposed on nose-up and nose-down pitch tilt. The yaw and roll axes of the head are represented by black arrows. (b)–(d) Representation of the induced roll rotation, the resulting head tilt estimate and the erroneous translation signal induced during steady-state TWR. The gravito-inertial acceleration (GIA) sensed by the otolith organs is represented by a pendulum. Only in the presence of a linear acceleration (translation signal, red arrow), such as shown in Fig. 1d, a pendulum would remain aligned with the head/body vertical axis while tilted.

Figure 2. Motion signals during TWR

(a) Illustration of the sequence of motion during the TWR stimulus: The head is initially tilted (A). The EVAR (red arrow) starts at t = 0 (B), and a tilt movement is performed immediately thereafter (t = 1s) (C). A second tilt movement is performed 30 s later (D). (b), (c) Sequence of pitch tilt position and velocity (green). (d), (e) Yaw and roll (induced) angular velocity (see also Supplementary Fig. 4). Black lines represent the rotation signal coded by the canals. Blue (d) or cyan (e) solid lines represent the simulated central rotation signal (velocity storage). Because the canals are sensitive to changes in velocity, each ±10° pitch tilt (green) induces a 15.6°/s roll rotation signal (cyan). Note that this induced roll rotation signal is smaller during the first tilt because of the larger yaw rotation signal (see Supplementary Modeling for details). (f) Simulated linear acceleration (translation) signal (red). Dotted lines: real motion of the head (egocentric reference frame).

Figure 3. Model schematic and simulations

(a) Model outline and (b) simulations of induced tilt velocity (left column, cyan), induced tilt position (middle column, black) and induced linear acceleration (right column, red) during steady-state TWR. Simulations in (b) are shown without (upper row, equation 1) or with (lower row, equation 3) the somatogravic feedback. Details about the model (with 5 parameters), which represents a synthesis of several previous studies^–, can be found in (see also Supplementary Modeling).

Figure 4. Response from a translation-selective Purkinje cell during

(a) actual lateral translation (0.5 Hz), (b) actual roll tilt (0.5 Hz), (c),(d) TWR (pitch tilt during leftward and rightward EVAR at 45°/s, respectively) and (e) pitch tilt alone. (f) shows tilt position timing. Note the transient horizontal VOR (red) and firing rate changes (FR, black) in (c) and (d) (but not e) after each tilt movement (vertical dashed lines). In (a), the cell fires maximally when the acceleration is negative. Because negative pitch movements (i.e. towards nose-up) during positive (leftward) EVAR (c) and positive pitch movements during rightward EVAR (d) induce a negative acceleration (see Fig. 1 and 2), these movements were defined as being along the cell’s preferred direction (PD). Accordingly, they resulted in a firing rate increase during TWR. Note the difference in response as a function of time: The first tilt resulted in a weak firing rate decrease, instead of large firing rate increases (c) and decreases (d) seen for later movements.

Figure 5. Quantification of firing rate changes during TWR at 45°/s in preferred and anti-preferred directions

(a) Schematic illustrating definition of response along preferred direction (PD, shown in b) and anti-PD (shown in c). PD is defined as the tilt/EVAR combination that would theoretically elicit an erroneous translation in the preferred direction of the cell (expected to increase firing rate); anti-PD is the opposite direction (b), (c), (d) Comparison of firing rate changes (difference between the firing rate averaged over 1 s intervals immediately before the onset and after the offset of each tilt movement) during steady-state TWR in PD (b) and anti-PD (c), as well as during tilt without EVAR (c) vs. the cell’s gain during actual translation. In (d), PD is not easily defined, thus we plotted the absolute difference between the responses to tilt in both directions. On panels (b–c), regression lines are shown in black and confidence intervals are represented by grey bands Blue: cerebellar nuclei cells (n = 31); Red: nodulus/uvula Purkinje cells (n = 27). When separated into cerebellar nuclei and nodulus/uvula, the slopes of the PD response were 1.6 [1.11,1.98, 95% CI] and 0.9 [0.6, 1.4] m.s⁻², respectively. For the TWR responses in anti-PD, the slopes were −0.7 [−1, − 0.5] and −0.7 [−1.3, −0.15] m.s⁻², respectively. Different symbols are used for different animals (V: squares, T: circles, K: triangles).

Figure 6. Population responses, eye movements and model simulations during steady-state TWR

(a) Average changes in firing rate (lines) and confidence interval (bands) following tilt in PD (red) or anti-PD (blue) (n = 21, 20 and 17 cells in animals V,T and K at 45°/s, n = 11 in animal T at 120°/s). Superimposed black lines show the simulated estimate of translational acceleration induced by TWR. Neural responses are shown in spikes per s and simulated translation is shown in units of m.s⁻² (left and right ordinate, respectively). Peak responses in each animal have approximately the same amplitude, ~1.5 m.s⁻², which matches the slope of the regression line shown in Fig. 5a (as both represent a decoded acceleration signal). (b) Horizontal eye velocity that reflects the induced erroneous translation signal. (c) Actual tilt aVOR (green) and induced (cyan) vertical aVOR. The confidence intervals are too narrow to be visible. Superimposed black lines show the simulated estimate of the induced tilt signal (scaled by a factor of 0.8 to be compared with the induced aVOR). The tVOR (b) and aVOR (c) were evaluated using 219, 780 and 1018 trials in animals V,T and K at 45°/s, and 495 in animal T at 120°/s.

Figure 7. Comparison between initial and steady-state TWR

(a) Induced aVOR, (b) induced tVOR and (c) average firing rate (n = 58 cells) following initial TWR (red, mean ± confidence interval) and steady-state TWR (blue, mean ± confidence interval), averaged across all animals. For consistency with Fig. 6, the time axis aligned with the beginning of the tilt movement. The period during which the EVAR axis accelerates to 45°/s is indicated by a black band (from t = −1 to t = 0s). The grey band represents the period during which the tilt movement is performed (from t = 0 to t = 1.4s). The simulated induced aVOR velocity signal (from Fig. 2e, cyan), scaled by a factor of 0.8 (in order to account for the gain of the vertical aVOR) is shown for comparison in (a) (cyan dashed lines). Similarly, simulations of linear acceleration are superimposed in (c) (cyan dashed lines). The sequence of rotation signals (A,B,C) induced at the beginning of TWR (see Fig. 2) is also shown in (a). Traces in (a) and (b) were constructed from 249 (initial TWR) and 2017 (steady-state) trials. The double arrow heads in (c) illustrate the 1 s interval used to measure mean firing rate in Fig. 8a.

Figure 8. Time dependence of normalized population responses as a function of the delay between the beginning of EVAR and the tilt movement

(a) Comparison of PD neuronal responses to initial TWR and steady-state responses (n = 58 cells; same color code as in Fig. 5). The black line and gray band represent the regression line and its 95% confidence interval. Dashed line: 62% prediction (no internal model). (b) Influence of the delay between the tilt movement and the constant velocity rotation onset on the amplitude of the response (n = 11 neurons). Data points show population slope (calculated from regressions as in a). The solid black line shows exponential fit: intercept = −0.01, CI = [−0.35, 0.3], time constant = 14s (similar to the time constant of decay of the horizontal aVOR: 12, 17 and 18 s in animals V, T and K, respectively). The solid red line shows internal model simulation (Fig. 3a). The dashed red line shows the prediction assuming that the response is proportional to the induced aVOR (and no internal model).