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Article  |   November 2019
The limits of fixation—Keeping the ametropic eye on target
Author Affiliations
  • Siegfried Wahl
    Institute for Ophthalmic Research, University of Tuebingen, Tuebingen, Germany
    Carl Zeiss Vision International GmbH, Aalen, Germany
  • Denitsa Dragneva
    Institute for Ophthalmic Research, University of Tuebingen, Tuebingen, Germany
  • Katharina Rifai
    Institute for Ophthalmic Research, University of Tuebingen, Tuebingen, Germany
    Carl Zeiss Vision International GmbH, Aalen, Germany
    katharina.rifai@medizin.uni-tuebingen.de
Journal of Vision November 2019, Vol.19, 8. doi:https://doi.org/10.1167/19.13.8
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      Siegfried Wahl, Denitsa Dragneva, Katharina Rifai; The limits of fixation—Keeping the ametropic eye on target. Journal of Vision 2019;19(13):8. doi: https://doi.org/10.1167/19.13.8.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Accurate positioning of the naked eye is a prerequisite in many ophthalmic measurement and intervention systems. In most of these procedures the eyeball is stabilized through fixation of a target. Fixation is initiated, as well as sustained by a whole set of oculomotor processes: saccadic movements and drift, as well as vergence. Target appearance influences sustained fixation, but the influence of target information on the initiation of fixation has not been evaluated in detail. The current study evaluates the accuracy of fixation initiation as well as sustained fixation under refractive error. Twenty-one ametrope subjects repeatedly fixated a laser speckle-based fixation target for an extended duration of 3 s. Fixational area, fixational saccade rate as well as fixational saccade amplitude were analyzed during two different time intervals, namely in fixation initiation and sustained fixation. Fixation initiation was evaluated within the first 500 ms of fixation, whereas sustained fixation was evaluated 1 s after the eyes were directed toward the fixation target. During fixation initiation, fixation accuracy decreased in comparison to sustained fixation; fixational saccades occurred more frequently, and with larger amplitudes. During sustained fixation, an impact of refractive error was shown. With increasing refractive error, fixational saccade amplitude and fixational saccade rate increased. Fixational area increased in high ametropia through larger and more frequent saccades, but at a rather moderate rate of 10 arcmin/diopter (dpt) radial increase.

Introduction
In natural behavior fixation is a complex interplay of different behavioral incentives. Upon saccading towards a target, the target is intended to be positioned onto the foveae of each eye. In parallel, it is brought into focus through accommodation. Each of the involved oculomotor processes, namely version, vergence, and accommodation, show their own kinematics after target appearance. Whereas saccade execution is temporally distinct, accommodation and vergence show slower time scales (Mordi & Ciuffreda, 2004; Yang, Bucci, & Kapoula, 2002). Thus, there is a phase, during which eyes are already on target, but accommodation and potentially corrective saccades or slow vergence are suspected to continue. In this phase, oculomotor behavior might deviate from sustained fixation. 
For fully corrected or emmetropic vision, properties of sustained fixation have been described in detail. Specifically, the properties of fixational saccades have been studied intensively (Cherici, Kuang, Poletti, & Rucci, 2012; Rolfs, 2009; Rucci & Poletti, 2015). Fixation underlies a certain degree of dispersion induced by a variety of fixational eye movements, commonly separated into fixational saccades, drift, and tremor. They determine the area covered by the foveola over time, while fixating a small defined target. Properties of sustained fixation vary with the appearance of the target (Hirasawa, Okano, Koshiji, Funaki, & Shoji, 2016; Thaler, Schütz, Goodale, & Gegenfurtner, 2013). 
In contrast, fixation initiation, namely fixation directly after an initial saccade toward a fixation target, has not been described in such detail. It is unclear what the actual impact of establishing fixation is, and if such a binary distinction between the two phases is justified. 
The present study intends to distinguish the two phases in terms of their dependency on the degree of ametropia, thus defining a fixation initiation phase followed by a phase of sustained fixation, during which target properties impact oculomotor behavior. Target properties are varied through blur of uncorrected visual error in ametropic vision. In uncorrected vision, a target shown at infinity might not necessarily be imaged onto the retina. Eyes in which a target from infinity is imaged correctly onto the retina are called emmetropic eyes. Eyes with focal planes either in front of the retina or behind the retina are termed ametropic. Thus, in ametropic eyes target blur scales with refractive error, and persists even after adjustment of accommodation (Leube, Kostial, Ochakovski, Ohlendorf, & Wahl, 2018). We hypothesize that fixation initiation shows properties distinct from sustained fixation in terms of its dependency on refractive error. 
In the current study, 21 uncorrected subjects with myopic and hyperopic refractive errors fixated a target repeatedly. Fixational properties, namely amplitude and rate of fixational saccades, together with fixational area were evaluated during fixation initiation as well as sustained fixation. 
Material and methods
Study approval
The study was approved by the Ethics Committee of the Medical Faculty of the Eberhard Karls University of Tübingen and the University Hospital. 
Subjects
Twenty-one subjects aged between 21 and 43 years participated in the study with a prior written consent in adherence to the Declaration of Helsinki. The study population consisted of 19 myopes (−2.0 to −7.75 dpt) and two hyperopes (3.0 to 6.0 dpt), and subjects were otherwise eye healthy. Subjects with a spherical refractive error of minimum ±2.0 dpt in the right eye were included. Refractive errors for each participant were measured objectively with a wave front-based auto refractor (ZEISS i.Profiler plus; Carl Zeiss Vision GmbH, Aalen, Germany). All participants were naïve to the purpose of the study and only four subjects had prior experience with fixation experiments. 
Experimental setup
The experimental setup is depicted in Figure 1. The study was conducted with the subjects sitting in a height-adjustable chair at a table. In doing so, they were facing an arrangement of the Eyelink 1000 plus eye tracker (SR Research, Ltd., Ottawa, Canada), a laser speckle-based fixation target as used, for example, in the Visufit 1000 spectacle centration device (Carl Zeiss Vision GmbH, Aalen, Germany). It produces a cross target projected to infinity. This cross target contains a laser speckle substructure. Laser speckles are the noise-like pattern occurring upon reflection of coherent laser light at a surface, for example, when shining a laser pointer onto a white wall. Speckles have an optically defined spatial scale that, in this fixation target, is independent from the refractive error of the subject and therefore is not objective to blurring. Thus, any subject is able to see the speckle structure clearly regardless of refractive error. It is therefore believed to support fixation under refractive error. The target was positioned on a specifically designed holder, and a chinrest. The fixation target holder placed the fixation target at a distance of 31 cm on top of the eye tracker. To present calibration targets during the calibration of the eye tracker, a 22 in. fast cathode ray tube (CRT) monitor with resolution of 1280 × 1024 pixels and refresh rate of 85 Hz (HM204Dt A; iiyama, Chuo, Japan) was positioned at 55 cm distance from the subject's eyes. During calibration the fixation target was removed allowing free sight onto the CRT monitor which was used for calibration only. During data acquisition, the fixation target was placed again 31 cm away from the subject's eyes on the holder. 
Figure 1
 
Experimental setup, (A) from the top, (B) from the side, (C) in the viewing direction of the participant. (D) Example eye trace together with a depiction of the sound indicator as trigger for start and end of fixation.
Figure 1
 
Experimental setup, (A) from the top, (B) from the side, (C) in the viewing direction of the participant. (D) Example eye trace together with a depiction of the sound indicator as trigger for start and end of fixation.
The fixation target consisted of a cross of an overall extent of ±10.2° horizontally and ±5.1° vertically with a cross arm width of approximately 30 arcmin and maximal overall power of 450 μW at 635 nm. The speckles of the fixation target were visible at any accommodative distance, the cross was in focus at accommodation to distance. Stereo geometry of the target was optimal at parallel gaze. Calibration, data acquisition, and sound were controlled by a custom-made Psychtoolbox-based software (Brainard, 1997) in MATLAB (MathWorks, Natick, MA) on a Macbook Pro (Apple Inc., Cupertino, CA). 
Experimental procedure
Subjects were led into the study room and seated in the chair. To ensure comfort and minimal movement during the experiment subjects were motivated to adjust the height of the chair so that they sat comfortably while placing their head in the chin rest. Subsequently, they were asked to remove their spectacles. Then, the light was turned off and subjects were given a few minutes for their eyes to adapt to the darkness. The subject's head was placed in the chin and forehead rest with foam cushions on the inner sides for a stable head support. Thereafter, the subject's head was fixed by an elastic band tightened around the forehead rest and the subject's head to reduce the head movements to a minimum. Viewing was binocular. Before each measurement the eye-tracker was calibrated with an automatic nine-point calibration procedure. The calibration targets were displayed on the monitor. Before the study began, head movements were recorded with the head free mode of the Eyelink eye tracker (SR Research) as a control for head stability. For a duration of 5 min the coordinates of the subject's head changed less than 1 mm horizontally and less than 2.5 mm in the vertical direction for any subject. Thereafter, the fixation target was placed covering the CRT monitor for the rest of the experiment. 
Participants were asked to alternate fixation between the centrally placed fixation target and two eccentric fixation markers at eccentricities of 21° at the pace of a two-part auditory cue. One trial would start at fixation on the left eccentric marker. After a random delay of 2–4 s, a sound occurred. Upon this sound subjects changed fixation to the central speckle-based fixation target. Subjects' fixation remained on target for 3 s. Thereafter, a lower-tone sound occurred. Upon this second sound, subjects changed fixation to the right eccentric fixation marker. The next trial was initiated with random waiting time between 2 and 4 s, after which the higher sound indicated the change from the right eccentric fixation marker back to central fixation on the fixation target. Subjects were instructed to keep the central fixation until the lower sound indicated a fixation change back to the left fixation target. From there, the described sequence started anew. The experiment consisted of 38 trials, each trial starting with eccentric fixation on one of the eccentric markers, changing fixation towards the centrally placed fixation target upon a sound, and changing fixation to the opposite fixation marker upon a second sound. 
Data analysis
Data analysis was carried out with MATLAB (MathWorks). Data of the dominant eye were analyzed. First, central fixation was identified within the time interval of the two sounds. Fixation was defined by the oculomotor behavior shown under the effort to maintain the gaze in a predefined region (Poletti & Rucci, 2016), eye positions after an initial saccade toward the target were considered as fixation. Within fixation, two phases were analyzed—a first phase representing the initiation phase of fixation, and a second phase of established fixational behavior. The first phase consisted of the first 500 ms of fixation, the second phase had a duration of 1,000 ms starting 1,000 ms after the execution of the saccade to the central fixation target. Blinks were automatically detected by the EyeLink system (SR Research) and removed, analysis windows were extended accordingly. Seven percent of trials were affected by blinks. Furthermore, the first trial of each dataset was removed. Trials with saccades from or to the fixation target unrelated to the sound were discarded from the analysis. 
Fixational saccades were extracted with a standard 2D velocity-based filtering algorithm (Kliegl & Engbert, 2003) with λ = 9, which determines the sensitivity of the algorithm. Events smaller than 1 arcmin or 1 arcmin/ms were excluded from the analysis. Furthermore, an intrasaccadic time interval of minimum 15 ms was applied. Over all subjects and trials, saccadic events larger than 4° were excluded, in all those were five events. Extraction of fixation phases as well as fixational saccades was controlled visually. Based on the main sequence, events with a peak velocity per amplitude fraction smaller than 260°/s−1 were discarded as drift events. Fixation stability was defined by the bivariate contour ellipse area (BCEA) of 63% proportional value (Castet & Crossland, 2012). BCEA is defined as  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}{\rm{BCEA}} = 2k\pi {\sigma _h}{\sigma _v} \times {\left( {1 - {\rho ^2}} \right)^{1/2}}\end{equation}
where σh is the standard deviation of point locations over the horizontal meridian, σv the standard deviation of point locations over the vertical meridian, and ρ the product-moment correlation of these two position components. The parameter k scales which fraction of the data is covered by the ellipse. With k = 1 used here, 63% of the data are covered by the calculated ellipse.  
In a first step, a linear mixed-model (LMM) considering the factors fixation phase (initiation or sustained) and refractive error was developed separately for the three measures, fixational saccade rate, fixational saccade amplitude, and fixational area, with zero refractive error (emmetropia) and the sustained phase as baseline. Fixational area was thereby considered an overall measure, which is affected by fixational saccade amplitude as well as rate, but might be affected by additional factors, such as drift. The AIC (Akaike information criterion) of two LMMs, one with the factor fixation phase included, and a second one with refractive error included separately, were compared to a model in which both factors are included. Analysis of distributions and inspection of residuals strongly suggested log-transformation to meet LMM assumptions. A LMM describing fixational saccade rate by fixation phase showed an AIC of 50.7, while including a refraction dependency as the only factor, AIC reduced to 13.8. A combined model showed a decreased AIC of 9.4. For fixational saccade amplitude, a LMM including only fixation phase as factor showed an AIC of 70.9, a LMM including a refraction dependency only showed an AIC of 60.7. The combined model reduced the AIC to 58.4. A LMM on fixational area with fixation phase as factor showed an AIC of 128.4. Inclusion of refractive error as only factor showed an AIC of 108.23. The proposed two-factor LMM showed a further decreased AIC of 96.6. Thus, three LMMs of the model  
\begin{equation}\text{rate/ampl/area} \sim 1 + (\text{refr} \times \text{phase}) + (1 \mid \text{subj})\end{equation}
 
were fitted with the maximum pseudo-likelihood method in MATLAB on log-scaled data, with 42 observations with four fixed-effect coefficients, and 21 random effects coefficients. 
In a second step, post hoc analysis significance of correlations with the degree of ametropia was estimated with bootstrapped confidence intervals on the basis of 10,000 samples. This analysis was performed separately for both phases—fixation initiation and sustained fixation. Confidence intervals, which did not overlap with zero, were reported. The average spherical refraction of the two eyes of each subject was taken as reference for the refractive status. Within the graphs, average values together with standard errors over trials are reported. Fixation initiation was expected not to vary with refractive error, but it was assumed to be somewhat driven by the general attempt to establish fixation. In contrast, fixational properties during sustained fixation were expected to depend on refractive error. Specifically, fixational saccade amplitude as well as fixational area were expected to increase with refractive error. 
Results
The present study evaluated the influence of refractive status on fixation, and specifically, its influence throughout establishment of fixation. Figure 2 shows a random selection of fixational eye movements for two example subjects. Subject A had a rather low average spherical equivalent refractive error of −2 dpt, whereas Subject B had high ametropia with an average spherical equivalent of −6 dpt. In Figure 2, the two fixation phases are marked in red and blue, respectively. The first 500 ms of fixation, termed fixation initiation, are depicted in red. Eye traces during fixation in the time interval between 1,000 ms and 2,000 ms are shown in blue, analyzed in the following as representative phase of sustained fixation. Subject A showed a clear difference between the initiation phase, and the sustained fixation phase, whereas Subject B showed stronger variations throughout the whole fixation phase. Differences in fixation between the two phases, as well as the impact of the refractive error in uncorrected fixation, were analyzed. Furthermore, fixational saccade properties and fixational area were analyzed. Fixational saccades were characterized via fixational saccade rate and average fixational saccade amplitude. Results of the LMM are summarized in Table 1
Figure 2
 
Eye movement samples from two example subjects, (A) with moderate myopia of −2 dpt, and (B) with high myopia of −6 dpt.
Figure 2
 
Eye movement samples from two example subjects, (A) with moderate myopia of −2 dpt, and (B) with high myopia of −6 dpt.
Table 1
 
Results summary of the linear mixed model analyses for the three parameters, amplitude, rate, and area. Bold p-values are cited in the text. Please note that the LMM is performed on log-scaled data.
Table 1
 
Results summary of the linear mixed model analyses for the three parameters, amplitude, rate, and area. Bold p-values are cited in the text. Please note that the LMM is performed on log-scaled data.
Differences between fixation initiation and sustained fixation
According to the fitted LMM, fixational saccade rate dropped significantly in sustained fixation, compared to fixation initiation (p < 0.0001), from an average of 3.2 Hz ± 0.68 Hz during initiation to 1.8 Hz ± 0.65 Hz under sustained fixation. Figure 3 shows fixational saccade rate in the two phases for all subjects, color coded for refraction with high ametropia more bluish and lower ametropia in more reddish. Fixational saccade amplitude dropped as well (p < 0.001), from 0.72° ± 0.23° in the initiation phase to 0.57° ± 0.4° during sustained fixation. Fixational areas were estimated through calculation of the BCEA. A general decrease of the fixational area was visible for most subjects, leading to a significant decrease in fixation area (p < 0.0001). It decreased from an average of 1,773 arcmin2 to 649 arcmin2
Figure 3
 
Comparison of fixational properties between fixation initiation and sustained fixation for each subject: (A) fixational area, (B) fixational saccade amplitude, and (C) fixational saccade rate. Subjects with high ametropia of refractive errors larger the 5 dpts myopia are marked in black.
Figure 3
 
Comparison of fixational properties between fixation initiation and sustained fixation for each subject: (A) fixational area, (B) fixational saccade amplitude, and (C) fixational saccade rate. Subjects with high ametropia of refractive errors larger the 5 dpts myopia are marked in black.
Refraction dependency
Systematic changes in fixation characteristics with the degree of ametropia were evaluated in a correlation analysis. Figure 4 shows fixation characteristics for the different refractive errors of the subjects: fixational saccade rate, amplitude, and fixational area. Fixation initiation and sustained fixation were represented separately. Each subgraph shows the respective fixational property for each subject, averaged over trials. This data is presented together with standard errors, and the standard deviation of spherical refractive error between both eyes. The black line indicates the result of the LMM. Please note that the data is presented on a semi-logarithmic scale. In the following, dependencies on refractive status were identified through analysis of main factor refraction in the LMM as well as a post hoc correlation analysis. Subfigures in the left column of Figure 4A, C, and E show potential dependencies on uncorrected refractive status in the initiation phase; Figure 4B, D, and F show the according dependencies in the sustained phase. 
Figure 4
 
Dependency of fixational properties on uncorrected refractive error during fixation initiation (A, C, E) and sustained fixation (B, D, F): Fixational area (A, B), fixational saccade amplitude (C, D), and rate (E, F) in fixation initiation, compared to sustained fixation.
Figure 4
 
Dependency of fixational properties on uncorrected refractive error during fixation initiation (A, C, E) and sustained fixation (B, D, F): Fixational area (A, B), fixational saccade amplitude (C, D), and rate (E, F) in fixation initiation, compared to sustained fixation.
Fixational saccade rate
Fixational saccade rate was not dependent on refractive status in the initiation phase (Pearson r = −0.17, not different from zero at 95%), and was at a constantly elevated level of 3.2 Hz. In contrast to that, significant correlations with average spherical refractive error occurred during sustained fixation. Fixational saccade rate, as shown in Figure 4B, was increased in subjects with high ametropia compared to subjects with moderate ametropia. This led to a significant correlation of fixational saccade rate with ametropia (Pearson r = 0.48, confidence interval 95%). The LMM showed a significant increase of fixational saccade rate with ametropia (p < 0.01). A compensating, significant interaction (p < 0.05) indicated that this relationship did primarily hold for the sustained phase. In fact, a model of the initiation phase only did not show a significant dependency on refraction (21 observations, two fixed effect coefficients, 21 random effect coefficients, p = 0.49). 
Fixational saccade amplitude
Fixational saccade amplitude, on average 0.72°, was not showing a dependency on refractive status in the initiation phase, (Pearson r = 0.14, not different from zero at 95% confidence), as visible in Figure 4C. In Figure 4D, fixational saccade amplitude increased with spherical refractive error at a rate of 4.8 arcmin/dpt in the sustained phase, when comparing subjects with different refractive errors. Figure 4F illustrates the correlation of Pearson r = 0.29 (confidence interval 95% confidence). The LMM showed a significant increase of fixational saccade amplitude with ametropia (p < 0.05). A compensating, significant interaction (p < 0.05) indicated that this relationship did primarily hold for the sustained phase. Like for the fixational saccade rate, a baseline model of the initiation phase only did not show a significant dependency on refraction (21 observations, two fixed effect coefficients, 21 random effect coefficients, p = 0.53). 
Fixational area
Fixational area, shown in Figure 4E, was not dependent on refractive status in the initiation phase, the correlation of Pearson r of −0.15 was not different from zero at 95%. Thus, fixational area was not different between subjects of moderate and high ametropia in the fixation initiation phase. In contrast, Figure 4F shows clearly a smaller fixation area in subjects with moderate ametropia (Pearson r = 0.67, confidence interval 99%) in sustained fixation. Fixational area increased by approximately 10 arcmin/dpt radius, with a more pronounced increase for higher ametropia. Subjects with moderate ametropia up to 4 dpt showed a constantly low fixation area with one exception. The LMM indicated a significant increase of fixational area with ametropia (p < 0.001). The interaction indicated a significantly smaller effect in the initiation phase (p < 0.001). A baseline model of the initiation phase only did not show a significant dependency on refraction (21 observations, two fixed effect coefficients, 21 random effect coefficients, p = 0.80), indicating that the fixational area was independent from refractive error in the initiation phase. 
Discussion
In the current study, properties of fixation under uncorrected vision have been analyzed during initiation of fixation as well as during sustained fixation. Fixational area, fixational saccade rates, and fixational saccade amplitudes showed characteristic differences between fixation initiation and sustained fixation. When comparing the two phases, fixational area was increased during initiation. Furthermore, fixational saccades occurred more frequently and had larger amplitudes, independent of the amount of uncorrected refractive error. During sustained fixation fixational area, fixational saccade rate, as well as fixational saccade amplitudes, increased with refractive error. Thus, accuracy of target information played a dominant role in sustained fixation but did not in fixation initiation. 
Myopia versus hyperopia
Two of the subjects in the present dataset were hyperopic. An object, locatable in a specific depth plane, would be brought back onto the retina by accommodation in hyperopic subjects. The laser-speckle target in this study provided fine spatial scale in those subjects as well. Thus, accommodation might have been discouraged. In this case, the stimulus would have been largely indistinguishable between a myopic defocus and a hyperopic defocus, and thus behavior might look similar. 
Difference between fixation initiation and sustained fixation
When evaluating the difference between fixation initiation and sustained fixation, a proper definition of fixation is a prerequisite. Fixation can generally be defined in two ways, either by an oculomotor behavior shown under the effort to maintain the gaze in a predefined region, or on the basis of oculomotor behavior, often defined by occurrence of microsaccades under certain amplitude thresholds (Poletti & Rucci, 2016). As fixational saccades' amplitude depended on refractive error in the present study, the earlier definition appeared more feasible. Within that definition, fixation initiation and sustained fixation would be expected not to differ, as in both phases the subject was intending to hold gaze onto the fixation target equally well. Nonetheless, fixation properties have been shown to vary in the two cases. The following analysis intends to explain the observed differences. 
Fixational saccade amplitude elevation during fixation initiation
In the initiation phase, fixational saccade amplitude was constantly elevated, independent from the refractive error of the subject. The task of initiation of fixation includes three subtasks, two of which are probably closely coupled. The first task is bringing the eyes onto the target with saccades. The second task is the alignment of the two eyes in the target plane with vergence, together with accommodation. After a large saccade onto a target, usually corrective saccades occur. The amplitude of those would depend on the accuracy of the large saccade toward the fixation target. The accuracy of a saccade depends, among other factors, on the noise of the visual input (van Beers, 2007). At large eccentricities, as used in the present study, correction saccades occur, independent of refractive error. Those might occur instead of or in parallel to fixational saccades. With a standard latency of approximately 200 ms, they fall well within the window of fixation initiation. Another potential source for additional eye movements during fixation initiation comes from the necessity to establish accommodation and vergence. As the time scale of vergence is usually slower than that of saccades (Collewijn, Erkelens, & Steinman, 1995), the initial fixation phase after the primary targeting saccade is very likely to include vergence, and potentially vergence saccades. Previous work has shown that vergence changes of 6°–8° result in vergence saccades of approximately 1.5° (Coubard & Kapoula, 2005). In our experiments, vergence changes were roughly 3°; assuming vergence saccades stay below vergence amplitude, the 0.72° average amplitude of saccades in the fixation initiation phase fits the existing description of vergence saccades. 
Fixational saccade rate during fixation initiation
The constantly elevated rate of fixational saccades might provide further evidence that fixation initiation includes motor behavior additional to fixation, probably in the form of corrective saccades, or vergence. While the amplitudes of the observed saccades, within fixation initiation, fit both types of eye movements well, the rate increase itself is surprising. There has been an intense debate if vergence saccades and disjunctive saccades underlie the same saccadic process, but evidence for parallel drivers for saccades of different types are rare. For disjunctive saccades, it is a matter of ongoing debate if they are controlled binocularly or not, with an increasing amount of evidence for a monocular component (for review, see King, 2011). 
We therefore postulate, that fixation initiation, defined by a voluntary direction of gaze onto a target, represents a process qualitatively different from sustained fixation. Fixational saccades generated during this process are not as strictly based on visual input, but process target information rather roughly. The phase might be characterized by a dense interaction of vergence and saccadic behavior. In that sense, stable fixation would be defined by the absence of vergence movements and corrective saccades. 
Fixational saccade amplitude during sustained fixation
Under sustained fixation, fixational saccade amplitudes as well as fixational area are influenced by target appearance (Cherici et al., 2012; Ghasia & Shaikh, 2015; Hirasawa et al., 2016; Thaler et al., 2013). Furthermore, saccadic amplitudes are well known to depend on high- as well as low-level image properties in free viewing (Castelhano, Mack, & Henderson, 2009; Foulsham, Teszka, & Kingstone, 2011; Henderson, 2003; Kowler, 2011; Tatler, Baddeley, & Vincent, 2006). Comparably, fixational eye movement distribution properties seem to depend on the given task (Mergenthaler & Engbert, 2010). In the current experiment, it is unclear if fixation target appearance was exactly identical. The speckle-based nature of the target left an identical substructure within the fixation target, independent of accommodation. However, the overall thickness of the fixation cross arms was increasing with deviation from far focus and would thus increase with increasing ametropia. Through a broadened target, fixational saccade amplitude during sustained fixation was dependent on the refractive error of the subject. The speckle-target substructure did not seem to support the establishment of fixation enough for fixational amplitude to be independent from refractive error. 
Fixational saccade rate during sustained fixation
More surprisingly, fixational saccade rates showed a dependency on refractive error as well and did not settle on the level known for sustained fixation from the literature. In general, saccades occur quite regularly, either as large-scale saccades or as microsaccades, at rates of 1–2 per second, with a distribution skewed to long latencies (Amit, Abeles, Bar-Gad, & Yuval-Greenberg, 2017; Cherici et al., 2012; Ko, Poletti, & Rucci, 2010). Under sustained fixation as well as high acuity tasks, saccades occur at lower rates than under free viewing (Amit et al., 2017; Ko et al., 2010). Saccade rate furthermore depends on training status and task (Bonneh et al., 2010; Cherici et al., 2012; Rolfs, Kliegl, & Engbert, 2008), but at a given task and target, fixational saccade rate is expected to be constant. The origin of this rhythmicity is a matter of constant debate (Amit et al., 2017). Nonetheless, saccade rates as well as amplitudes of highly ametropic subjects, of 5 dpts spherical equivalent refractive error or more, stayed on the level of fixation initiation. Thus, the increased fixational saccade rate with increasing ametropia can instead be interpreted as a continued effort to initiate fixation. One might therefore ask if, in high ametropia, sustained fixation is ever reached. 
Fixational area—sustained fixation
Increases in fixational area observed are approximately in line with the literature, when a 8.9 arcmin/dpt increase in standard deviation is reported (Ukwade & Bedell, 1993). Fixational area increased together with fixational saccade rate and amplitude. The measured increase indicates that drifts do not compensate for a spread through fixational saccades. Furthermore, the possibility that fixational saccades reposition more with increasing ametropia to prevent a larger spread in fixation area, is ruled out. In highly ametrope subjects with refractive errors of 5 dpts or more, fixational area was not reduced from fixation initiation to sustained fixation. This supports the abovementioned hypothesis, that sustained fixation is never fully established; instead, the behavioral pattern of fixation initiation stays valid for them. 
Do highly ametropic subjects actually fixate?
In the current study, highly ametropic subjects showed oculomotor behavior during sustained fixation, which was comparable to initiation of fixation. Specifically, fixational saccade rate and fixational saccade amplitude stayed elevated. Figure 4 shows color codes for refractive error for the different phases; highly ametropic subjects are coded more blueish and they tended not show a large gradient between the phases. The question arises if highly ametropic subjects actually fixate, or if they remain in the attempt to establish fixation. Whereas a constantly elevated level of fixational saccade amplitude and fixational area might just be attributed to target widening through blur, the increased fixational saccade rate can be treated as a clear indicator that in high ametropia, additional oculomotor processes take place, comparable to the fixation initiation phase. 
Accuracy of fixation in ophthalmology and optometry
Nonetheless, many medical examinations and measurements of the eye properties rely on stable fixation during measurement (Abell et al., 2014; Mrochen, Kaemmerer, Mierdel, & Seiler, 2001; Porter et al., 2005; Thibos, Applegate, Schwiegerling, & Webb, 2002). Thus, as most examinations are performed under uncorrected vision, fixation does represent a challenge for measurement accuracy in case of refractive error. 
These results shed light on fixational saccade accuracy to be expected in optometric and ophthalmic devices used with the naked eye. Specifically, in subjects with high ametropia, increased measurement errors might be expected due to inaccurate fixation in ophthalmic and optometric devices. Specifically, oculomotor behavior indicates that sustained fixation is never established in highly ametropic subjects, which might affect their visual performance in addition to experiencing blur through uncorrected refractive errors. 
In summary, in this study, fixational saccade rate, fixational saccade amplitude, and fixational area were increased during fixation initiation in comparison to sustained fixation, independent of refractive error. During sustained fixation, fixational saccade amplitude and rate increased together with fixational area under uncorrected refractive error. Furthermore, in high ametropes with refractive errors of 5 dpts and more, fixational saccade rate, amplitude as well as area stayed elevated. 
Acknowledgments
This work was funded by the German Federal Ministry of Education and Research (BMBF) through the Bernstein Center for Computational Neuroscience Tuebingen (FKZ 01GQ1002). We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of University of Tübingen. 
Commercial relationships: Siegfried Wahl and Katharina Rifai are employed at Carl Zeiss Vision International GmbH. 
Corresponding author: Katharina Rifai. 
Address: Institute for Ophthalmic Research, University of Tuebingen, Tuebingen, Germany. 
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Figure 1
 
Experimental setup, (A) from the top, (B) from the side, (C) in the viewing direction of the participant. (D) Example eye trace together with a depiction of the sound indicator as trigger for start and end of fixation.
Figure 1
 
Experimental setup, (A) from the top, (B) from the side, (C) in the viewing direction of the participant. (D) Example eye trace together with a depiction of the sound indicator as trigger for start and end of fixation.
Figure 2
 
Eye movement samples from two example subjects, (A) with moderate myopia of −2 dpt, and (B) with high myopia of −6 dpt.
Figure 2
 
Eye movement samples from two example subjects, (A) with moderate myopia of −2 dpt, and (B) with high myopia of −6 dpt.
Figure 3
 
Comparison of fixational properties between fixation initiation and sustained fixation for each subject: (A) fixational area, (B) fixational saccade amplitude, and (C) fixational saccade rate. Subjects with high ametropia of refractive errors larger the 5 dpts myopia are marked in black.
Figure 3
 
Comparison of fixational properties between fixation initiation and sustained fixation for each subject: (A) fixational area, (B) fixational saccade amplitude, and (C) fixational saccade rate. Subjects with high ametropia of refractive errors larger the 5 dpts myopia are marked in black.
Figure 4
 
Dependency of fixational properties on uncorrected refractive error during fixation initiation (A, C, E) and sustained fixation (B, D, F): Fixational area (A, B), fixational saccade amplitude (C, D), and rate (E, F) in fixation initiation, compared to sustained fixation.
Figure 4
 
Dependency of fixational properties on uncorrected refractive error during fixation initiation (A, C, E) and sustained fixation (B, D, F): Fixational area (A, B), fixational saccade amplitude (C, D), and rate (E, F) in fixation initiation, compared to sustained fixation.
Table 1
 
Results summary of the linear mixed model analyses for the three parameters, amplitude, rate, and area. Bold p-values are cited in the text. Please note that the LMM is performed on log-scaled data.
Table 1
 
Results summary of the linear mixed model analyses for the three parameters, amplitude, rate, and area. Bold p-values are cited in the text. Please note that the LMM is performed on log-scaled data.
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