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FIGURE 1. Simulated example of why decreasing the gain of the VOR can help minimizing the retinal image slip. (A) A perfect compensatory VOR (gain = 1, no phase shift, no noise) results in zero retinal image slip (eye velocity in red, head velocity in blue, retinal image slip in dashed yellow). (B) In the presence of a 45° phase shift between eye and head rotation, a strong retinal image slip is present despite a gain of 1. (C) Reducing the VOR gain to 0.72 minimizes the overall retinal image slip. (D) Variance of the retinal slip plotted with respect to the VOR gain for phase shifts as in (B,C) (blue), and noise as in (E,F) (red). The plots have been obtained from numerical simulations. The optimal gain in both cases is ~0.72 (red and blue dots). (E) Signal-dependent motor noise, proportional to eye velocity introduces a strong retinal image slip even if the gain = 1. (F) Reduction of the VOR gain also reduces the motor noise and thus the overall retinal image slip; retinal image slip plotted with respect to the VOR gain for this example is shown in (D) (red curve).
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FIGURE 2. Sequential steps of the methodological approach. (A) Example of raw head (blue) and eye (red) motion data obtained from one animal at a stimulus frequency of 0.1 Hz. (B) gain (B1) and phase values (B2) derived from raw data (blue) for a single animal along with the model simulation of the best fitting model; the gain value for this animal is 0.075. (C) Simulated cost function (retinal image slip variance) plotted with respect to the gain factor using the model derived in (B) for different noise factors; here, the simulation is shown for the optimal gain factor that matches the animal's actual gain factor. (D) Averaged eye position response during rotation at 0.5 Hz for the data (D1) and the simulation (D2); note that the average eye position trace of data and simulation (mean, red line) is similar due to the prior model fitting, but that the variability, which is also similar (shaded red) is a prediction based on the noise level and gain factor.
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FIGURE 3. Comparison of experimental and modeling data. (A,B) Experimental gain (A) and phase values (B) for eye movements of Xenopus tadpoles (n = 6) (colored small dots) along with average gain and phase values (large black dots); the red curve depicts simulated gain and phase values obtained from the computational model (red line) of the tadpole aVOR that have been fitted to the average results (black); note that an eye movement with a phase shift of −180° [dashed line in (B)] and a gain of 1 would perfectly compensate for the head rotation at all frequencies. (C,D) Example experimental data (blue circles) obtained from one animal (same as in Figure 2) and modeling results (red curve, yellow circles), with gain and phase values plotted in (C,D), respectively.
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FIGURE 4. Determination of the noise level. (A) Simulated cost surface [as log (cost)] for one animal (same as in Figures 3C,D) and for eye movements at a stimulus frequency of 0.5 Hz and ±10° head motion amplitude plotted as function of the noise factor of sensorimotor noise and model gain factor; the cost of the aVOR has been quantified by calculating the variance of the retinal image slip; yellow dots indicate the minimum cost for each noise factor; the dashed horizontal line is the experimentally determined gain value, and the vertical dashed line the noise factor at which the optimal gain value is closest to the experimental gain value (2.6 in this case). (B) Section through the cost surface for a noise factor of 2.6; the optimal gain is 0.075, as shown in (A). (C) Simulated eye movements (red) for the optimal gain value determined in (A,B). (D) Raw eye movement recordings (red) for this animal for comparison.
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FIGURE 5. Comparison of experimentally measured and simulated eye movements. (A–H) Eye movements (raw traces in black, red lines show average) shown for all stimulus motion cycles at different stimulus frequencies depicted for the representative animal in Figure 4 (A–D), along with corresponding simulations (E–H) of the data using the aVOR model (with appropriately adapted parameters) and optimal gain values to minimize the retinal image slip caused by signal-dependent sensor and motor noise (see also Figure 4); shaded areas indicate one standard deviation of the mean response.
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FIGURE 6. Relation between aVOR gain and variability of eye movements. (A) For comparison, aVOR gain values (see Figure 2) are shown for each animal separately in different colors. (B) Variability of eye movements for each animal plotted with respect to the predicted variability, i.e., the variability of the model simulations. (C) Variability of eye movements for each animal plotted with respect to the aVOR gain. Each data point in (B,C) represents the mean across four stimulus frequencies; error bars denote the standard deviation; note that for the animals depicted in yellow and orange only data for three frequencies were available.
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FIGURE 7. Relation between aVOR gain and variability of eye movements. (A) Replot of data from Figure 6C (red) together with model simulations of the standard aVOR model with varying sensorimotor noise and optimized aVOR gain (all other model parameters kept constant); the model predicts that for low aVOR gain values, the eye position variability will increase with increasing gain values (<0.5), but decrease again for higher values (>0.5). (B) Variability of eye movements for each animal plotted with respect to the retinal image slip [same simulations as in (B)]. (C) Optimal aVOR gain values [model simulations as in (A,B)] decreases with the sensorimotor noise factor.
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Figure 1. Simulated example of why decreasing the gain of the VOR can help minimizing the retinal image slip. (A) A perfect compensatory VOR (gain = 1, no phase shift, no noise) results in zero retinal image slip (eye velocity in red, head velocity in blue, retinal image slip in dashed yellow). (B) In the presence of a 45° phase shift between eye and head rotation, a strong retinal image slip is present despite a gain of 1. (C) Reducing the VOR gain to 0.72 minimizes the overall retinal image slip. (D) Variance of the retinal slip plotted with respect to the VOR gain for phase shifts as in (B,C) (blue), and noise as in (E,F) (red). The plots have been obtained from numerical simulations. The optimal gain in both cases is ~0.72 (red and blue dots). (E) Signal-dependent motor noise, proportional to eye velocity introduces a strong retinal image slip even if the gain = 1. (F) Reduction of the VOR gain also reduces the motor noise and thus the overall retinal image slip; retinal image slip plotted with respect to the VOR gain for this example is shown in (D) (red curve).
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Figure 2. Sequential steps of the methodological approach. (A) Example of raw head (blue) and eye (red) motion data obtained from one animal at a stimulus frequency of 0.1 Hz. (B) gain (B1) and phase values (B2) derived from raw data (blue) for a single animal along with the model simulation of the best fitting model; the gain value for this animal is 0.075. (C) Simulated cost function (retinal image slip variance) plotted with respect to the gain factor using the model derived in (B) for different noise factors; here, the simulation is shown for the optimal gain factor that matches the animal's actual gain factor. (D) Averaged eye position response during rotation at 0.5 Hz for the data (D1) and the simulation (D2); note that the average eye position trace of data and simulation (mean, red line) is similar due to the prior model fitting, but that the variability, which is also similar (shaded red) is a prediction based on the noise level and gain factor.
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Figure 3. Comparison of experimental and modeling data. (A,B) Experimental gain (A) and phase values (B) for eye movements of Xenopus tadpoles (n = 6) (colored small dots) along with average gain and phase values (large black dots); the red curve depicts simulated gain and phase values obtained from the computational model (red line) of the tadpole aVOR that have been fitted to the average results (black); note that an eye movement with a phase shift of −180° [dashed line in (B)] and a gain of 1 would perfectly compensate for the head rotation at all frequencies. (C,D) Example experimental data (blue circles) obtained from one animal (same as in Figure 2) and modeling results (red curve, yellow circles), with gain and phase values plotted in (C,D), respectively.
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Figure 4. Determination of the noise level. (A) Simulated cost surface [as log (cost)] for one animal (same as in Figures 3C,D) and for eye movements at a stimulus frequency of 0.5 Hz and ±10° head motion amplitude plotted as function of the noise factor of sensorimotor noise and model gain factor; the cost of the aVOR has been quantified by calculating the variance of the retinal image slip; yellow dots indicate the minimum cost for each noise factor; the dashed horizontal line is the experimentally determined gain value, and the vertical dashed line the noise factor at which the optimal gain value is closest to the experimental gain value (2.6 in this case). (B) Section through the cost surface for a noise factor of 2.6; the optimal gain is 0.075, as shown in (A). (C) Simulated eye movements (red) for the optimal gain value determined in (A,B). (D) Raw eye movement recordings (red) for this animal for comparison.
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Figure 5. Comparison of experimentally measured and simulated eye movements. (A–H) Eye movements (raw traces in black, red lines show average) shown for all stimulus motion cycles at different stimulus frequencies depicted for the representative animal in Figure 4
(A–D), along with corresponding simulations (E–H) of the data using the aVOR model (with appropriately adapted parameters) and optimal gain values to minimize the retinal image slip caused by signal-dependent sensor and motor noise (see also Figure 4); shaded areas indicate one standard deviation of the mean response.
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Figure 6. Relation between aVOR gain and variability of eye movements. (A) For comparison, aVOR gain values (see Figure 2) are shown for each animal separately in different colors. (B) Variability of eye movements for each animal plotted with respect to the predicted variability, i.e., the variability of the model simulations. (C) Variability of eye movements for each animal plotted with respect to the aVOR gain. Each data point in (B,C) represents the mean across four stimulus frequencies; error bars denote the standard deviation; note that for the animals depicted in yellow and orange only data for three frequencies were available.
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Figure 7. Relation between aVOR gain and variability of eye movements. (A) Replot of data from Figure 6C (red) together with model simulations of the standard aVOR model with varying sensorimotor noise and optimized aVOR gain (all other model parameters kept constant); the model predicts that for low aVOR gain values, the eye position variability will increase with increasing gain values (<0.5), but decrease again for higher values (>0.5). (B) Variability of eye movements for each animal plotted with respect to the retinal image slip [same simulations as in (B)]. (C) Optimal aVOR gain values [model simulations as in (A,B)] decreases with the sensorimotor noise factor.
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