Post-Saccadic Oscillations of the Pupil and Lens Reduce Fixation Stability in Retinitis Pigmentosa and Age-Related Macular Degeneration

Abstract

Purpose: Post-saccadic oscillations (PSOs) reflect movements of gaze that result from motion of the pupil and lens relative to the eyeball rather than eyeball rotations. Here, we analyzed the characteristics of PSOs in subjects with age-related macular degeneration (AMD), retinitis pigmentosa (RP), and normal vision (NV). Our aim was to assess the differences in PSOs between people with vision loss and healthy controls because PSOs affect retinal image stability after each saccade.

Methods: Participants completed a horizontal saccade task and their gaze was measured using a pupil-based eye tracker. Oscillations occurring in the 80 to 200 ms post-saccadic period were described with a damped oscillation model. We compared the amplitude, decay time constant, and frequency of the PSOs for the three different groups. We also examined the correlation between these PSO parameters and the amplitude, peak velocity, and final deceleration of the preceding saccades.

Results: Subjects with vision loss (AMD, n = 6, and RP, n = 5) had larger oscillation amplitudes, longer decay constants, and lower frequencies than subjects with NV (n = 7). The oscillation amplitudes increased with increases in saccade deceleration in all three groups. The other PSO parameters, however, did not show consistent correlations with either saccade amplitude or peak velocity.

Conclusions: Post-saccadic fixation stability in AMD and RP is reduced due to abnormal PSOs. The differences with respect to NV are not due to differences in saccade kinematics, suggesting that anatomic and neuronal variations affect the suspension of the iris and the lens in the patients’ eyes.

In a recent study,¹ we compared saccadic eye movements in patients with late age-related macular degeneration (AMD), patients with retinitis pigmentosa (RP), and age-matched control subjects with normal vision (NV) to explore how they are affected by central and peripheral vision loss, respectively. Apart from interesting differences in saccade reaction times, end point accuracy, end point variability, and changes in the relationship between the amplitude, duration, and peak velocity of saccades, we also noticed differences in post-saccadic oscillations (PSOs). In the present study, we investigated this serendipitous finding in more detail to better understand the potential impact of post-saccadic fixation instability on high-acuity vision and the possible implications for planning the saccade trajectory. 

The PSOs measured with pupil-based eye tracking reflect motion of the pupil center inside the iris after a saccadic eye movement.² Measurements with a Dual Purkinje Image (DPI) eye tracker, which can independently measure the movements of the lens, indicate that this back-and-forth motion results from inertial and viscoelastic forces acting on the crystalline lens and iris as the eyeball comes to a halt³ (Fig. 1). However, because the lens oscillation alone cannot fully account for the observed pupil wobbles, it was later proposed that fluid oscillations in the aqueous and vitreous cavity may also influence the pupil oscillations.⁴ 

Figure 1.

 

Top-down cross-sectional view of the right eye with green arrows illustrating post-saccadic oscillations of the optical and visual axes due to back-and-forth movements of the pupil, iris, and lens relative to the eyeball after a horizontal saccade to the right. Pupil-based eye tracking measures gaze by detecting the edges of the pupil with a video camera and records the center of the pupil relative to a glint (yellow star) from an infrared light source in the image plane. Post-saccadic oscillations in the signal thus reflect the back-and-forth motion of the pupil relative to the eyeball. The damped oscillatory motion results from inertial and viscoelastic forces acting on the crystalline lens and iris.

PSOs are usually too small to be seen with the naked eye. In certain patients, however, larger wobbles may be observed under slit lamp examination and are known as “phakodonesis.”⁵ PSOs are easily observed with a pupil-based eye tracker, which we used in this study, because they measure gaze by determining the location of the pupil center with respect to a glint from an external infra-red light-source (see Fig. 1). This so-called pupil-glint vector better reflects movements of the visual axis (gaze) than scleral search coils, which are more precise, but are fixed to the eyeball and therefore do not account for movements of the pupil and lens relative to the eyeball.⁶ 

A previous study characterizing PSOs with a DPI tracker found that lens oscillations after horizontal, abducting saccades of 9 degrees are well-characterized by a model of a classically damped harmonic oscillator.³ They found that the PSOs typically took approximately 50 to 60 ms to decay and that the lens oscillated with a frequency of about 15 to 20 hertz (Hz). The authors surmised that these temporal characteristics of the oscillation reflect the dynamic response of the zonular fibers, which connect the lens to the ciliary body. The authors also noted that the wobble duration is comparable to the interval of post-saccadic suppression and postulated that a PSO that lasts longer than this would have detrimental perceptual consequences. 

Many factors, such as age, saccade amplitude, and peak velocity, may affect the amplitude of a PSO. A study that examined saccades in young (18–26 years old) and old (50–80 years old) participants found that PSO amplitudes increased with increasing peak velocity and that this effect was more pronounced in older participants.⁷ Top-down cognitive processes, which can cause an increase in pupil size, have been shown to increase PSO amplitudes.⁸ It has also been suggested that the PSO amplitudes of Chinese versus Caucasian students may differ significantly in their relation to saccade amplitude and pupil size.⁹ 

The PSO amplitude may also relate to the deceleration magnitude near saccade offset. A study investigated what they called the “gentle braking hypothesis,” which states that smaller saccades stop more abruptly and therefore cause larger oscillations, whereas larger saccades brake more gradually, leading to smaller PSOs.¹⁰ Although they demonstrated an inverse relationship between PSO amplitude and saccade amplitude, they did not verify whether this was indeed due to differences in deceleration. Moreover, Kimmel et al. (2012) observed the opposite effect in rhesus monkeys and suggested that saccade deceleration could be the best predictor of PSO amplitude. However, neither of these studies directly quantified the relationship between saccade amplitude and saccade deceleration, or between PSO amplitude and saccade deceleration. 

It is known that fixation stability varies considerably among those with central vision loss (AMD), peripheral vision loss (RP), and no vision loss (healthy controls)^11,12 but, to our knowledge, there are no prior studies that have investigated the contribution of PSOs to fixation instability in participants with visual impairment due to retinal disease. Most studies have investigated differences in PSO amplitude, but they have not reported on the temporal characteristics of the PSOs, such as their frequency or decay time constant.^7–10,13 In addition, these studies only included healthy participants. Our study aims to fill this gap by presenting a quantitative analysis of the amplitude, frequency, and decay constant of PSOs in patients with degenerative vision loss due to RP (loss of peripheral vision) and late stage AMD (loss of central vision) compared to healthy control subjects (no vision impairment). 

Methods

Subjects

We recruited five patients with moderately advanced RP, six patients with bilateral geographic atrophy due to late atrophic AMD^14,15 in both eyes, and seven healthy control subjects with NV. Their ages ranged from 41 to 84 years old. Three of the subjects with AMD and one of the subjects with RP had cataract surgery and an intraocular lens (IOL) implanted in the past (Supplementary Table S1). The study was reviewed and approved by the human research ethics committees of the Royal Victorian Eye and Ear Hospital and the Bionics Institute, and conducted in accordance with the Declaration of Helsinki, with all participants providing written informed consent. 

For detailed descriptions of the subjects’ ophthalmic screening, the experimental procedures, and the visual field deficits of the included patients, readers are referred to our previous paper,¹ which reports on the saccade performance of these subjects. Here, we duplicate part of the methods’ descriptions for clarity. 

Equipment

Participants were seated 60 cm in front of a 30-inch computer screen (Dell U3011, 2560 × 1600 pixels) in a dark, soundproof room. Stimuli were generated with a laptop computer equipped with an open GL graphics card. The stimulus program was written in MATLAB (version 2014) using the Psychophysics Toolbox extension.¹⁶ Binocular gaze was measured with a remote eye-tracking system (Eyelink 1000 Plus, SR Research) at a sampling rate of 500 Hz per eye. The head-referenced signals from each eye were calibrated with a 13-point monocular calibration procedure prior to each measurement. Subjects were asked to keep their head still during the measurements with the help of a chin rest. 

Task

Subjects performed a center-out pro-saccade task. Each trial began by fixating at the center of the screen. A tone was played simultaneously with the appearance of the fixation point. The fixation period varied pseudorandomly between 800 and 1800 seconds from trial to trial. A target was presented at either 2, 5, 9, 12, 16, 20, 22, or 25 degrees to the left or right of the center of the screen at the same time that the fixation point disappeared. Subjects were asked to make a saccade as quickly and accurately as possible to the target. The target remained on the screen until the participants pressed a button to indicate that they had found it or until 10 seconds had passed. The central fixation point and the peripheral target were both white filled circles with a 1-degree diameter. The inter-trial interval was 500 ms. 

Each target was presented at the same location twice in a row. On the first presentation, the target location could not be predicted (unpredictable trials). On the second presentation (predictable trials), a triangle embedded in the fixation point cued the subject to the direction of the target jump and subjects typically knew the target location from the previous trial. This allowed the patients with RP to make a single saccade to the target on the second presentation even if it appeared outside their intact visual field. Subjects typically completed 4 blocks, each consisting of 128 trials with 4 repetitions per condition. 

Saccade Detection

Saccade onset was determined using a velocity and acceleration criterion of 40 deg/s and 7500 deg/s², respectively. Saccade end was marked based on a velocity and acceleration criterion of 30 deg/s and −5000 deg/s² and a direction-reversal criterion. All onset and offset markings were visually inspected using custom software and corrected if deemed necessary. 

Visual Field Estimation

Prior to the experiments, subjects had an ophthalmic examination encompassing Goldmann perimetry for the subjects with RP (size V and III targets) or Macular Integrity Assessment (MAIA; CentreVue) for the subjects with AMD, and multi-modal imaging including non-dilated color fundus photography (CFP; Canon CR6-5NM non-mydriatic camera, Japan), near infrared (NIR; Heidelberg Spectralis, Germany), and optical coherence tomography (OCT; Heidelberg Spectralis, Germany) imaging. This allowed us to map the areas of impaired vision in each of the two eyes with respect to the measured gaze directions as described before.¹ 

Analysis

Data analysis was done in Matlab (version 2022a) using the Statistics and Machine Learning Toolbox. We analyzed PSOs following the first saccades made in the unpredictable trials and repeated the analyses for the predictable trials. Because the findings were comparable for these trial types, we first show data from the unpredictable trials only, and pool across trial types later. Responses with blink artifacts were discarded. As a measure of pre-saccadic and post-saccadic fixation stability, we determined the root mean square deviation (RMSD) from the mean eye position before and after the saccade, respectively. To calculate the RMSDs, we took a 60 ms section of data immediately preceding and immediately following the saccade and computed the RMSD as follows:  

\begin{eqnarray}RMSD = \sqrt {\frac{1}{n}\mathop \sum \nolimits_{i = 1}^{i = n} {{\left( {{x_i} - \bar x} \right)}^2}} \end{eqnarray}

(1)

where \(\bar x\) is the mean of the horizontal eye position samples, x_i. We chose a 60 ms time window because this typically captured at least one post-saccadic oscillation cycle with little chance of another saccade occurring in this time window. For further evaluation, we then took the median of the RMSD values for each subject per eye and per saccade direction (movements to the left or to the right), so we had four values per subject. We used the Wilcoxon rank sum test to compare the medians between groups. In addition, we applied general linear mixed effects (GLME) analysis to compare the pre- and post-saccadic means across groups. These GLME models assumed that the data were Gamma distributed to accommodate the rightward skewness of the RMSD distributions. The random effects accommodated the between-subject variability. They were grouped by subject, by movement direction within the subject, and by recorded eye within movement direction to account for the nested structure of the repeated measurements. 

We also fitted a heuristic damped oscillation model to each PSO. The time window we used to fit the model ranged between 80 and 200 ms. It differed for each subject to accommodate the actual durations of their PSOs (see Supplementary Table S1 that lists the default time windows per subject). For trials in which a subsequent saccade occurred during this time window, the window was either shortened by 20 or 40 ms or the trial was excluded (see Supplementary Methods for details). The damped oscillation equation was:  

\begin{eqnarray}&& d\left( t \right) = \nonumber \\ && A \cdot {e^{ - \frac{t}{\tau }}} \cdot \cos \left( {\frac{{2\pi \cdot t}}{{T + \alpha \cdot t}} + \phi } \right) + {\rm{\;}}B \cdot t + C{\rm{\; \quad for\;}}t \ge 0\end{eqnarray}

(2)

where d(t) is the gaze displacement as a function of time t (saccade offset occurs at t = 0), A is the initial oscillation amplitude, τ is the decay time constant, T is the initial oscillation period (f = 1/T is the initial frequency), α· t is a time-dependent modulation of the oscillation period, and ϕ is the phase. B describes a constant drift and C a displacement offset. The time-dependent frequency modulation was included, somewhat heuristically, to accommodate the observed decrease in oscillation frequency over time. Least-square fits were obtained by a Fletcher's version of the Levenberg-Maquardt algorithm.¹⁷ A brute-force method was used to obtain suitable initial guesses of the fit parameters (see Supplementary Methods for details). Because of the frequency modulation, we also measured the time to first peak (tfp), that is, the time between the saccade end and the first peak of the PSO. The location of this first peak was determined numerically by evaluating d(t) at 1 ms resolution with B and C set to zero. Statistical evaluation of the PSO parameters was performed in the same way as the RMSD values. 

To study how PSO parameters relate to the preceding saccade, we first examined how saccade deceleration varies with saccade amplitude and peak velocity. The amplitude and peak velocity of each saccade was quantified as described in our prior publication.¹ Saccade deceleration was calculated as the average acceleration during the last 10 ms of the saccade, where acceleration was the time derivative of the vectorial track velocity. Note that decelerations have negative values as they indicate decreases in eye velocity. We decided not to use peak deceleration because this is a noisier measure and its timing varied with respect to the end of the saccade, especially in saccades with more than one velocity peak. At this stage, we pooled data across first saccades in the unpredictable and predictable trials. This was to leverage the fact that, in predictable trials, patients with RP made more and larger saccades to eccentric locations in their impaired visual field than they did in the unpredictable trials. 

To make boxplots of saccade deceleration as a function of saccade amplitude and peak velocity, we first determined the median deceleration for each eye and each saccade direction (left or right) per amplitude or velocity bin, giving us typically (but not always) four values per subject per bin. We did this to account for the fact that each subject had a different number of responses in each bin. Boxplots of the PSO parameters as a function of saccade deceleration were generated in the same way. We also made boxplots of Spearman's rank correlations that were computed across the un-binned data from each participant and used sign tests to determine if the signs of correlations were consistent. To measure the strength and direction of the associations, we used either LME or GLME regression. LME regression was applied to characterize saccade deceleration as a function of saccade amplitude and peak velocity, respectively. The (nonlinear) relationships between PSO parameters and saccade deceleration were quantified with GLME regression using a Gamma distribution (to accommodate skewness of the PSO parameter distributions) and a log link function. Random effects on the intercepts and slopes were again grouped by subject, by movement direction within subject, and by recorded eye. A log likelihood ratio test was used to compare the fits between the groups. This is calculated by taking the log of the ratio of the maximum likelihood of the two models being compared. 

In all analyses, outliers of more than three scaled median absolute deviations (MADs) away from the group median were excluded, but only if they were also three scaled MADs away from the median computed across all 18 participants. Statistical tests were always two-sided and P values less than 0.05 (type I error) were considered statistically significant. 

Results

Figure 2 shows gaze displacement traces of a series of rightward saccades made by a subject with NV, a subject with AMD, and a subject with RP. All traces are aligned with saccade end at t = 0. The color code represents saccade amplitude. Note the presence of PSOs in each of the three participants. The subject with RP showed remarkably large, long-lasting PSOs. In the participant with AMD, the post-saccadic fixation stability seems to be worse than in the control subject, but these differences are subtler and will be quantified further below. The amplitude of the PSOs exhibits no consistent relationship with saccade amplitude in these examples. If anything, the PSO amplitude seems to increase with saccade amplitude in the subject with RP, but it decreases with saccade amplitude in the subject with AMD, whereas there is no obvious relationship in the subject with NV. 

Figure 2.

 

Examples of post-saccadic oscillations in a subject with NV (subject 12), a subject with AMD (subject 3), and a subject with RP (subject 7) after horizontal saccades to the right. Gaze displacement is aligned with saccade end (t = 0) and is referenced with respect to gaze at the end of the analysis time window. Colors correspond to the amplitude of the saccade preceding the oscillation (in degrees). All traces are from first saccades of the right eye made during unpredictable trials.

As a model-free measure of PSO intensity, we first quantified the RMSD from the mean post-saccadic eye position in a 60 ms time window after the saccade (Equation 1). In addition, we computed the RMSDs for a 60 ms pre-saccadic time window to control for possible variations in noise in the gaze position measurements, and to exclude that elevated post-saccadic RMSD values do not reflect fixation instability in general. The boxplot in Figure 3 shows the results for both time windows in each of the three groups. 

Figure 3.

 

Pre-saccadic and post-saccadic fixation instability in the three groups. Data are from first saccades made in the unpredictable trials only. Fixation instability was quantified as the root mean square deviation (RMSD) from the mean horizontal gaze position signal, and median RMSD values were computed separately for saccades to the left and right in each eye for each subject, yielding four values per subject. On each box, the central horizontal mark indicates the resulting group median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers, and the outliers (defined as any value that is more than 1.5 times the interquartile range away from the bottom or top of the box) are plotted individually using the “+” symbol. The larger RMSD values for the post-saccadic time window (60 ms) compared with the pre-saccadic time window (60 ms) are indicative of significant PSOs. The P values are from the two-sided Wilcoxon rank sum test. Pentagrams: marginal means estimated with mixed effects regression.

Note that there was a significant difference between the pre- and post-saccadic RMSD, with the post-saccadic RMSD being about four to seven times larger (Wilcoxon rank sum test AMD_pre < AMD_post: P < 0.001; RP_pre < RP_post: P < 0.001; and NV_pre < NV_post: P < 0.001). This confirms the presence of elevated fixation instability right after saccades in each of the three groups. There was also a significant difference between the controls and patients. In both patient groups, the post-saccadic fixation instability was significantly larger than in the control group (Wilcoxon rank sum test AMD > NV: P < 0.001; and RP > NV: P < 0.001). In addition, the post-saccadic RMSD was significantly larger in the RP group compared with the AMD group (Wilcoxon rank sum test RP > AMD: P < 0.01). Marginal group means ± 1 standard error (SE) and the 95% confidence intervals (95% CIs) estimated with GLME analysis were: AMD_post = 0.236 ± 0.017 deg (95% CI = 0.2041 to 0.269 degrees); RP_post = 0.348 ± 0.018 degrees (95% CI = 0.312 to .384 degrees); NV_post = 0.151 ± 0.015 degrees (95% CI = 0.121 to 0.180 degrees). 

Model-Based PSO Analysis

To further quantify the nature of the PSO differences among the three groups, we adopted a damped oscillation model (Equation 2). Figure 4A illustrates how this oscillation model combined an exponential amplitude decay with a frequency decline and a constant drift. Figures 4B and 4C show example fits for a PSO measured in a subject with AMD, a subject with RP, and a subject with NV. The need to include a frequency decline in the model is well-illustrated by the PSO in the participant with RP; the duration of the first cycle is clearly shorter than the second. For most PSOs, this damped oscillation model gave a good description of the PSO: for more than 95% of the responses, Pearson's correlations between data and fit exceeded r = 0.85 in all three groups. Median correlations between data and fit were r = 0.98, r = 0.96, and r = 0.99 for PSOs in the AMD, RP, and NV groups, respectively. 

Figure 4.

 

Illustration of the applied PSO model. (A) Underdamped oscillation (dashed bold black) and its components: a frequency-decreasing oscillation (dashed orange), an exponential decay of the oscillation amplitude (dashed orange) and a constant drift (dashed purple). Parameters taken from a PSO following a 9-degree saccade made by a subject with NV. (B) Example AMD. (C) Example RP. Note that this example has a different time scale. (D) Example NV. The black dashed line denotes model fit and the solid line represents the measured gaze displacement relative to gaze at the end of the analysis time window. Saccade end is at t = 0 ms.

Figure 5 and Table 1 compare the relevant PSO parameters among the three groups. One may notice that the medians (central horizontal mark in each box) are typically different from the means (pentagrams). This is due to skewness of the parameter distributions. The pattern of the results, however, was similar for both central tendency measures. Note that compared with the NV group the initial oscillation amplitude (A) is significantly larger for the RP group but not for the AMD group (Wilcoxon rank sum test AMD_A > NV_A: P > 0.05; and RP_A > NV_A: P < 0.001). The oscillation amplitude is also larger for the RP group than for the AMD group (Wilcoxon rank sum test RP_A > AMD_A: P < 0.05). The decay time constant (τ), however, is significantly longer for both the AMD group and RP group (Wilcoxon rank sum test AMD_τ > NV_τ: P < 0.01; and RP_τ > NV_τ: P < 0.001). This explains why the post-saccade RMSD is significantly higher for the AMD group than for controls (Wilcoxon rank sum test AMD_post > NV_post: P < 0.001) even though the initial oscillation amplitude is not (Wilcoxon rank sum test AMD_A > NV_A: P > 0.05). Typically, patients with RP have even longer time constants than patients with AMD (Wilcoxon rank sum test RP_τ > AMD_τ: P < 0.01). The initial oscillation frequency (f) and the time to first peak (tfp) differ between patients and controls too. Both measures indicate lower PSO frequencies in the patients compared with the controls (Wilcoxon rank sum test AMD_f < NV_f : P < 0.001; RP_f < NV_f : P < 0.001; AMD_tfp > NV_tfp: P < 0.05; and RP_tfp > NV_tfp: P < 0.001) even though the decrease in oscillation frequency over time (as indexed by α) is lower in both patient groups compared with the control group (AMD_α < NV_α: P < 0.001; RP_α < NV_α: P < 0.001; Supplementary Fig. S1). The initial PSO frequency in the RP group is even lower than in the AMD group (Wilcoxon rank sum test RP_f < AMD_f: P < 0.01) and the time to first peak is significantly longer for the RP group compared with the AMD group too (Wilcoxon rank sum test RP_tfp > AMD_tfp: P > 0.01). Post-saccadic drifts are not significantly different from zero in the patient groups (sign test AMD_B ≠ 0: P > 0.5; and RP_B ≠ 0: P > 0.1; see Supplementary Fig. S1). 

Figure 5.

 

Boxplot comparing the PSO parameters between the three groups. Shown are the initial oscillation amplitude (A), decay time constant (τ), initial frequency (f) and time to first peak for primary saccades in the unpredictable trials. Two-sided Wilcoxon rank sum tests: not significant (n.s.) P > 0.05. Pentagrams: marginal means estimated with mixed effects regression.

Table 1.

 

Means of the PSO Parameters Estimated With GLME Regression

 

There was no significant difference between PSOs following saccades into the intact visual field and saccades into the impaired visual field (Supplementary Fig. S2). It made no difference whether the target was in the intact or impaired visual field either (Supplementary Fig. S3). In the above analyses, we therefore pooled the data. We also duplicated the analyses from Figures 3 to 5 for PSOs following first saccades in the predictable trials. The results showed similar effects (Supplementary Fig. S4). That is, both datasets revealed an ordering in PSO behavior from NV to AMD to RP with PSOs in subjects with NV having the smallest and shortest-lasting PSOs and subjects with RP having the largest and longest-lasting PSOs. 

Three of our patients with AMD and one of our subjects with RP had an IOL implanted during a cataract surgery (see Supplementary Table S1). We therefore assess whether this might have impacted their PSOs (Supplementary Fig. S5). Interestingly, the three subjects with AMD and IOLs had less post-saccadic fixation instability and smaller oscillation amplitudes than the three without IOLs. They more closely resembled the controls with NV in this regard. The subjects with IOLs and AMD did have longer time constants, however, which more closely resembles the decay of the PSOs in the patients with RP. The one patient with RP and an IOL only showed differences from the patients with RP and without an IOL in terms of PSO frequency and time to first peak. For both of these parameters, that subject had even more “abnormal” values compared to control subjects with NV. 

Relationship With Saccade Parameters

The above reported group differences in PSO characteristics could be of mechanical origin, neural origin, or a combination of the two. The notion that post-saccadic wobbles of the pupil-lens assembly are a mechanical consequence of the lens inertia and the braking of saccades^3,18 would suggest that the changes in PSOs are linked to changes in saccade kinematics. Indeed, we previously found the largest changes in saccade kinematics in the patients with RP¹ who show the largest and longest lasting PSOs as well (see Figure 5). In this group, saccade durations were longer and peak velocities lower compared to amplitude-matched saccades of subjects with NV.¹ We wondered, therefore, if the observed differences in PSO behavior between the groups might be explained by the differences in saccade kinematics. 

As illustrated in Figure 2 above, there was no clear evidence of a systematic relationship between PSO amplitude and saccade amplitude. However, if the PSOs are related to the braking of the saccade, we would instead expect a correlation with deceleration of the saccade. Unlike assumed by others,¹⁰ in our experiments, saccade deceleration was not significantly related to saccade amplitude in any of the groups (Marginal means of LME regression line slopes, NV: −24 ± 48 deg/s² per deg, 95% CI = −119 to 70 deg/s² per degree, P > 0.6; RP: −159 ± 115 deg/s² per degree, 95% CI = 386 to 67 deg/s² per degree, P > 0.1; AMD: −51 ± 51 deg/s² per degree, 95% CI = −152 to 48 deg/s² per degree, P > 0.3). This is shown in Figure 6A. Instead, saccade deceleration increased significantly with larger peak velocity values in all three groups (Marginal means of LME regression line slopes, NV: −5.9 ± 1.8 deg/s² per deg/s, 95% CI = −9.5 to −2.4 deg/s² per deg/s, P < 0.01; RP: −13.9 ± 3.5 deg/s² per deg/s, 95% CI = −20.9 to −6.9 deg/s² per deg/s, P < 0.001; AMD: −6.6 ± 1.4 deg/s² per deg/s, 95% CI: = −9.3 to −3.8 deg/s² per deg/s, P < 0.001; Figure 6B). In these analyses, we pooled saccades across unpredictable and predictable trials to maximize the number of responses in each amplitude bin. 

Figure 6.

 

Saccade deceleration as a function of saccade amplitude (A) and peak velocity (B). Lines are LME regression lines fitted to the data. Median decelerations are significantly different between the three groups (C). Note that deceleration is a negative valued acceleration. We therefore reversed the drawing direction along the ordinate for clarity. Data from first saccades in the unpredictable and predictable trials together. Green = AMD; red = RP; and blue = NV. Two-sided Wilcoxon rank sum tests: not significant (n.s.) P > 0.05. Pentagrams: marginal means estimated with mixed effects modeling.

Interestingly, the average and median deceleration toward the end of the saccade differed significantly between patients and controls (Figure 6C); in both patient groups, the saccade braking was significantly weaker than in controls (Wilcoxon rank sum test AMD < NV: P < 0.01; and RP < NV: P < 0.001). The mean difference in saccade deceleration with respect to the control group was 2.3e3 ± 1.0e3 deg/s² (95% CI = 0.5e3 to 4.2e3 deg/s²) for the AMD group, and 2.4e3 ± 1.0e3 deg/s² (95% CI = 0.4e3 to 4.3e3 deg/s²) for the RP group. For the RP group, but not the AMD group, this difference in mean deceleration was linked to differences in peak velocity as indicated by the steeper LME regression line for the RP group in Figure 6B (slope difference AMD versus NV: −7 ± 3 deg/s² per deg/s; P < 0.02; 95% CI = −14 to −1 deg/s² per deg/s; slope difference RP versus NV: −1 ± 3 deg/s² per deg/s; P > 0.5; 95% CI = −7 to 5 deg/s² per deg/s). The present findings thus bolster our previous conclusion that the saccade kinematics are different among the three groups.¹ But because the saccade decelerations are typically weaker in the patients whereas their PSOs are larger and last longer, it is not immediately clear if differences in PSO behavior relate to the differences in saccade braking. 

To explore whether the PSO parameters were related to the saccade kinematics, we made scatter plots of the PSO parameters as a function of the amplitude, peak velocity, and deceleration of the preceding saccade (see Supplementary Fig. S6 for examples), and we computed Spearman rank correlations for each participant. Of all relationships examined, only the rank correlation between PSO amplitude and saccade deceleration had a consistent sign in all subjects of all three groups (Supplementary Fig. S7). 

Figure 7A shows boxplots of the relationship between the initial PSO amplitude and saccade deceleration for each of the groups. Note that the PSO amplitude increases significantly with larger deceleration values in all three groups albeit not at a constant rate, and not along the same curve for each group. The latter is quantified in Table 2A, which lists the fixed-effects regression coefficients of the GLME model that we fitted to the data (see the solid lines in Figure 7A). This model assumed that the initial PSO amplitude changed with saccade deceleration (DEC) in the following way:  

\begin{eqnarray} A\left( {DEC} \right) = e^{{\beta _0} + {\beta _1} \cdot\, DEC + {\beta _2} \,\cdot\, DE{C^2}} \end{eqnarray}

(3)

 

Figure 7.

 

Initial amplitude (A), time constant (B), initial frequency (C), and time to first peak (D) of the post-saccadic oscillations as a function of saccade deceleration. Curves are GLME regression lines fitted to the data. Green = AMD; red = RP; and blue = NV. Drawing direction along the abscissa is reversed for clarity.

Table 2.

 

GLME Regression Coefficients of the Relations Between PSO Parameters and Saccade Deceleration

 

We used a similar regression model for the other PSO parameters. This analysis showed that the time constant (Figure 7B), initial frequency (Figure 7C), and time to first peak (Figure 7D) also changed significantly with saccade deceleration in one or more of the groups. Note, however, that in the control group, these relations are not monotonous. This is most clearly observed in Figure 7C, where the initial oscillation frequency in the NV group shows a maximum of about 50 Hz for saccade decelerations of approximately −24,000 deg/s². Both stronger and weaker braking of the saccade was associated with lower oscillation frequencies. Even so, the relationships differed significantly between the groups on all of these parameters (Likelihood Ratio Tests: P < 0.001; Tables 2B–2D), indicating that differences in saccade deceleration alone cannot account for the differences in PSO behavior. A similar conclusion was reached by analyzing the PSO parameters as a function of saccade amplitude (Supplementary Fig. S8) or peak velocity (Supplementary Fig. S9). 

Discussion

In summary, we found that PSOs in patients with AMD and patients with RP are different from those in subjects with NV. In patients with RP, they were larger, longer lasting, and of lower frequency. In patients with AMD, they were longer lasting and had a lower frequency. Overall, post-saccadic fixation stability (quantified by the RMSD) was poorer in both groups. The differences in initial PSO amplitude between groups relate partly to differences in saccade kinematics, and more specifically to the braking deceleration of the saccades. But the observed increases in PSO amplitude with increasing saccade deceleration cannot fully account for the observed group differences in PSO amplitude. The differences in temporal characteristics of the PSOs are also not immediately accounted for by differences in amplitude, peak velocity, or deceleration of saccades. Below we will discuss the implications of our findings. 

Implications for Vision

Our video-based eye tracking method measured the so-called pupil-glint vector, which reflects movements of the eye's optical axis,^19,20 and therefore the yoked visual axis, rather than movements of the eyeball per se (see Fig. 1). This is because the inner part of the iris, which defines the pupil, and the crystalline lens undergo inertial movements within the eyeball; these movements manifest at the end of saccades in the form of PSOs.³ Our findings that PSOs are larger (59% and 15% as indexed by the oscillation amplitude A) and last longer (191% and 92% as indexed by the time constant τ) in RP and AMD thus imply greater and longer instability of the retinal images after each saccade. Saccadic suppression that lasts beyond the duration of the saccade may help suppress the impact of this retinal motion on visual stability,³ but saccadic suppression is incomplete even during the saccade itself.²¹ Either way, it seems likely that the increased post-saccadic instability of the retinal images contributes to the visual difficulties that patients with RP and patients with AMD experience. To our knowledge, this possibility has not been reported previously. Especially in patients with RP, who have a relatively preserved fovea, such visual instability was unexpected. 

Patients with AMD are known to exhibit problems with fixation. The level of instability they exhibit is usually tied to how long they have had AMD and the extent of their foveal vision loss. They may compensate for this by developing a so-called preferred retinal locus (PRL) or eccentric fixation (see, e.g. Ref. 11 for a comprehensive review). Macular Integrity Assessment (MAIA) performed prior to the experiments indicated that most of the subject in our AMD group used eccentric fixation and that they all had impaired fixation stability.¹ Yet, in the experiments, the pre-saccadic fixation stability of the patients with AMD was not significantly different from that in the patients with RP (see Fig. 3). Note, however, that in our saccade task, the fixation point had already disappeared during the pre-saccadic fixation window, whereas the fixation point remained present in the central fixation monitor of the MAIAs. 

Implications for Saccade Planning

Our finding that the initial PSO amplitude increases with increasing saccade deceleration in all three groups (see Fig. 7) is consistent with a previous study in rhesus macaques²² and suggests that saccade deceleration is the “driving force” of PSOs as hypothesized before by Hooge et al.¹⁰ In addition, we found that saccade deceleration is on average lower in the RP and AMD group compared with the NV group irrespective of saccade amplitude (see Fig. 6). Building on the “gentle braking hypothesis,”¹⁰ this leads us to speculate that one reason for the altered main sequence relationships in RP and AMD¹ might be to limit the effect of PSOs on vision. Indeed, in subjects with AMD, PSOs were smaller compared with the PSOs of subjects with RP (see Fig. 3), and so were the increases in saccade duration and the decreases in peak velocity.¹ 

Optimal control theories suggest that the stereotyped relationships between saccade amplitude, duration, and peak velocity reflect an optimal tradeoff between speed and accuracy of saccades,^23–26 but they primarily consider the consequences of (constant and signal-dependent) noise in the control signals sent to the extra-ocular muscles. Our present findings would suggest that this does not suffice because saccade accuracy cannot be defined by what the eyeball is doing alone. Motion of the pupil-lens system with respect to the eyeball needs to be considered as well because it too determines movement of the visual axis. More broadly speaking, one might argue that optimal gaze control theories need to consider nested control of the head, eyes, and pupil lens. 

Differences in PSO Characteristics

The time constant as well as the initial frequency and time to first peak did not correlate consistently with saccade amplitude, peak velocity, or deceleration (see Fig. 7, Supplementary Figs. S7–S9). This is to be expected for a classical damped mass-spring system because an increase in its step-input would not change its mechanical properties (mass, viscosity, and stiffness). However, if the mechanics of the eyeball-iris link were unchanged, we would not expect the temporal characteristics of the PSOs to differ between patients and controls. Yet they do: the time constant is systematically longer, the initial frequency systematically lower, and the time to first peak is systematically longer in both patient groups, especially the RP group (see Fig. 5). These differences are not a simple consequence of differences in amplitude, peak velocity, or deceleration of saccades because their relationships with PSO parameters differ between groups (see Fig. 7, Supplementary Figs. S7–S9). Our findings suggest, therefore, that the mechanics of the iris and lens suspension in the eyeball are somehow different between patients and controls. Reduced viscous damping, for instance, would account for a slower decay of the oscillations and lower oscillation frequencies in an underdamped spring-mass system. 

Deubel and Bridgeman also suggested a link between PSOs and the mechanical properties of the zonular fibers.^18,27 They found lower oscillation amplitudes in older adults and suggested that this relates to the decreased accommodation in older participants. There are also different conditions that would cause weakened zonular fibers, such as pseudoexfoliation syndrome and Marfan's syndrome, but, to the best of our knowledge, our subjects did not have these conditions. 

It is possible that PSO parameters are related to changes in the mass of the lens. The PSOs that we saw in the RP group are reminiscent of the PSOs reported by Nyström et al.²⁸ for a subject with aphakia who had both her lenses removed at an early age. It is possible that less mass in the lens could lead to larger oscillation amplitudes and longer decay constants. We do not know whether the mass of the lens in patients with RP is different than that of healthy controls. Implanted lenses are known to be lighter than natural lenses, however. But the patients with RP and IOLs did not display any differences in oscillation amplitude or decay constant from the patients with RP and no IOL. The patients with AMD and IOLs had smaller oscillation amplitudes and resembled the control subjects more than the patients with AMD and no IOL, but their time constants were longer and more closely resembled those of patients with RP. A difference in the mass of the implanted lens may not entirely explain differences in PSOs in patients with and without an IOL. Although the PSOs of subjects with and without IOLs may differ, our main finding that post-saccadic fixation stability is affected in both groups of patients is unchanged by the inclusion of subjects with IOLs. 

What might cause changes in the mechanical properties of the eyeball-iris link? In principle, these changes could be due to structural anatomic differences, differences in the neural drive to the intra-ocular eye muscles, or a combination of the two. Furthermore, differences in the neural drive to the intra-ocular eye muscles could be independent of the occurrence of a saccade,²⁹ or they may be linked to the neural control of the saccade.³⁰ It is difficult to rule out any of these possibilities based on our current data, for one because we have no independent measure of the eyeball movements. 

One possibility that we investigated was that differences in pupil size may be contributing to differences in the patient's oscillations. It has been shown that larger pupil sizes correlate with larger oscillation amplitudes.^4,7 We did find a small difference between the pupil areas of the RP and the NV group (Supplementary Fig. S10): patients with RP tended to have smaller pupils than subjects with NV. This would support the notion of altered mechanics, but note that the differences in pupil size are opposite to what one would expect from the studies by Mardanbegi et al. (2018) and Nyström et al. (2015) because patients with RP had larger PSOs than controls. There were no pupil size differences between the AMD and the RP group or between the AMD and the NV group. Taken together, we believe that differences in the pupil size alone do not account for differences in PSOs. 

Another possibility is that visual or internal feedback loops are affected in RP and AMD. As pointed out by Robinson and colleagues,^31,32 the saccadic system is potentially unstable due to delayed internal feedback and may oscillate if the burst neurons, which generate saccades, are not adequately inhibited by the omnipause neurons. Although saccadic oscillations may occur transiently in healthy subjects too (for example, around the orthogonal axis of a purely vertical saccade, during combined saccade-vergence gaze shifts, or during blinks), inhibition of pontomedullary burst neurons by omnipause neurons would normally prevent the saccadic mechanism from oscillating when steady fixation is desired.³³ Perhaps the post-saccadic re-engagement of this fixation control is perturbed in RP and AMD. 

The decrease in PSO frequency over time remains puzzling too. Maybe the larger frequency decreases in controls with NV (see Supplementary Fig. S1) indicate that their eyeball-iris link is closer to a critically damped state. But note that a frequency change is not expected in a linearly damped mass-spring system to begin with. We are the first to admit that the way in which we modeled this effect was only heuristic. The resulting fits, however, were quite good. They typically accounted for 92% to 98% of the variance in the post-saccadic time window. We believe, therefore, that the model parameters adequately captured the PSO characteristics. Other studies have adopted more sophisticated methods to model the time course of PSOs by considering the impact of the entire preceding saccade profile.^34,35 However, as noted in these studies too, the viscoelastic properties of the eyeball-iris link are probably not stationary. Whether these changes are force-dependent or due to active neural control of the circular and radial muscles of the iris remains an open question. 

Strengths and Limitations

To our knowledge, this is the first study to quantify PSOs in patients with AMD and patients with RP and compare them to PSOs in a control group of elderly participants with NV. We should point out, however, that the average age of the patients with AMD (75 ± 8 years) was higher than that of the controls with NV (65 ± 11 years) and those with RP (62 ± 15 years). An age difference between the RP and AMD group can be expected because late AMD tends to occur later in life than moderate RP, which can develop earlier. A previous study reported that PSO amplitudes are larger in the elderly,⁷ but it is not clear if this could actually be due to different saccade decelerations. A strength of the current study is that we took saccade deceleration into account. Saccade deceleration appears to be a key factor in modulating initial PSO amplitude, more consistently so than saccade amplitude or peak velocity. Another strength of our study is that we have quantified the temporal characteristics of the PSOs whereas most previous studies have only investigated differences in oscillation amplitude. Our results for the AMD group illustrate the importance of considering the temporal aspects. In this group, the increases in PSO amplitude were small, but the changes in the time constant revealed prolonged post-saccadic fixation instability. An interesting open question is how the impairments in post-saccadic fixation stability affects visual acuity, and how this would compare to the effects of different forms of pathologic nystagmus. Still, we believe PSOs may be relevant to consider in the clinical evaluation of patients with visual impairments. 

A limitation of our study is the small number of included participants. In two out of six subjects with AMD, the location of the scotomas was such that we ended up with only a small number of trials in which the target appeared within the subjects’ bilateral scotoma(s). In a third patient with AMD, we blocked vision of the left eye because she had a parafoveal scotoma, with good remaining central vision (see Ref. 1 for details). Therefore, future studies should be cautious about pooling PSO data from patients with AMD for saccades to targets presented in their scotomas and their intact visual field. We cannot exclude that we did not have enough power to detect differences in PSO characteristics between these conditions in our group of patients with AMD. Another limiting factor was that three of the patients with AMD and one patient with RP had IOLs. Due to the low number of included subjects, it remains somewhat uncertain how the presence of IOLs influenced our results. Nevertheless, with or without IOLs, PSOs appear abnormal in both groups of patients. Future studies may shed more light on the effects of IOLs in the different groups. 

Conclusions

Post-saccadic oscillations in patients with AMD and in patients with RP are abnormally large and long lasting. The resulting excess image motion on the retina may contribute to the patients’ functional vision loss. Previously reported changes in saccade kinematics may reflect a strategy of the brain to minimize these effects. 

Acknowledgments

Supported by the European Union Program FP7-PEOPLE-2013-ITN “HealthPAC,” grant 604063 - IDP, the RadboudUMC, and the EU Horizon 2020 program, ERC advanced Grant, “Orient,” nr. 693400. 

Disclosure: L. Guadron, None; S.A. Titchener, None; C.J. Abbott, None; L.N. Ayton, None; A.J. van Opstal, None; M.A. Petoe, None; J. Goossens, None