October 2014
Volume 14, Issue 12
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Article  |   October 2014
The impact of higher-order aberrations on the strength of directional signals produced by accommodative microfluctuations
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Journal of Vision October 2014, Vol.14, 25. doi:10.1167/14.12.25
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      Sangeetha Metlapally, Jianliang L. Tong, Humza J. Tahir, Clifton M. Schor; The impact of higher-order aberrations on the strength of directional signals produced by accommodative microfluctuations. Journal of Vision 2014;14(12):25. doi: 10.1167/14.12.25.

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Abstract

It has been proposed that the accommodation system could perform contrast discrimination between the two dioptric extremes of accommodative microfluctuations to extract directional signals for reflex accommodation. Higher-order aberrations (HOAs) may have a significant influence on the strength of these contrast signals. Our goal was to compute the effect HOAs may have on contrast signals for stimuli within the upper defocus limit by comparing computed microcontrast fluctuations with psychophysical contrast increment thresholds (Bradley & Ohzawa, 1986). Wavefront aberrations were measured while subjects viewed a Maltese spoke stimulus monocularly. Computations were performed for accommodation or disaccommodation stimuli from a 3 Diopter (D) baseline. Microfluctuations were estimated from the standard deviation of the wavefronts over time at baseline. Through-focus Modulation Transfer, optical contrast increments (ΔC), and Weber fractions (ΔC/C) were derived from point spread functions computed from the wavefronts at baseline for 2 and 4 cycles per degree (cpd) components, with and without HOAs. The ΔCs thus computed from the wavefronts were compared with psychophysical contrast increment threshold data. Microfluctuations are potentially useful for extracting directional information for defocus values within 3 D, where contrast increments for the 2 or 4 cpd components exceed psychophysical thresholds. HOAs largely reduce contrast signals produced by microfluctuations, depending on the mean focus error, and their magnitude in individual subjects, and they may shrink the effective stimulus range for reflex accommodation. The upper defocus limit could therefore be constrained by discrimination of microcontrast fluctuations.

Introduction
Reflex accommodation is a response purely to a change in the vergence of light reaching the eye (Heath, 1956). Blur, as an indicator of defocus, is necessary as a cue for accommodation, but is not sufficient (Smithline, 1974), as defocus blur does not provide odd-error information about the direction to accommodate. The eventual goal of the system would be to reduce the imposed defocus and optimize retinal image contrast (Alpern, 1958; Fender, 1964; Heath, 1956; Kotulak & Schor, 1986a; Manny & Banks, 1984; Owens, 1980; Raymond, Lindblad, & Leibowitz, 1984; Switkes, Bradley, & Schor, 1990; Toates, 1972). Fincham (1951) was one of the earliest researchers to study the characteristics of the stimulus that produces reflex accommodation and observed that reflex accommodation has an upper limit for its operating range of about 2 D, which we refer to here as the upper defocus limit. Chromatic aberration and monochromatic higher-order aberrations (HOAs), such as spherical aberration were investigated as potential directional cues by Fincham (1951), Campbell and Westheimer (1959), and later by several others (Chin, Hampson, & E. Mallen, 2009; Chin, Hampson, & E. A. Mallen, 2009; Fernandez & Artal, 2005; Kruger, Nowbotsing, Aggarwala, & Mathews, 1995). These studies either eliminated or altered chromatic aberration, or monochromatic HOAs, and indicate that there is intersubject variability in which of these cues may be used to determine the correct direction of accommodation. It is as yet unclear what characteristics of the stimulus, the optics of the eye or the behavior of the accommodation system, are used to decipher the sign of defocus and drive reflex accommodation. 
Steady state accommodative responses are characterized by accommodative microfluctuations that may offer a temporal cue for detecting the correct direction and magnitude of the accommodation stimulus. Several researchers (Alpern, 1958; Campbell, Westheimer, & Robson, 1958; Charman & Heron, 1988; Kotulak & Schor, 1986a) have proposed that odd-error information could be derived from spatial-even-error signals, i.e., contrast changes from defocus fluctuations could drive the system to stop or turn away from the direction, which causes a decrease in contrast and toward the direction that causes an increase in contrast. The magnitude of microfluctuations have been reported to increase with target vergence (Charman & Heron, 1988; Kotulak & Schor, 1986b), and in some instances have been reported to be high near the middle of the accommodative range and minimal at the near and far point (Leahy, Leroux, Dainty, & Diaz-Santana, 2010; Miege & Denieul, 1988). In addition to information about the response direction that favors an increased retinal image contrast, they could provide an estimate of the magnitude of the lag of accommodation (Kotulak & Schor, 1986b). Indeed, observations from our laboratory suggest that adding uncorrelated contrast noise on the stimulus screen to simulate and/or exaggerate the effect of defocus fluctuations due to microfluctuations reduced the percentage of correct responses to a polychromatic stimulus in some young subjects, favoring a role for microfluctuations in providing directional information (Metlapally, Tong, Tahir, & Schor, 2014). 
Spatial variations, together with temporal variations of the retinal image contrast, therefore, have been studied with great interest. MacKenzie, Hoffman, and Watt (2010) have most recently investigated the contribution of various spatial frequency components to the accommodation response computationally. They computed the retinal contrast changes produced by ±0.25 D accommodative microfluctuations at a range of baseline defocus levels up to 1.5 D and spatial frequencies 2, 4, 8, 12 and 16 cpd. They assumed that microfluctuations provide directional information over time and additionally, that a greater magnitude of contrast change brought about by microfluctuations corresponded to a greater signal for accommodation. They concluded that spatial information above the 4–6 cpd range would not be useful in driving the accommodation response based on retinal contrast changes brought about by microfluctuations. The literature would therefore support the assumption that microfluctuations of accommodation could provide temporal directional information to the system and allow contrast signals to be computed from these, particularly for spatial information below the 4–6 cpd range. 
Additionally, while monochromatic HOAs (≥ third order) within normal magnitudes only negligibly impact the overall contrast of objects compared to lower-order aberrations (LOAs; Salmon & van de Pol, 2006; Thibos, Hong, Bradley, & Cheng, 2002), they could interact with accommodative microfluctuations and have a significant impact on the resultant microcontrast changes. Our analysis also seeks to answer questions about what limits the range of initial focus errors that can be detected by the reflex accommodation system, i.e., what constrains the upper defocus limit. MacKenzie et al. (2010) estimated the changes with all the monochromatic aberrations of one subject, but did not compute the effect of individual aberrations on the contrast changes produced by the microfluctuations. In this study, we extend their analysis to higher levels of baseline defocus within the upper defocus limit and analyze the effects of certain aberrations on contrast changes brought about by microfluctuations. We also compare the contrast increments produced by oscillations of accommodation as computed from wavefront measures, with psychophysical contrast increment thresholds from Bradley and Ohzawa (1986). 
Our study asks new and important questions about the influence of HOAs on the strength of directional signals and what influences the upper defocus limit for reflex accommodation. This approach ignores the influence of motor factors (for e.g., where voluntary accommodation would be used to negate most of the defocus until the system gets close to optimal focus) and concentrates on sensory or retinal image driven accommodation while emphasizing reflex to avoid the influence of cognitive factors. Also, in the older literature, investigations focused on how the system responded to spatial information in the image by looking at the lag of accommodation (Kotulak & Schor, 1987). We have investigated what spatial information is available at a focus error within the upper defocus limit that could be used to drive the initial responses. The lag of accommodation was incorporated in our calculations as we shifted our baseline from far (0 D) to 3 D. Our pragmatic hypothesis was that the upper defocus limit may be set by the availability of detectable directional signals from microcontrast fluctuations caused by accommodative microfluctuations. 
Methods
Subjects
Young adult prepresbyopic volunteers with normal, general ocular health, low refractive errors, and astigmatism correctable to within 0.5 D (n = 25) were screened for the study. Among the subjects who met these inclusion criteria, we recruited only those from whom we could elicit accommodation responses to our polychromatic, spatially broadband, high-contrast Maltese spoke target viewed through a Badal imaging lens. Most of the subjects screened had sluggish or no reflex accommodation responses in the above laboratory condition, possibly due to the dissociation of focus change from size cues and lack of habitual binocular cues. The subjects selected (n = 4) were 18–25 years old (M = 23 years). The study procedures followed the tenets of the Declaration of Helsinki. 
Empirical methods
Subjects were stabilized with a headrest and chin-cup and viewed a 4.5 cd/m2 Maltese spoke target on a high resolution LCD screen (Totoku Electric Co., Ltd., Tokyo, Japan) monocularly in a dimly lit room through a Maxwellian view system (see schematic in Figure 1A). A Badal lens ensured that there were no size changes with the step changes in optical vergence. Wavefront aberrations were measured using a custom built Shack-Hartmann wavefront sensor with a 4-mm artificial pupil imaged at the entrance pupil of the eye. They were sampled dynamically over each 8-s trial at 50 Hz and fitted with Zernike polynomials up to the sixth order. The high sampling rate ensured that we obtained accurate estimates of the mean and standard deviation of wavefront aberrations of the subject that are known to change dynamically in general (Hofer, Artal, Singer, Aragon, & Williams, 2001; Mira-Agudelo, Lundstrom, & Artal, 2009), or with the tear film, (Koh et al., 2002; Montes-Mico, Alio, Munoz, Perez-Santonja, & Charman, 2004; Tutt, Bradley, Begley, & Thibos, 2000), or perhaps in micro eye movements. 
Figure 1
 
Schematic representations of the stimulus and recording set-up and stimulus time sequence used in the study. (A) Plan view of the apparatus used in the experiments, and the inset illustrates the broadband Maltese spoke stimulus as displayed on the monitor; (B) time sequence of the step stimulus, showing the change of the baseline from far (0 D) to 3 D at 1 s and sustenance of the stimulus/response at that level for 4 s. The mean response level is indicated by the dashed lines, showing a lag of accommodation. Accommodative microfluctuations about the mean are schematized as a simple sinusoidal fluctuation.
Figure 1
 
Schematic representations of the stimulus and recording set-up and stimulus time sequence used in the study. (A) Plan view of the apparatus used in the experiments, and the inset illustrates the broadband Maltese spoke stimulus as displayed on the monitor; (B) time sequence of the step stimulus, showing the change of the baseline from far (0 D) to 3 D at 1 s and sustenance of the stimulus/response at that level for 4 s. The mean response level is indicated by the dashed lines, showing a lag of accommodation. Accommodative microfluctuations about the mean are schematized as a simple sinusoidal fluctuation.
Refractive errors were compensated so that all the spokes of our stimulus appeared sharp at the beginning of the trial. Each experimental trial involved a 3 D accommodative step stimulus that changed the baseline from far to 33 cm. The stimulus represented a typical near viewing distance, and was held constant as a baseline for 4 s after the presentation (see time sequence in Figure 1B). This allowed us to compute the effects of simulated accommodative and disaccommodative step stimuli of magnitudes up to 3 D from the baseline. Data were collected in two experimental sessions, with over 20 trials in each session. 
Computational methods
Microfluctuations were estimated from the standard deviation of the wavefront aberrations collected from at least two trials within one experimental session (for ∼3 s each, after subjects had completed their responses to the 3 D step). In order to simulate optical changes due to different magnitudes of defocus stimuli presented from this 3 D baseline, the average wavefronts at the 3 D baseline from at least two trials were used to generate sequential sets of Zernike coefficients, where the defocus term was changed in 0.25 D steps from −5 D to 5 D. The wavefront estimates for analysis included refractive error and lag of accommodation, defined here as the difference between the stimulus and the response at the 3 D baseline. Monochromatic (550 nm) Modulation Transfer Functions (MTFs), defined as the absolute value of the Optical Transfer Functions (OTFs) were derived from Point Spread Functions (PSFs) computed from Zernike polynomials at each of these defocus steps, for 2 and 4 cpd components, and radially averaged for all orientations using a Matlab software program (The MathWorks, Inc., Natick, MA). Modulation transfer through-focus was graphed giving us the contrast (C) available at each of the defocus steps. This analysis was done separately with native aberrations up to the sixth order included (with HOAs) or aberrations ≥ third order excluded (without HOAs). We calculated the unsigned piece-wise differences in computed MTF at 0.25 D defocus increments, based on the minimum magnitude of microfluctuations exhibited by our subjects over the entire range of defocus values. This calculation provided estimates of contrast increments produced by microfluctuations (delta C or ΔC) as predicted by the optics or optical contrast increments. Weber fractions (ΔC/C) were calculated from retinal image contrast changes due to microfluctuations. 
Psychophysical measures of contrast increment thresholds were estimated from Bradley and Ohzawa's (1986) data corresponding to the contrast range of our stimuli. The data were fit with a linear regression after scaling the background screen contrasts to the peak retinal contrasts in our subjects. Our computed optical contrast increments produced by microfluctuations were then compared to the predicted psychophysical contrast increment thresholds to estimate whether the contrast changes caused by microfluctuations were above the contrast increment threshold. Weber fractions for the psychophysical estimates were compared to the computed optical estimates. Computed optical contrast increments that were greater than the predicted psychophysical contrast increment thresholds were considered detectable by the accommodative system. Finally, we investigated the impact of HOAs by including or eliminating some or all HOAs (≥ third order). The LOAs, defined as aberrations correctable using spectacle lenses comprising mainly of defocus and astigmatism, were tested for their effects. The notable HOAs tested were the third-order aberrations, particularly coma, and among the fourth-order coefficients, spherical aberration. These were expected to have a bigger impact than other HOAs on the overall dioptric magnitude of the microfluctuations by affecting the contrast increments they produce, as they are known to change or influence dynamic accommodation (Atchison, Collins, Wildsoet, Christensen, & Waterworth, 1995; Cheng et al., 2004; Gambra, Sawides, Dorronsoro, & Marcos, 2009; Ivanoff, 1947; Lopez-Gil et al., 2008; Plainis, Ginis, & Pallikaris, 2005). 
Results
The higher-order root-mean-square (HORMS) of the aberrations in our four subjects DD, JY, KS, and VBS at the 3 D baseline were 0.08, 0.16, 0.23, and 0.10 μm, respectively. The peak-to-peak magnitude of microfluctuations in our subjects ranged from 0.25 to 0.29 D based on the standard deviation of the defocus term in our subjects' wavefront measures. The calculated magnitude of the range is different (0.29–0.33 D) if the spherical aberration and the secondary spherical aberration terms are included using the following equation (Thibos, Hong, Bradley, & Applegate, 2004):  where M = equivalent defocus (D); Display FormulaImage not available = Zernike defocus (μm); Display FormulaImage not available = Zernike spherical aberration (μm); Display FormulaImage not available = Zernike secondary spherical aberration (μm); R = pupil radius (mm).  
We used the lower end of the range of contrast increments calculated from the defocus term as it was a conservative estimate of the total (peak to peak) variability of the defocus term and would result in the smallest estimate of available optical contrast signals in our subjects. The computed contrast changes due to microfluctuations were based on the optics of the eye when response to the 3 D step-stimulus was completed because we were interested in how the aberration profile at this baseline would impact the responses to subsequent changes in the accommodation stimulus. We assumed that microfluctuations available at the baseline condition were used to determine the correct direction to accommodate. 
Through-focus modulation transfer
The through-focus modulation transfers for each of our subjects at 2 and 4 cpd are shown in Figure 2. The graphs depict the functions with native aberrations up to the sixth order included or with HOAs (≥ third order) eliminated. The through-focus functions show peak modulation at defocus levels between −0.75 to −1 D for all subjects, which indicates a shift in the optimum focus position due to the lag of accommodation (defined here as the difference between the accommodative stimulus and accommodative response). More specifically, it is the monocular refractive state of the subjects at the 3 D baseline, defined as the peak of the through-focus function obtained using simulated changes in the Zernike defocus term or Display FormulaImage not available alone, for a spatially broadband high-contrast Maltese spoke target, and a 4-mm pupil. For a better understanding, note that increasing values in the negative range of the through-focus function (left of the peak) denote a relative hyperopic state, a lag of accommodation, or an accommodative stimulus. Increasing values in the positive range (right of peak) of this function denote a relative myopic state, a lead of accommodation, or a disaccommodative stimulus.  
Figure 2
 
Through focus MTFs for our subjects for spatial frequencies 2 (black squares) and 4 cpd (green triangles). The through-focus functions are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The peak of the function is shifted to the left of 0 D and represents the best focus position for each subject based on the best modulation transfer for our experimental conditions and measurements.
Figure 2
 
Through focus MTFs for our subjects for spatial frequencies 2 (black squares) and 4 cpd (green triangles). The through-focus functions are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The peak of the function is shifted to the left of 0 D and represents the best focus position for each subject based on the best modulation transfer for our experimental conditions and measurements.
Comparisons of optical and psychophysical contrast increments
Unsigned piece-wise differences in modulation transfer (ΔC vs. D) computed for all the above variables, namely 2 and 4 cpd, with and without HOAs in 0.25 D steps are shown in Figure 3 and Weber fractions (ΔC/C vs. D) in Figure 4. Psychophysical contrast thresholds and Weber fractions predicted from Bradley and Ohzawa (1986) for the corresponding spatial frequencies are also plotted for comparison (thick solid lines). Values above the psychophysical thresholds are assumed to be visible to the accommodation system. 
Figure 3
 
Absolute differences in modulation transfer from through-focus functions (Figure 2) estimate the optical contrast increments available to the system due to microfluctuations in each subject. Data for spatial frequencies of 2 cpd (black squares) and 4 cpd (green triangles). The optical contrast increments are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The asterisk (*) refers to the psychophysical contrast increment thresholds for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) from Bradley and Ohzawa (1986).
Figure 3
 
Absolute differences in modulation transfer from through-focus functions (Figure 2) estimate the optical contrast increments available to the system due to microfluctuations in each subject. Data for spatial frequencies of 2 cpd (black squares) and 4 cpd (green triangles). The optical contrast increments are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The asterisk (*) refers to the psychophysical contrast increment thresholds for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) from Bradley and Ohzawa (1986).
Figure 4
 
Weber fractions for two subjects JY (left) and KS (right) for 2 cpd (top, black squares) and 4 cpd (bottom, green triangles). Results with native aberrations included (solid symbols/solid lines) and HOAs eliminated (open symbols/dashed lines) are shown. The psychophysical Weber fractions for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) have been computed from Bradley and Ohzawa (1986). The asterisk (*) in the legend refers to this psychophysical threshold data.
Figure 4
 
Weber fractions for two subjects JY (left) and KS (right) for 2 cpd (top, black squares) and 4 cpd (bottom, green triangles). Results with native aberrations included (solid symbols/solid lines) and HOAs eliminated (open symbols/dashed lines) are shown. The psychophysical Weber fractions for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) have been computed from Bradley and Ohzawa (1986). The asterisk (*) in the legend refers to this psychophysical threshold data.
The computed increments at 2 cpd rise above threshold for defocused stimuli ranging from about 3.25 to 1.5 D away to the left (accommodation) and right (disaccommodation) of the best focus position. The lower spatial frequencies could thus be important for initiating the accommodation response at the upper defocus limit of reflex accommodation. The computed increments at the higher spatial frequency of 4 cpd rise above threshold for defocused stimuli that are about 1.5 to 0.5 D away on both sides of the best focus position as the system brings the object closer to optimal focus. Thus the data in Figures 3 and 4 point to a baton relay effect, where useful directional information to initiate the accommodative response is available at lower spatial frequencies near the upper defocus limit, and higher spatial frequencies are “handed the relay baton” as the defocus is reduced. In other words, the system is more sensitive to lower spatial frequencies at larger defocus blur levels and higher spatial frequencies at smaller defocus blur levels. In some cases, there may also be assistance from the 4 cpd spatial frequency due to the notches in the through-focus function (Atchison, Woods, & Bradley, 1998; Tahir, Parry, Pallikaris, & Murray, 2009) causing ΔC to rise above threshold at baseline defocus ranges where the 2 cpd otherwise dominates. Overall, the data are in line with previous observations (Charman & Tucker, 1978; Kotulak & Schor, 1986a; MacKenzie et al., 2010) that illustrate that the lower spatial frequencies are important at larger levels of mean defocus error, while the higher spatial frequency takes precedence in driving responses to smaller levels of defocus near the optimal focus position. 
The baton relay effect is better visualized in Figure 5, where computed data above the psychophysical threshold for 2 and 4 cpd with potentially useful directional information are highlighted using gray shading for 2 cpd and light green shading for 4 cpd. Thus, at certain baseline levels of defocus, microfluctuations are more robust at bringing about a perceptible change in contrast at low-mid spatial frequencies. Assuming a significant role for temporal microcontrast fluctuations in providing a directional signal in the 2–4 cpd range, this finding provides a rationale for an upper defocus limit and highlights that the stimulus operating range for reflex accommodation closely matches the availability of detectable contrast signals. Frequencies above 6 cpd were not studied here, and have not been thought to be useful due to the steep fall-off of higher spatial frequencies with relatively small magnitudes of defocus (MacKenzie et al., 2010). 
Figure 5
 
The baton relay effect highlighted with the data from subject KS. The computed contrast increments rise above the psychophysical contrast increment thresholds near the upper defocus limit for the 2 cpd component (black lines, shaded gray), and at smaller defocus levels for the 4 cpd component (green lines, shaded light green). The asterisk (*) represents psychophysical contrast increment threshold data from Bradley and Ohzawa (1986).
Figure 5
 
The baton relay effect highlighted with the data from subject KS. The computed contrast increments rise above the psychophysical contrast increment thresholds near the upper defocus limit for the 2 cpd component (black lines, shaded gray), and at smaller defocus levels for the 4 cpd component (green lines, shaded light green). The asterisk (*) represents psychophysical contrast increment threshold data from Bradley and Ohzawa (1986).
Role of aberrations
The expected impact of aberrations is to lower the peak of the transmitted contrast. This impact is expected to be more prominent at higher spatial frequencies. We only present analysis to investigate the role of aberrations for the 4 cpd spatial frequency, although two of our four subjects showed similar, but less prominent differences at 2 cpd. We find, in general, that HOAs measured at the 3 D baseline serve to raise the lower defocus threshold and reduce the upper defocus limit, thereby potentially shrinking the range of mean defocus error values over which the contrast changes from microfluctuations could provide a useful directional signal by 0.25–0.5 D. 
The individual HOA make-up of our subjects measured for accommodation responses to far (infinity) and to the 3 D baseline were unique, giving individual variations that we see with respect to the impact of HOAs (Figure 6). All computations of contrast increments in Figure 6 were based on the Zernikes at the 3 D baseline. The role of aberrations was investigated based on the static aberrations measured during these response states. In two of our subjects (DD and VBS) there were no differences in the outcomes when computations included or excluded any of the major aberrations, owing to the low HORMS magnitude and good optical quality. In JY and KS, HOAs had an impact on the magnitude of contrast increments where the most significant impact at reducing the contrast increments at 4 cpd came from the third-order aberrations, particularly coma. 
Figure 6
 
The impact of various combinations of LOAs and HOAs on optical contrast increments at 4 cpd. Astigmatism was removed (Defocus only), or specific HOAs (coma, or spherical aberration [SA]) or combinations of HOAs (e.g., third-order aberrations) were added to LOAs to find the HOAs that have the greatest impact at reducing the contrast increments to the same level as including all native aberrations (All HOAs). Graphs for JY and KS demonstrate the impact of the third-order aberrations, while data from DD and VBS did not show any impact of aberrations on the computed contrast increments. Comparisons were made with all aberrations included (All HOAs) or without HOAs.
Figure 6
 
The impact of various combinations of LOAs and HOAs on optical contrast increments at 4 cpd. Astigmatism was removed (Defocus only), or specific HOAs (coma, or spherical aberration [SA]) or combinations of HOAs (e.g., third-order aberrations) were added to LOAs to find the HOAs that have the greatest impact at reducing the contrast increments to the same level as including all native aberrations (All HOAs). Graphs for JY and KS demonstrate the impact of the third-order aberrations, while data from DD and VBS did not show any impact of aberrations on the computed contrast increments. Comparisons were made with all aberrations included (All HOAs) or without HOAs.
The dynamic process of accommodation is known to change spherical aberration in particular, both in sign and magnitude from far to the 3 D baseline (Atchison et al., 1995; Cheng et al., 2004; Ivanoff, 1947; Lopez-Gil et al., 2008; Plainis et al., 2005). Three out of our four subjects had positive spherical aberration while viewing the target at infinity with a M ± SD for 4-mm pupils of 0.04 ± 0.04 μm, indicating a higher value and greater variability than published ranges for normal subjects (Salmon & van de Pol, 2006). This changed to relatively negative values at the 3 D baseline as expected (0.005 ± 0.03 μm; M ± SD), although the magnitude of these changes varied among our subjects. The presence or absence of the resultant low magnitudes of static spherical aberration at the 3 D baseline therefore did not have an effect on the magnitude of optical contrast increments, due specifically to the design of our experiment. 
Discussion
The computations performed in this study provide evidence that the upper defocus limit suggested by Fincham (1951) for reflex accommodation closely matches the availability of detectable contrast signals from microfluctuations. This analysis explains the fall-off of correct initial accommodative responses at the upper defocus limit. We found that aberrations might have an impact on the magnitude of contrast changes brought about by equal dioptric magnitudes of microfluctuations, depending on the aberration make up of individuals. A range of HORMS values were seen in our four subjects that were within published reports of the normal range seen in larger population studies of aberrations (Salmon & van de Pol, 2006; Thibos et al., 2002). Larger magnitudes of HOAs seen within normal ranges in some of our subjects chiefly served to reduce the detectability of microcontrast fluctuation signals based on psychophysical contrast increment thresholds published by Bradley and Ohzawa (1986). 
The baton relay effect demonstrated in Figure 5 is in agreement with older literature that describes the theory and supporting evidence (Charman & Tucker, 1977, 1978; Kotulak & Schor, 1986b) and most recently with the analysis by MacKenzie et al. (2010). The importance of low spatial frequency information for driving accommodation responses at higher mean levels of defocus and higher spatial frequency information for driving the response at lower mean levels of defocus as the system fine-tunes its focus responses is evident from these studies and our analysis. Not only is the optical contrast increment signal robust for progressively lower spatial frequencies as defocus increases, but additionally, our study suggests that the signal is useful as it rises above estimates of perceptual contrast increment thresholds at each of the spatial frequencies. The eye is maximally sensitive to contrast increments at mean defocus levels about 0.75–1 D away from the position of best focus for 4 cpd and 2.5–3 D away for 2 cpd. Walsh and Charman (1988) and others (Hamerly & Dvorak, 1981; Watt & Morgan, 1983) have similarly concluded that sensitivity to small increments is maximal when the target is slightly blurred. Additionally, as the mean defocus level increases, the best sensitivity shifts from high spatial frequencies to progressively lower spatial frequencies. We would therefore expect the baton relay to continue until the overall contrast is too low and increments are too small or invariant with focus changes, resulting in a less useful contrast signal from microfluctuations. 
We did not analyze spatial frequencies higher than 4 cpd as our goal was to study how a directional signal is derived from microcontrast fluctuations occurring with defocus levels near the upper defocus limit for accommodation and disaccommodation stimuli, and not for the maintenance of accommodative responses near optimal focus. Also, as described by MacKenzie et al. (2010), contrast signals from microfluctuations from high spatial frequencies are much lower. The demand on accommodation could vary vastly with natural world scenes and tasks of widely varying spatial content due to the 1/f fall-off of contrast with spatial frequency (e.g., fine tasks vs. object recognition), affecting the responses we have predicted. Based on the empirical result showing the upper defocus limit of ∼ 2 D estimated by Fincham (1951), and an extension of computations by MacKenzie et al. (2010) for higher levels of mean defocus, we also conclude that spatial frequencies lower than 2 cpd perhaps do not play a significant role. This is because with increasing magnitudes of defocus, we expect lower optical contrast increments at these frequencies. 
The role of aberrations was specific to individual subjects' aberration make-up, and specifically the static aberration make-up measured at the 3 D baseline. By design, spherical aberration did not have an impact on the magnitude of optical contrast increments, as all computations were based on the Zernikes at the 3 D baseline, where we expected and found low magnitudes of static spherical aberration as reported previously (Atchison et al., 1995; Cheng et al., 2004; Ivanoff, 1947; Lopez-Gil et al., 2008; Plainis et al., 2005). We did not perform computations to investigate the effects of large amounts of positive or negative spherical aberrations that are known to reduce or enhance the accommodative response, respectively (Gambra et al., 2009). We have therefore minimized any impact that spherical aberration might have had on providing an odd-error cue to accommodation (Campbell & Westheimer, 1959; Fernandez & Artal, 2005; Fincham, 1951; Thibos, Bradley, Liu, & Lopez-Gil, 2013; Wilson, Decker, & Roorda, 2002; Wu & Jiang, 2011). However, the impact of third-order aberrations, particularly coma is notable with our analysis. When they were removed, the optical contrast increments were elevated to nearly the same levels as those without HOAs (see Figure 6, JY and KS), suggesting that they had the largest impact. Gambra et al. (2009) studied the effect of correcting all aberrations using an Adaptive Optics (AO) system and found that it improved responses to accommodative stimuli. They also showed that third-order aberrations, namely coma and trefoil, could be detrimental to the accommodative response. Additionally, Lopez-Gil et al. (2007) reported that while asymmetric aberrations may not provide a directional cue per se, when present in large enough magnitudes to affect the contrast, they may deter performance by affecting the gain and masking other useful directional cues. These findings are in line with the predictions from our study that removing the influence of HOAs would improve the strength of the contrast signal from the accommodative microfluctuations. They are also consistent with the detrimental impact of third-order aberrations on microcontrast fluctuation signals. Precision of initial direction is likely to be more problematic in normal subjects showing higher magnitudes of third-order aberrations, and keratoconic subjects with abnormal magnitudes of coma. Furthermore, natural world scenes would vary in contrast at different orientations and accommodation responses would suffer or be spared due to astigmatism or coma (Tahir et al., 2009). However, this is not reflected in our study as we radially averaged MTFs and did not analyze this. 
It is important to consider how changes in some of the variables in the experiment would lead to altered results and interpretations. As noted earlier, the magnitude of fluctuations of accommodation varies between subjects, and it is well known that it varies with the accommodative demand (Arnulf & Dupuy, 1960; Charman & Heron, 1988; Kotulak & Schor, 1986b; Leahy, Leroux, Dainty, & Diaz-Santana, 2010; Miege & Denieul, 1988), and the magnitude of aberrations (Gambra et al., 2009). The dioptric estimate of the fluctuation magnitude based on Zernike coefficients would vary depending on the inclusion of certain higher order terms and the formula used, and we chose a constant 0.25 D as a conservative estimate at the 3 D baseline for simplicity and to evaluate the effect of the same magnitude of oscillations on the contrast signals based on individual subjects' aberrations. Computed contrast increments for larger magnitudes of accommodative fluctuations show insignificant differences between the contrast increments with HOAs and without HOAs at a number of mean defocus levels, due to the reduced sampling resolution (data not shown). Increased fluctuations may thus help compensate for the reduced contrast increments that are otherwise produced with HOAs. We also used natural pupils, which were stopped using a 4-mm pupil imaged at the entrance pupil of the eye. We did not consider accommodative miosis with accommodation and our calculations of optical contrast increments are for a 4-mm pupil. It may make a significant difference if the natural pupil miosis caused the diameter to be less than the size of our artificial pupil stop for larger values of accommodation and disaccommodation simulated in our analysis (dioptric range of 0 to 6 D), unlike the small focus errors evaluated by MacKenzie et al. (2010). The smaller pupil diameter would increase the depth of focus and reduce changes in contrast produced by microfluctuations. 
Furthermore the accuracy of accommodation is likely to be different in the monocular condition that was necessary in this study compared to the binocular response, based on the type of heterophoria (Goss & Rainey, 1999; Hasebe, Nonaka, & Ohtsuki, 2005; Momeni-Moghaddam, Goss, & Sobhani, 2014; Schor, 1999; Sreenivasan, Irving, & Bobier, 2012; Tassinari, 2002). In addition, our estimates of the lag of accommodation are based on through-focus modulation transfer at a specific pupil size, a specific wavelength, and not a metric of retinal image quality. In general, Zernike defocus tends to underestimate the accommodative response (Tarrant, Roorda, & Wildsoet, 2010), probably resulting in the larger lags than generally reported in the literature. Also worth mentioning is the fact that the lag and spherical aberration would create opposite biases depending on the accommodative state. The lag is greater at near than at far, making the blur from accommodative stimuli larger (and perhaps less detectable) than disaccommodative stimuli. Positive spherical aberration, on the other hand, is greater at far than at near as described before. Thus, spherical aberration would be more positive (or less negative) at the far end of fluctuation than at the near end, therefore creating a bias that would reduce the magnitude of defocus blur from accommodative stimuli (making them more detectable) and vice versa for disaccommodative stimuli. 
Finally, due to intersubject and methodological variations, results from studies of HOAs in determining the initial direction, the gain, and the dynamics of accommodation arrive at varied conclusions (Chen, Kruger, Hofer, Singer, & Williams, 2006; Chin, Hampson, & E. Mallen, 2009; Chin, Hampson, & E. A. Mallen, 2009; Fernandez & Artal, 2005; Hampson, Chin, & Mallen, 2010; Wilson et al., 2002). These investigations have focused on the presence of certain or all HOAs, the time frame of correction of HOAs during the dynamic accommodation process on the initial direction, dynamics, and gain of the accommodative response. Our efforts were focused purely on the directional cue from contrast changes predicted from temporal accommodative microfluctuations and how HOAs within physiological ranges might affect this. Our study suggests that HOAs could potentially reduce the contrast signal due to accommodative microfluctuations. We speculate that other beneficial signals offered by the presence of HOAs could then be used to determine the direction of reflex accommodation. Thus, the redundancy of a number of cues such as chromatic aberrations, monochromatic HOAs, and temporal cues such as microfluctuations may be useful both individually, and in combination in different individuals depending on their specific optical and neural profile. 
Acknowlegements
We acknowledge grant support from NIH R01 EY01678 to CMS and NEI K12 EY017269 via the Berkeley Clinical Scientist Development Program (BCSDP) to SM. We thank all our subjects, Ms. Allison Mina Choi for help with some of the data collection, and Dr. Zhi-Lei Zhang for technical/programming assistance. The authors are grateful for custom written programs from Dr. Austin Roorda upon which some of our programs were built. This work has been presented in part at the Association for Research in Vision and Ophthalmology (ARVO) 2013 conference, Seattle, WA, USA. 
Commercial relationships: none. 
Corresponding author: Sangeetha Metlapally. 
Email: smetlapally@berkeley.edu. 
Address: School of Optometry, University of California, Berkeley, CA, USA. 
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Figure 1
 
Schematic representations of the stimulus and recording set-up and stimulus time sequence used in the study. (A) Plan view of the apparatus used in the experiments, and the inset illustrates the broadband Maltese spoke stimulus as displayed on the monitor; (B) time sequence of the step stimulus, showing the change of the baseline from far (0 D) to 3 D at 1 s and sustenance of the stimulus/response at that level for 4 s. The mean response level is indicated by the dashed lines, showing a lag of accommodation. Accommodative microfluctuations about the mean are schematized as a simple sinusoidal fluctuation.
Figure 1
 
Schematic representations of the stimulus and recording set-up and stimulus time sequence used in the study. (A) Plan view of the apparatus used in the experiments, and the inset illustrates the broadband Maltese spoke stimulus as displayed on the monitor; (B) time sequence of the step stimulus, showing the change of the baseline from far (0 D) to 3 D at 1 s and sustenance of the stimulus/response at that level for 4 s. The mean response level is indicated by the dashed lines, showing a lag of accommodation. Accommodative microfluctuations about the mean are schematized as a simple sinusoidal fluctuation.
Figure 2
 
Through focus MTFs for our subjects for spatial frequencies 2 (black squares) and 4 cpd (green triangles). The through-focus functions are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The peak of the function is shifted to the left of 0 D and represents the best focus position for each subject based on the best modulation transfer for our experimental conditions and measurements.
Figure 2
 
Through focus MTFs for our subjects for spatial frequencies 2 (black squares) and 4 cpd (green triangles). The through-focus functions are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The peak of the function is shifted to the left of 0 D and represents the best focus position for each subject based on the best modulation transfer for our experimental conditions and measurements.
Figure 3
 
Absolute differences in modulation transfer from through-focus functions (Figure 2) estimate the optical contrast increments available to the system due to microfluctuations in each subject. Data for spatial frequencies of 2 cpd (black squares) and 4 cpd (green triangles). The optical contrast increments are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The asterisk (*) refers to the psychophysical contrast increment thresholds for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) from Bradley and Ohzawa (1986).
Figure 3
 
Absolute differences in modulation transfer from through-focus functions (Figure 2) estimate the optical contrast increments available to the system due to microfluctuations in each subject. Data for spatial frequencies of 2 cpd (black squares) and 4 cpd (green triangles). The optical contrast increments are plotted using subjects' native aberrations (solid symbols/solid lines, with HOAs) or with the HOAs removed (open symbols/dashed lines, without HOAs). The asterisk (*) refers to the psychophysical contrast increment thresholds for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) from Bradley and Ohzawa (1986).
Figure 4
 
Weber fractions for two subjects JY (left) and KS (right) for 2 cpd (top, black squares) and 4 cpd (bottom, green triangles). Results with native aberrations included (solid symbols/solid lines) and HOAs eliminated (open symbols/dashed lines) are shown. The psychophysical Weber fractions for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) have been computed from Bradley and Ohzawa (1986). The asterisk (*) in the legend refers to this psychophysical threshold data.
Figure 4
 
Weber fractions for two subjects JY (left) and KS (right) for 2 cpd (top, black squares) and 4 cpd (bottom, green triangles). Results with native aberrations included (solid symbols/solid lines) and HOAs eliminated (open symbols/dashed lines) are shown. The psychophysical Weber fractions for 2 cpd (thick solid black line) and 4 cpd (thick solid green line) have been computed from Bradley and Ohzawa (1986). The asterisk (*) in the legend refers to this psychophysical threshold data.
Figure 5
 
The baton relay effect highlighted with the data from subject KS. The computed contrast increments rise above the psychophysical contrast increment thresholds near the upper defocus limit for the 2 cpd component (black lines, shaded gray), and at smaller defocus levels for the 4 cpd component (green lines, shaded light green). The asterisk (*) represents psychophysical contrast increment threshold data from Bradley and Ohzawa (1986).
Figure 5
 
The baton relay effect highlighted with the data from subject KS. The computed contrast increments rise above the psychophysical contrast increment thresholds near the upper defocus limit for the 2 cpd component (black lines, shaded gray), and at smaller defocus levels for the 4 cpd component (green lines, shaded light green). The asterisk (*) represents psychophysical contrast increment threshold data from Bradley and Ohzawa (1986).
Figure 6
 
The impact of various combinations of LOAs and HOAs on optical contrast increments at 4 cpd. Astigmatism was removed (Defocus only), or specific HOAs (coma, or spherical aberration [SA]) or combinations of HOAs (e.g., third-order aberrations) were added to LOAs to find the HOAs that have the greatest impact at reducing the contrast increments to the same level as including all native aberrations (All HOAs). Graphs for JY and KS demonstrate the impact of the third-order aberrations, while data from DD and VBS did not show any impact of aberrations on the computed contrast increments. Comparisons were made with all aberrations included (All HOAs) or without HOAs.
Figure 6
 
The impact of various combinations of LOAs and HOAs on optical contrast increments at 4 cpd. Astigmatism was removed (Defocus only), or specific HOAs (coma, or spherical aberration [SA]) or combinations of HOAs (e.g., third-order aberrations) were added to LOAs to find the HOAs that have the greatest impact at reducing the contrast increments to the same level as including all native aberrations (All HOAs). Graphs for JY and KS demonstrate the impact of the third-order aberrations, while data from DD and VBS did not show any impact of aberrations on the computed contrast increments. Comparisons were made with all aberrations included (All HOAs) or without HOAs.
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