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

Transcription

1 Computation of linear acceleration through an internal model in the macaque cerebellum Jean Laurens, Hui Meng and Dora E. Angelaki Supplementary Materials Legends of Supplementary Movies 1 and 2 Supplementary Modeling Supplementary Fig. 1 Recording Map Supplementary Fig. 2 Eye Movements Supplementary Fig. 3 and Supplementary Table 1 Yaw velocity signals during steady-state TWR Supplementary Fig. 4 Comparison between the nodulus/uvula and cerebellar nuclei populations Supplementary Fig. 5 Temporal relationship between VOR components and neuronal responses Supplementary Fig. 6 1

2 Legends of Supplementary Movies 1 and 2 Supplementary Movie 1: Illustration of the TWR stimulus. (a) Real stimulus (Rotator). Only the head of the subject is represented (the rail system onto which the rotator is mounted is not shown). The red and green arrows represent the rotation axis of the EVAR and tilt movement, respectively. For better visibility, this movie shows a tilt angle of 25 and one tilt movement every 5s. (b) Decomposition of the EVAR velocity in an egocentric frame of reference. This panel shows the head tilting back and forth relative to the EVAR velocity vector (red). The projection of this vector onto the yaw and roll axes (black arrows) are shown in blue and cyan. Note that the projection onto the yaw axis is approximately constant (but see Supplementary Fig. 4), whereas the projection onto the roll axis switches direction with each tilt movement (as in Fig. 2d,e; broken lines). (c) Rotation signals sensed by the semicircular canals: yaw (blue) and roll (cyan) (comparable to black lines in Fig. 2d,e). The brain continuously reconstructs the net rotation by summing the yaw and roll velocity signals (red arrow). During initial TWR, the yaw and roll velocity components are correctly sensed by the canals and, even though these signals decay with a time constant of 4s, the resulting net rotation signal (red) remains aligned with the earth-vertical. During steady-state TWR, however, yaw velocity (blue) changes minimally and the net rotation signal (red) is closely aligned with the roll axis (cyan) following each pitch tilt. Supplementary Movie 2: Motion perception during TWR. (a) Real stimulus (Rotator) showing actual pitch tilt (10 ). (b) Simulated motion perception generated by the internal model. During initial TWR, head tilt is perceived accurately, but the perception of rotation decreases over time. During steady-state TWR, each forward and backward tilt movement induces erroneous sideward tilt and translation signals. Note that the amplitude and duration of illusory motion shown in this movie were scaled for better visibility and are not accurate. 2

3 Supplementary Modeling The inputs to the model of Fig. 3 are the head angular velocity vector, ω(t), the gravity vector, g(t), and the linear acceleration (translation) vector, a(t). These inputs are all expressed in head (egocentric) coordinates. The model computes the following sensory signals: the three-dimensional rotation transduced by the canals, V(t), and the gravitoinertial acceleration, GIA(t), measured by the otolith organs, according to: dv(t)/dt = -1/τ c V(t) +dω(t)/dt (S1) GIA(t) = g(t)-a(t) In these simulations, the canals are modeled as a high-pass filter with a time constant τ c, as in previous studies (Laurens and Droulez, Biol. Cybern. 2007; Laurens and Angelaki, Exp. Brain Res. 2011; Merfeld, Exp. Brain Res. 1995). The model then computes the internal estimate of head rotation Ω(t), as well as the internal representation of gravity G(t) and linear acceleration A(t) through the following equations: Ω(t) = V(t)+VS(t) (S2) dvs(t)/dt = -1/τ VS VS(t) + k v V(t) + k F GIA(t)xG(t) (S3) dg(t)/dt = G(t)x Ω(t) -1/τ S (G(t)-GIA(t)) (eq 3 in main text) A(t) = GIA(t)-G(t) (eq 2 in main text) In these equations, VS is the velocity storage (Raphan et al., in: Control of gaze by brain stem neurons, 1977), which controls the time constant of the central rotation signals and which is modeled as a leaky integrator with time constant τ VS, and k v being a gain term. The internal rotation estimate, Ω(t), is the sum of the vestibular signal, V(t), and the velocity storage signal, VS(t). The term k F GIA(t)xG(t) implements the velocity feedback (Fig. 3a; see Laurens and Angelaki, Exp. Brain Res for details), which influences the time constant of the induced rotation signal (as in Fig. 6c). The central integration of rotation signals and the corresponding otolith influence on this integration have been extensively studied previously (Hess and Angelaki, Exp. Brain Res. 1993; Angelaki and Hess, Exp. Brain Res. 1994; J. Neurophysiol. 1995; Laurens and Droulez, Biol. Cybern. 2007; Laurens et al., J. Neurophysiol. 2010; J. Neurosci. 2011; Laurens and Angelaki, Exp. Brain Res. 2011; Merfeld et al., J. Vestib. Res. 1993; Merfeld, Exp. 3

4 Brain Res. 1995); thus, they are simply represented by a single central processing element in Fig. 3a. The internal estimate of gravity, G(t), is computed by integrating the rotation signal over time according to the vector product G(t) x Ω(t) (equation 1). In addition, the term -1/τ S (G(t)-GIA(t)) is responsible for the somatogravic feedback (equation 3), which continuously realigns the internal estimate of gravity with the GIA vector. This loop allows the model to be stable; otherwise discrepancies between g(t) and G(t), which can arise due to noise and incorrect rotation signals, would not be corrected and cause severe disorientation (see Laurens and Angelaki, Exp. Brain Res for details). To better illustrate the role of the somatogravic feedback loop, we show simulations performed without (Fig. 3b, upper row) or with (Fig. 3b, lower row) this feedback. In both simulations, the induced rotation signal is the same. In the absence of the somatogravic feedback, the tilt estimate would be computed by a straightforward integration of the induced rotation signal. As a consequence, it would reach a high value (~50 ) and remain at that level when the induced signal reaches zero. The resulting acceleration estimate would exceed 7.5 m/s 2 and remain indefinitely. The term -1/τ S (G(t)-GIA(t)) creates a feedback loop which brings G(t) in alignment with GIA(t). As a consequence, the simulated induced tilt signal would have both smaller amplitude (with a peak of ~5.3 ) and would decay to zero. Similarly, the induced translation estimate would reach a lower magnitude (~1 m/s 2 ) and also decrease to zero. The time constants of decay for both induced tilt and translation signals are both governed similarly by the term k F only. Our goal was to use model simulations to predict neuronal responses. Thus, we set 4 model parameters either based on existing literature or by considering behavioral measures (eye movements) in each animal, and only one was set to a common value across all animals based on neuronal responses. The model of Fig. 3a has 5 parameters, which were set to the following values: (1) The time constant of the canals (τ c ) and (2) the gain of the velocity storage (k v ) were set to τ c =4s and k v =0.2 respectively. (3) The time constant of the velocity storage (τ VS ) was set to 12, 17 and 18s in animal V, T and K, respectively, values chosen to simulate the time constant of the yaw avor in each animal. Note that this 4

5 time constant governs only the rate at which the EVAR signal decreases and plays a negligible role in the steady-state stimulations shown in Fig. 3. (4) The gain of the velocity feedback (k F ), which governs the rate at which induced signals decay, was set to 0.023, 0.06 and 0.1 in animal V, T and K, respectively. These values were determined to match the induced avor (Fig. 6c). (5) Finally, the time constant of the somatogravic feedback was set to τ S =0.5s. The simulations shown in Fig. 2, 3b and 7 were performed with τ VS = 14 s and k F = 0.05, values obtained by fitting the average responses across animals. Note that in the simulations of Fig. 2d, the central yaw rotation signal (blue) does not decay exactly to zero, but rather maintains a small non-zero steady-state value (see also Supplementary Fig. 4). This steady-state yaw velocity signal is theoretically the result of the velocity feedback (governed by k F ) in the model and has been measured experimentally as a steady-state slow phase eye velocity component during TWR (Hess and Angelaki, Exp. Brain Res. 1993; Raphan et al., Brain Res. 1983). Finally, in order to interpret the responses to initial TWR, one must take into account the fact that the amplitude of the induced rotation signal differs compared to steady-state. Specifically, note that if the first tilt movement occurred simultaneously as the onset of EVAR, then the induced signal would be half as large as in subsequent tilts. This can be explained by considering Fig. 2a,e. When the EVAR starts (from 1 to 2 in Fig 2a), the roll component of head velocity, i.e. the projection of the EVAR vector on the naso-occipital axis, changes from 0 to sin(10º) = -7.8º/s. This velocity is correctly encoded by the canals (Fig. 2e, black line). When the first tilt movement occurs immediately afterwards (from 2 to 3 in Fig. 2a), roll velocity changes by 15.6º/s (i.e., from -7.8 to +7.8º/s), which is detected by the canals and added to the previous velocity, resulting in a net roll velocity signal of = +7.8º/s. In contrast, the next movement (from 3 to 4 in Fig. 2a) occurs 30 s later, i.e. in the steady-state, when the canal signals have decreased to 0. These subsequent tilt movements cause the roll velocity to change from 0 to ±15.6º/s, i.e., twice the magnitude of the signal induced during the first tilt. In our experiment, the roll rotation signal during initial TWR is ~62% (rather than 50%) of the steady-state value. This because of the filtering that takes 5

6 place in the 2.4 s interval between the beginning of the EVAR and the completion of the first tilt. The difference in induced tilt amplitude between initial and steady-state responses can be seen in the induced avor (Fig. 7a, red vs. blue), as well as in model simulations (Fig. 7a, cyan; which reflect the simulated rotation signal, scaled by 0.8 to account for the avor gain). In line with theory and simulations, the induced avor following the first tilt is initially negative (when the EVAR starts, black band) and then reverts sign when the first tilt movement is completed (grey band). The induced tvor during the first tilt movement was small and short-lasting, such that it never became significantly different from 0 (as shown by the 95% confidence intervals; Fig. 7b, red band). These early tilt avor and tvor responses contrast with the corresponding eye velocity evoked during steady-state (Fig. 7a,b, compare red vs. blue curves). Note that, although the tilt and induced avor can be directly measured as the vertical and torsional eye velocity, the tvor (horizontal eye velocity) is superimposed on the yaw (horizontal eye velocity) avor. Fortunately, it is possible to separate the tvor from the yaw avor using a similar method as in Merfeld et al. (Nature 1999), which is based on an important property of the tvor: lateral linear accelerations evoke a horizontal tvor in darkness, whereas conjugate horizontal slow phase eye velocity is negligible for fore-aft movements (Angelaki, J. Neurophysiol. 1998; Angelaki and Hess, Nat. Rev. Neurosci. 2005; McHenry and Angelaki, J. Neurophysiol. 2000; Paige and Tomko, J. Neurophysiol. 1991; Telford et al., J. Neurophysiol. 1997). Thus, only pitch TWR runs (Supplementary Fig. 3b), but not roll TWR runs (Supplementary Fig. 3e), produce an observable conjugate tvor in darkness. As shown in Supplementary Fig. 3a,b, EVAR with a positive sign induces a yaw avor with a negative sign. Accordingly, we used a variable S yawavor = -1 during positive EVAR and S yawavor = 1 during negative EVAR. Furthermore, a positive pitch movement during positive EVAR induces a negative tvor (Supplementary Fig. 3a,b). Accordingly, we created a variable S tvor equal to -1 or 1, depending of the sign of the tvor during pitch TWR. S tvor is equal to 0 during roll TWR, which doesn't induce any tvor, and to ±cos(45) during TWR along intermediate axes (see Merfeld et al., Nature 1999). Using these variables, the net horizontal VOR follows the equation: 6

7 horizontalvor(t) = yawavor(t)*s yawavor + tvor(t)*s tvor + bias(t), where yaw avor(t) and tvor(t) are the amplitude of the yaw and translational VOR, respectively. The term bias(t) was added to account for any spontaneous nystagmus (e.g. a rightward bias). We solved this equation by performing a linear regression at each point in time (t), across all trials: the regression coefficients are yaw avor(t) and tvor(t). A similar approach was used to compute the tilt and induced avor (Supplementary Fig. 3c). Note that the tvor, particularly when measured in darkness, varies in both peak response and dynamics among animals (Paige and Tomko, J. Neurophysiol. 1991; Angelaki, J. Neurophysiol. 1998; Fig. 6b, magenta; see also Supplementary Table 1). The largest peak eye velocity for 45º/s rotations was seen in animal V (12±1.6 /s). Because of the small horizontal eye velocity, animal T was also tested with 120º/s rotations. Although TWR at 120 /s induced the same tilt avor (because the tilt movement was the same), the induced avor was larger (see Supplementary Table 1; peak at 29±1.2 /s, as compared to 11±0.7 /s during TWR at 45 /s). As illustrated in the rightmost column of Fig. 6b, the horizontal eye velocity, which reflects the erroneous translation estimate (Merfeld et al., Nature 1999), was larger and rose faster to its peak than those for 45º/s rotations (see Supplementary Table 1). Because of the small number of trials with tilts at 5-20s after TWR onset, we have not analyzed the time course of the tvor, although comparisons between the tvor during steady-state and first tilt have been illustrated (Fig. 7b). The model described above follows the structure of an inverse sensory model, in which the physical laws are inverted. For example, the actual physical relationship GIA(t) = G(t)-A(t) is inverted by equation (2), i.e., A(t) = G(t)-GIA(t). Other authors (Borah et al., Ann. NY Acad. Sci. 1988; Merfeld et al., J. Vestib. Res. 1993; Merfeld, Exp. Brain Res. 1995) have formulated hypotheses similar to ours on the basis of a forward model, whose principle is outlined in Supplementary Fig. 1. In this alternate formulation, signals are fed though forward models, which simulate the physical laws that govern these organs. The outputs of these alternate, forward sensory models are predicted sensory signals (GIA'(t) and V'(t)) which are then compared to the actual sensory inputs. Any difference between the predicted and actual signals creates a 7

8 feedback, which corrects the motion estimates. The somatogravic feedback is not represented explicitly in these models; instead the somatogravic effect is produced by the design and weight of the feedback loops. Although there are conceptual differences between forward and inverse sensory models, these two approaches have always yielded largely equivalent predictions. Indeed, the common components of both approaches (i.e., the forward model of Supplementary Fig. 1 and the inverse model of Fig. 3) are the brain s ability to integrate rotation information into an internal estimate of gravity, and to solve the GIA ambiguity based on an internal model of the equivalent principle. Therefore, they both provide identical predictions of the effects of TWR, and either one could be used as the theoretical basis in the present study. We have chosen the inverse sensory model because it is easier conceptually and easier to implement using Bayesian inference (Laurens and Angelaki, Exp. Brain Res. 2011). An important difference between sensory and motor internal models is also worth mentioning. In a motor system, an internal model generates a motor command and operates in closed loop by receiving sensory feedback, hence one can distinguish a forward model, which predicts a sensory consequence, from an inverse model, which generates the motor command. In contrast, the sensory internal model in the vestibular system generates estimates of head motion in open loop. These estimates do indeed ultimately drive a motor command (e.g., the tvor) but this motor action doesn't influence the motion of the head, and doesn't have any sensory consequences (since the animal is moved in darkness). Therefore, the brain never receives any feedback or sensory input as a result of the motor command. This is a notable conceptual difference between the sensory internal model of the equivalence principle, as presented here, and the more traditional application of forward/inverse models in motor control. 8

9 EVAR Peak ( /s) t peak Peak ( /s) Time cst. t peak Peak ( /s) Time cst. t peak Peak (m/s 2 )Time cst. Animal V 45 /s 19 +/ s 12 +/ s 3.9 s 12 +/ s 3.0 s 1.5 +/ s Animal T 45 /s 17 +/ s 11 +/ s 6.5 s 5 +/ s 2.4 s 1.2 +/ s Animal K 45 /s 19 +/ s 10 +/ s 5.4 s 5 +/ s 2.4 s 1.5 +/ s Animal T 120 /s 17 +/ s 29 +/ s 3.0 s 13 +/ s 2.4 s 1.8 +/ s Supplementary Table 1. Peak amplitude and time course of eye velocity and population response during steady-state TWR. The table indicates the time to peak (t peak ), the peak value (±95% confidence interval) and the time constant (Time cst.) of the response variables shown in Fig. 6. The time to peak is defined as the time between the beginning of the tilt movement (t = 0) and the moment where the response rises to 90% of its maximum value. The time constant of decay was computed by fitting a single exponential to the response following the peak. VOR: n = 219, 780 and 1018 in animals V,T and K at 45 /s, and 495 in animal T at 120 /s. Population response: n = 21, 20 and 17 cells in animals V,T and K at 45 /s, n = 11 in animal T at 120 /s. GIA(t) V(t) Feedback signals comparison W(t) Motion estimates A(t) G(t) Forward model of otoliths Forward model of canals GIA (t) V (t) Predicted sensory signals Supplementary Fig. 1: Processing of vestibular information with a forward internal model (alternative formulation to the model of Fig. 3) This schematic model is representative of the models of Borah et al. (Ann. NY Acad. Sci. 1988), Merfeld (Exp. Brain Res. 1995) and Glasauer and Merfeld (in: Three-dimensional kinematics of eye, head and limb movements, 1997). 9

10 Supplementary Fig. 2: Reconstructed three-dimensional locations of the recorded cells in stereotaxic coordinates, shown separately for each animal. (a)-(c) Frontal views; (e)-(g) Saggital views. Each square corresponds to a single neuron, color-coded according to its type/location: Blue: cerebellar nuclei cells; Red: nodulus/uvula Purkinje cells, separated into identified (filled symbols, where each complex spike was followed by a pause in simple spike firing for at least 10ms) and "putative" (open symbols). Black (cerebellar nuclei) and grey (nodulus/uvula) circles illustrate cells recorded from the same animals using different experimental protocols. Eye movement-sensitive cells in the cerebellar nuclei are shown with cyan crosses. In each animal, the position of the abducens nuclei was first identified and then used to reconstruct the stereotaxic coordinates of the recorded cells. Recording locations of cerebellar nuclei neurons include mostly the rostral fastigial, but potentially also the anterior interposed, nuclei. Most nodulus/uvula cells were recorded within a 6 mm region directly ventral to the rostral fastigial nuclei, which includes mostly the nodulus and to a lesser extent the ventral uvula. In (d) and (h), the boundaries of the corresponding anatomical structures were reconstructed based on the macaque brain atlas (brainmaps.org: Mikula et al., in: Interactive Educational Media for the Neural and Cognitive Sciences, Brains, Minds & Media, 2008; Paxinos et al.: The rhesus monkey brain in stereotaxic coordinates. 3rd ed. Academic Press, San Diego, 2000). Int.: interposed nuclei, Med.: medial (fastigial) nuclei, Nod.: Nodulus. The presence of complex spikes recorded simultaneously with simple spikes provides strong support that we have recorded from Purkinje cells, rather than cerebellar interneurons.. 10

11 Supplementary Fig. 3. Example eye movements during (a-c) pitch and (d-f) roll TWR. (a), (d) EVAR velocity and pitch/roll tilt stimuli. (b), (e) Horizontal and (c), (f) vertical eye velocity. All runs were performed with the same EVAR direction (leftward), but initial tilt positions were opposite (a,d: red vs blue curves). The curves shown in (b,c) and (e,f) represent averages across 3 trials in animal V. Pitch TWR elicits an induced roll (torsional) avor and an induced horizontal tvor, whose direction reverses with the pitch direction, and which is superimposed on the yaw avor (b, black). Roll TWR elicits an induced vertical avor (f), but no conjugate horizontal tvor in darkness (e). (g, h) Average torsional VOR (red and blue curves) during pitch and roll TWR (n=73 and 77 trials, respectively) recorded in two additional animals implanted with a dual eye coil (details of implantation and analyses in Klier et al., J. Neurosci. 2006) in order to measure three-dimensional (horizontal, vertical and torsional) eye movements. For comparison, the average induced avor and tilt avor curves obtained by measuring vertical eye movements are shown in grey. During pitch TWR (g), the induced avor is around the roll axis. During roll TWR, the tilt avor is around the roll axis. Thus, torsional and vertical avor mirror each other. This confirms that no additional torsional eye velocity beyond the expected avors occurred during steady-state TWR. 11

12 Supplementary Fig. 4: Yaw velocity signals. (a) Projection of the EVAR vector (red) on the yaw axis of the head (black arrow, the projection is shown as a dashed blue line). During each tilt movement, the projection of the EVAR vector on the yaw axis increases when the head nears upright and decreases when the head tilts away. The tilt angle used in this panel is 30. (b-c) Same simulations as in Fig. 2d. The yaw signals following TWR in the steady-state (b, red rectangle) are detailed in (c). During tilt (grey band), a small yaw velocity component transiently activates the horizontal canals during each tilt (black, peak velocity 0.6 /s). This components is visible in the central yaw signal (blue). Due to central processing (Merfeld et al., J. Vestib. Res. 1993; Merfeld, Exp. Brain Res. 1995; Angelaki and Hess, J. Neurophysiol. 1995; Laurens et al., J. Neurophysiol. 2010; Laurens and Angelaki, Exp. Brain Res. 2011), a longer-lasting yaw signal develops after each tilt movement. This component doesn't reach zero between two successive tilts. (d): yaw avor measured during steady-state TWR, which matches the simulation accurately. Note the small amplitude. Importantly, by following the analysis of Merfeld et al. (Nature 1999), the tvor and yaw avor components can be satisfactorily distinguished (see also Supplementary Fig. 3). 12

13 Supplementary Fig. 5: Response profile of individual neurons and comparison between neuronal populations. (a), (b) Average (±95% CI) steady-state responses of all 'confirmed' (red) and 'putative' (black) Purkinje cells and cerebellar nuclei neurons (blue) during TWR in the preferred direction (PD), anti-pd or during control. (c,d) Acceleration signals decoded from the populations of nodulus/uvula (n=27) and cerebellar nuclei (n=31) neurons (c) and 'putative' (n=13) and 'confirmed' (n=14) Purkinje cells in the nodulus/uvula (d), computed as follows: Decoded Acceleration = (FR PD -FR Control )/S, where FR PD is the average firing rate across the population during TWR in the PD (in spikes per s), FR Control is the average response during control (in spikes per s), and S is the gain to sinusoidal translation in (spikes per s)/(m.s -2 ). The black bands below the curves indicate the points in time at which the decoded signals are significantly different (t-test, with p < 0.01). Although cerebellar nuclei responses were generally more modest in amplitude, there were no significant differences in nodulus/uvula and cerebellar nuclei population responses, particularly during peak amplitude (c). There were no differences in population responses of 'confirmed' and 'putative' Purkinje cells (d). Thus, data have been presented together in the main text. Note that cerebellar neurons show a larger firing rate response in their "on direction" than firing rate decrease in their "off direction". This is not unexpected and often described in other cerebellar areas (e.g., the primate flocculus; Miles et al., J. Neurophysiol. 1980). 13

14 Supplementary Fig. 6: Temporal relationships between various motion variables. (a) During each tilt movement (grey band), the tilt avor (green) occurs in nearly perfect synchrony with stimulus velocity (black). The induced avor (cyan) grows steadily and peaks after the tilt avor peak. The neuronal response (red, decoded acceleration signal) follows the induced avor with roughly similar dynamics and reaches a peak ~ ms later. The tvor (magenta) builds up even more slowly and peaks at ~3s, well after the neuronal response. Data shown represent normalized averages of all cells/sessions recorded (a.u.: arbitrary units). (b) Model simulations of the same variables. Note the striking match with actual response profiles in (a). Even while ignoring transmission delays of neuronal signals, this analysis demonstrates that temporal differences observed experimentally between tilt avor, induced avor, tvor and neuronal responses can be explained by the dynamic relationships between motion variables in the model. (c) Schematic representation of the filtering relationships between various variables. The model in Fig. 3 can be approximated by filtering the tilt velocity though a series of leaky integrators, but it has been expanded to also simulate the tvor under the assumption it is driven by a mixture of acceleration and velocity signals (Angelaki, J. Neurophysiol. 1998). Though curve fitting, we estimated that the tvor could be approximated by: d(tvor)/dt = 0.63 da/dt A -1/12.5 tvor, where A is the decoded acceleration signal. The corresponding transfer function is H(ω) = (4+8.4*j*ω)/(1+12.5*j*ω). These successive filters are responsible for the observed temporal differences in tilt avor, induced avor, tvor and neuronal responses. 14