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The authors have declared that no competing interests exist.

Model development: IO. Revised the manuscript: GB AAT SM CJB EK KPW DS. Conceived and designed the experiments: GB SM KPW AAT DS. Performed the experiments: GB AAT. Analyzed the data: GB. Contributed reagents/materials/analysis tools: GB CJB EK. Wrote the paper: GB.

Eccentric gaze in darkness evokes minor centripetal eye drifts in healthy subjects, as cerebellar control sufficiently compensates for the inherent deficiencies of the brainstem gaze-holding network. This behavior is commonly described using a leaky integrator model, which assumes that eye velocity grows linearly with gaze eccentricity. Results from previous studies in patients and healthy subjects suggest caution when this assumption is applied to eye eccentricities larger than 20 degrees. To obtain a detailed characterization of the centripetal gaze-evoked drift, we recorded horizontal eye position in 20 healthy subjects. With their head fixed, they were asked to fixate a flashing dot (50 ms every 2 s)that was quasi-stationary displacing(0.5 deg/s) between ±40 deg horizontally in otherwise complete darkness. Drift velocity was weak at all angles tested. Linearity was assessed by dividing the range of gaze eccentricity in four bins of 20 deg each, and comparing the slopes of a linear function fitted to the horizontal velocity in each bin. The slopes of single subjects for gaze eccentricities of ±0−20 deg were, in median,0.41 times the slopes obtained for gaze eccentricities of ±20−40 deg. By smoothing the individual subjects' eye velocity as a function of gaze eccentricity, we derived a population of position-velocity curves. We show that a tangent function provides a better fit to the mean of these curves when large eccentricities are considered. This implies that the quasi-linear behavior within the typical ocular motor range is the result of a tuning procedure, which is optimized in the most commonly used range of gaze. We hypothesize that the observed non-linearity at eccentric gaze results from a saturation of the input that each neuron in the integrating network receives from the others. As a consequence, gaze-holding performance declines more rapidly at large eccentricities.

Most healthy human subjects display a physiological centrifugal horizontal nystagmus at extreme lateral gaze in darkness (

In general, however, gaze shifts to moderate horizontal eccentricities evoke, even in darkness, only very weak centripetal eye drift in healthy subjects, as cerebellar control sufficiently compensates for the inherent leakiness of the brainstem gaze-holding network (

To better understand the physiological and pathological manifestations of the inherent deficiencies of the gaze holding system, it is crucial to clarify how the centripetal horizontal eye drift grows in relation to eccentric gaze position. Several studies reported drift velocity for only one or very few specific horizontal gaze eccentricities (typically 30, 40 or 50 deg) (

The purpose for the present study is, therefore, to characterize the relation between centripetal eye drift velocity and gaze eccentricity in healthy human subjects and clarify the limit of applicability of the single time constant leaky integrator model.

Twenty healthy human subjects (8 females; mean age ±1 SD: 41±11 years; range 24–67 years) participated in the study. Informed consent of all participants was obtained in written form after full explanation of the experimental procedures. The protocol was approved by the Ethics Committee of the Canton of Zurich, Switzerland (Protocol N° E-33/2007), and was in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki for research involving human subjects.

Participants were comfortably seated upright on a chair mounted on a two servo-controlled motor-driven axes turntable system (Tönnies D561, Freiburg i.Br., Germany; control system: Acutrol® ACT2000, Acutronic, Switzerland Ltd.). The two independent motor-driven axes are coincident and earth vertical. One rotates the chair and the other a cylinder (Optokinetik Drum, radius: 74 cm) mounted concentrically to the chair. Remotely controlled LEDs are attached to the cylinder at the level of subject's eyes. Safety belts around the feet and the shoulders restrained the subject. An adjustable chin rest and a forehead strap were used to stabilize the subject's head.

Horizontal eye movements were recorded at 220 Hz with a head-mounted video-oculography (VOG) device (“EyeSeeCam”) (

Participants were asked to fixate a briefly flashing (50 ms every 2 s) red LED without moving the head. The LED was positioned at the level of the eyes in the range of horizontal gaze eccentricity from 40 deg left to 40 deg right. Each subject was tested in two subsequent runs, changing the order of presentation of the requested gaze eccentricities. Specifically the LED always started straight-ahead and slowly displaced (0.5 deg/s) up to 40 deg of eccentricity in one of the two possible directions (the initial direction was in one run leftward and in the other rightward, randomly selecting the first one), then the direction was reversed, continuing the displacement until the 40 deg of eccentricity of the opposite side was reached, when it was reversed again to return to straight ahead position, where it stopped. Both eyes were recorded simultaneously.

Data analysis was done offline on a PC using interactive programs written in MATLAB (The Mathworks, Natick, MA), version 7.5. Velocity traces were obtained as the derivative of horizontal eye position traces. Saccades and blinks were interactively removed using a custom program that identifies all the data points exceeding by a given threshold the median velocity calculated over a time window moving in steps of one third of its width. The data points that exceeded the threshold at least two times were considered part of a saccade. The beginning and the end of each saccade were identified by searching for the closer reversals of the velocity. All data points belonging to saccades were removed. The threshold was set to 3 deg/s and the width of the window was 0.5 sec. Missing data or unreliable data (due to blinks, or if the pupil could not be reliably found at eccentric positions due to coverage by the eye lids) were not interpolated. We calculated median eye velocities recorded from every single subject within 4 non-overlapping, 20 deg wide bins of eye eccentricity (i.e. 0–20 deg and 20–40 deg for both sides), keeping the two runs (which differ by the starting direction), the two directions of target displacement and the two eyes, separated. Individual median eye velocities from all subjects were compared within each bin using a repeated measures three-way analysis of variance (Matlab function RMAOV33.m) (

The behavior of eye drift as a function of gaze eccentricity was analyzed using two separate procedures, one focused on single subject data and the other on pooled whole population data. The first provided a test of the reliability of the linear modeling, by testing the consistency of the parameters estimated by linear fit of different ranges of gaze eccentricity. The second allowed defining the general behavior of gaze holding, evaluating which function can best represent the growth of the drift velocity with gaze eccentricity.

Instantaneous eye velocity values from both eyes, directions of target displacement and runs were pooled for each subject. The resulting data were sorted according to their eye eccentricity and then split in four 20 deg wide bins from 40 left to 40 right (i.e. 0–20 and20–40 for both sides). Under the assumption of linear behavior, the slopes obtained fitting the data from different bins should be the same within each subject. Calling

The linear slope _{1}

Each subject's instantaneous velocity was smoothed as a function of eye eccentricity using a weighted linear least-squares robust regression method (Matlab function smooth.m with “rloess” algorithm) based on a second order polynomial model (

_{2}_{3}

To illustrate how the underlying nonlinearities in the brainstem gaze holding networks could surface at extreme eye positions, we used a mathematical model of a network of neurons. The network simulates eye drift velocity by mimicking the physiological behavior of the neurons. The equations of the model were derived as follows.

Electrophysiological data have shown that, during eye fixations, neurons thought to be part of the neural integrator network in the brainstem (

The subscript indexes the neurons; _{i}_{i}

A similar relation can be written for neurons in the left side population. To model the synaptic current that each neuron creates in the postsynaptic neuron, we use a synaptic activation function

That is, the dynamics of the network can be reduced to a single equation. This network will maintain stable fixations for any values of Δ for which the right hand side of _{R}_{L}

Comparing with _{∞}

In the dark, the ability to maintain fixations depends on the behavior of

Panel A - Left eye position plotted as function of time. Positive angles correspond to right eccentricities as seen by the subject. In this trial the dot was moving (0.5 deg/s) rightward at first. Inset 1: At extreme eccentricities the centrifugal beating nystagmus is clearly visible and the slow phase shows the tendency of the eye to return toward the primary position. Inset 2 and 3: Difference in the slope of the position trace at the same eccentricities when the dot is moving outward or inward. Panel B and C - Position (panel B) and velocity (panel C) of the eye at eccentricities larger than 10 deg right. The eye velocity begins to decrease from its baseline before the onset of the nystagmus, showing the growing centrifugal drift. Note that the baseline velocity is not zero but is positive between 10 and 25 deg of gaze eccentricity. When returning to 10 deg, however, the velocity is negative, showing the asymmetry in the baseline velocity showing the subject's attempt to match the target displacement velocity.

The constant value of the velocity signal observed in

Panel A - Black lines: Medians of the eye velocity within a 1 deg-wide bin plotted as a function of gaze eccentricity keeping the direction of target displacement separated. Gray lines: velocity of the target as a function of its position during the whole recording period. Note that the eye velocity matches the target velocity from the beginning suggesting that the offset observed around the straight ahead gaze is not due to a memory effect. The arrows show the directions of target and eye displacement. Panel B - Black lines: Velocity traces from the left panel after subtracting the correspondent target velocity. Gray line: Medians of the eye velocity within a 1 deg-wide bin plotted as a function of gaze eccentricity after pooling data from different directions of target displacement.

To find out whether the direction of target displacements or other factors affect the recorded instantaneous eye velocity, we calculated for each subject the medians of the velocity within four non-overlapping 20 deg wide bins in the range tested, keeping the different runs (which differ by the starting direction), the different directions of the target displacement and the two eyes separated. Pooling the data of all our subjects, we used a repeated measure three-way ANOVA within each bin separately. After applying Bonferroni correction for multiple comparisons, we found that the run (i.e. initial displacement toward right vs. initial displacement toward left) and the eye (i.e. left vs. right eye) were not associated with a significant difference in any bin. The direction of target displacement was instead a significant factor (p<0.001; F(1,38) = 32.5 and F(1,38) = 39.8 for left and right eccentricities, respectively) in the two central bins (between 0 and 20 deg on both sides), with higher horizontal eye velocities when the target was moving toward the subjects' straight-ahead position, but not in the two outer bins (between 20 and 40 deg on both sides). The median difference between eye velocities recorded at the same eccentricity with the target moving in opposite directions was 0.45 deg/s; approximately half of the value expected considering the two opposite velocity offsets needed to match the target displacement in both directions. This finding, together with the statistical analysis, suggests that the velocity signal observed in

To investigate the behavior of eye drift as a function of gaze eccentricity we applied two separate analyses to our data: one fitting single subject data and the other evaluating the average of the whole population.

In the first of these analyses, instantaneous velocity recorded from each single subject was sorted as a function of gaze eccentricity, pooling data from the two eyes, the two runs and the directions of target displacement. Data of each subject were separated in four bins defined as for the statistical analysis above. We fitted a linear function of the eye eccentricity (Eq.1) to the values in each bin separately.

Gray dots: Instantaneous velocity plotted as a function of the eye eccentricity. Light gray dots: Velocity in the 0–20 deg bins; dark gray dots: Velocity in the 20–40 deg bins; black line: linear fit of the velocity in the 0–20 deg bins and in the 20–40 deg bins.

Using a paired Wilcoxon signed rank test we found that the slopes fitted from each subject for gaze eccentricities between 0 and 20 on one side were significantly (p<0.05) smaller than those obtained in the same subject for gaze eccentricities between 20 and 40 deg on the same side. The median ratios (median absolute deviation in square brackets) of the paired slopes were 0.44 [0.29] on the left side and 0.32 [0.25] on the right side, respectively. Ratios of the slopes were significantly lower than 1 (p<0.01), confirming the significant increase of the rate of growing of the eye drift velocity with gaze eccentricity and therefore indicating a non-linear behavior. Pooling both sides the median ratio was 0.41 [0.29]. The mean slopes of the fitted subjects are reported in

0 deg to 20 deg | 20 deg to 40 deg | |

Median | 0.33 ± 0.17 | 1.21 ± 0.56 |

Slope | −0.021±0.014 | −0.045±0.022 |

Ratio to 20–40 deg | 0.44 [0.29] | 1 |

Median | −0.36 [0.30] | −1.31 [0.54] |

Slope | −0.020[0.010] | −0.047[0.045] |

Ratio to 20–40 deg | 0.32 [0.25] | 1 |

The second analysis aimed at identifying a function that better represents the drift behavior, showing an improvement over the linear one. We characterized the behavior of the whole population by smoothing the individual instantaneous eye velocity traces of all subjects as a function of gaze eccentricity and interpolating them for all angles between 40 deg left to 40deg right in steps of 0.1 deg. Lilliefors test (^{−1} on the right and −0.029 s^{−1} on the left side, corresponding to a time constant of 28 and 34 s, respectively. The values of

Dashed gray lines: Individual position-velocity curves obtained after smoothing and interpolating instantaneous velocity as a function of eye eccentricity; solid thick gray line: mean of the smoothed individual position-velocity curves; solid medium gray line: mean ±1 standard deviation of the smoothed individual position-velocity curves; dashed black line: tangent fit of the mean of the smoothed individual position-velocity curves.

To show how the observed behavior can stem from the nonlinearities that affect the integrator network at the neuronal level, we simulated a mathematical model of a network incorporating some of the known characteristics of those involved in the velocity signal integration in the goldfish and showing the effect of different tuning of the free parameters (see Methods). Without tuning (using the same value for every

Panel A shows the output of the synaptic activation function of each neuron (thin lines) as a function of the internal representation of eye eccentricity (Δ), the zero of each line indicates the neuron threshold, i.e. the eccentricity at which the inhibitory cutoff takes place. The thick lines are the cumulative output of both sides of the network, obtained by combining all the synaptic activation functions with their factor

As the eye spends less time in the most eccentric positions than in the center, we considered a reasonable assumption to use a non-uniform tuning procedure, which will weigh more the drifts occurring in more central eye positions. This can be simulated by weighting Eq.10, which represents the error to minimize, with a Gaussian function of the eye eccentricity Δ (

The Gaussian function of the eye eccentricity represented by the solid line in the central panel has been multiplied to the resulting drift, i.e. the error to minimize, during the optimization procedure. The contents of the panels are as in

If we set a gaze-dependent tolerance for allowed drift in the minimization procedure, simulating a mechanism favoring leakiness over instability, it is possible to obtain a plot (

The inverse of the Gaussian function of the eye eccentricity shown by the solid line in the central panel has been used as a gaze-dependent threshold for the non-penalized drift during the optimization procedure. The contents of the panels are as in

To illustrate that different hypotheses in the network design can also generate simulations that mimic the experimental data, we include a simulation with saturating synaptic activation functions and a partial overlap of the activation thresholds in the center of the eye position range (

The contents of the panels are as in

Random perturbations around the weights used in

Effect of modifying the tuned values of

Centripetal drift of the eye in eccentric positions is a known phenomenon possibly caused by non-ideal integration of the eye velocity command when generating the position command for the motoneurons driving the eyes (

In this study we investigated gaze-holding performance in healthy human subjects by measuring eye drift velocity as a function of gaze eccentricity over a ±40 deg range. Pooling all subjects, we found a clear drift pattern with approximately linear behavior only within the central 20 deg of gaze eccentricity. For larger eccentricities the slope increased gradually, resulting in a curve that was better fit by a tangent function (

It may be argued that the specific characteristics of our paradigm affected the recorded drift velocity. In contrast to the reported earlier studies that used large gaze shifts between different recorded positions, we slowly displaced the target to obtain a sequence of quasi-continuous position steps. This allowed us to minimize the distance between the evaluated gaze eccentricities, keeping the recording time short, and not sacrificing the range tested. A saccade-based paradigm usually requires the subject to rapidly look eccentrically to elicit centripetal drift, and to look back to straight head after each trial to guarantee the same starting condition in each trial. This approach is inefficient if one aims at acquiring the same number of eccentric gaze positions as we recorded, as it would require two gaze shifts for every gaze eccentricity and only the very first second of every eccentric fixation could be used. For our experimental setup, which uses LEDs embedded in a motorized drum surrounding the subject, a quasi-static displacing target was a good compromise between the efficient data acquisition and recording time. We reasoned that, considering the characteristics of the system, there is, in principle, no reason to prefer one method over the other, as both require the integration of a velocity command to reach the desired eccentricity and none of the two guarantees that the possible nonlinearity of the integration network will not affect the estimation of the centripetal drift. For small gaze angles we found an evident velocity offset of a magnitude similar to the velocity of target displacement. This offset caused a significant difference between the instantaneous velocities recorded when the eye moved rightward or leftward (

Although the nonlinear behavior emerges clearly when considering the population mean curve, the high variability across subjects, shown by the light gray dashed lines in

Considerable variability between subjects is a common observation in both the papers on end-point nystagmus reporting eye drift velocities (

Although previous studies did not measure eye drift velocities in a continuous range of gaze eccentricities, a decrease of the integrator time constants at large angles of gaze was already reported (

Although the tangent function is not derived from a specific model, the described behavior is consistent with the biological constraints that the brain has to overcome to hold gaze steady (

We conclude that gaze holding in healthy humans does not follow a linear function, but is much better characterized by a tangent. The nonlinearity of the gaze holding behavior in healthy subjects is well grounded on neuronal physiology and the use of a tangent function provides a compact and simple characterization of healthy behavior to be used as a reference when investigating pathological conditions of gaze holding, e.g. in patients with progressive degenerative vestibulo-cerebellar disease.

The authors thank M. Penner for technical assistance.