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Accommodation and the Stiles±Crawfordeffect: theory and a case study
P. B. Kruger1, N. LoÂpez-Gil2 and L. R. Stark1
1Schnurmacher Institute for Vision Research, State College of Optometry, State University of NewYork, 33 West 42nd Street, New York, NY 10036, USA and 2Departamento de FõÂsica, Universidadde Murcia, 30071 Murcia, Spain
Summary
Purpose: Fincham (The accommodation re¯ex and its stimulus. Br. J. Ophthalmol. 35, 381±393)was the ®rst to suggest that the Stiles±Crawford effect (Type I) might provide a stimulus foraccommodation, but the possibility has not been investigated experimentally. The present paperoutlines a theoretical basis for such a mechanism, and includes a case study on a subject with anasally decentred Stiles±Crawford (S±C) function.
Methods: Accommodation to a monochromatic sine grating was monitored continuously with thenatural S±C function intact, or with apodising ®lters imaged in the subject's pupil to neutralise,reverse or double the natural S±C function.
Results: Mean accommodative gain was not reduced signi®cantly when the normal S±C functionwas either neutralised or reversed.
Conclusions: For the present subject, the average S±C effect does not mediate the accommo-dation response to defocus, but more subjects should be examined. Other methods by whichdirectionally sensitive cone receptors could detect light vergence are discussed. q 2001 TheCollege of Optometrists. Published by Elsevier Science Ltd. All rights reserved.
Introduction
The human eye has the remarkable ability to change its
power to focus objects at different distances clearly on the
retina. An early view of the stimulus for re¯ex accommo-
dation was that blur from inaccurate focus provides a non-
directional (even-error) stimulus. Feedback from changes
in blur is an essential part of this type of focusing system
because blur increases for defocus both behind and in
front of the retina (Carter, 1962; Troelstra et al., 1964;
Stark et al., 1965). In an early experiment, Fincham
(1951) noted that some subjects accommodated in the
correct direction without trial-and-error changes in
focus, and he suggested that the eye responds to the
vergence of light by using colour fringes from the long-
itudinal chromatic aberration of the eye or by using
changes in brightness across blur circles that result from
the Stiles±Crawford (1933) effect (Type I). Light
vergence refers to the curvature of radiating wavefronts
of light, measured in dioptres, and is the reciprocal of the
distance in metres from the measurement plane to the
target. Vergence also refers to the convergence or diver-
gence of bundles of rays, where the rays are perpendicular
to their corresponding wavefronts. The idea that accom-
modation responds to the vergence of light was dismissed
by Heath (1956), and over several decades it became
routine to model accommodation as a contrast-maximis-
ing negative-feedback system in which blur from defocus
provides a non-directional (even-error) stimulus (Carter,
1962; Troelstra et al., 1964; Stark et al., 1965). Although
there is evidence for such a non-directional stimulus
(Phillips and Stark, 1977), current research weighs against
the postulate (Stark and Takahashi, 1965; Ciuffreda,
1991) that even-error blur provides the sole true stimulus
to accommodation.
Longitudinal chromatic aberration of the eye provides a
powerful directional stimulus for accommodation (Finc-
ham, 1951; Smithline, 1974; Kruger and Pola, 1986;
339
Ophthal. Physiol. Opt. Vol. 21, No. 5, pp. 339±351, 2001q 2001 The College of Optometrists. Published by Elsevier Science Ltd
All rights reserved. Printed in Great Britain0275-5408/01/$20.00
www.elsevier.com/locate/ophopt
PII: S0275-5408(01)00004-7
Received: 20 October 2000
Revised form: 6 December 2000
Accepted: 26 January 2001
Correspondence and reprint requests to: P. B. Kruger.
E-mail address: [email protected] (P. B. Kruger).
Flitcroft and Judge, 1988; Flitcroft, 1990; Kruger et al.,
1993, 1995; Aggarwala et al., 1995a,b; Kotulak et al.,
1995; Lee et al., 1999). However, there must be another
directional stimulus besides longitudinal chromatic aberra-
tion because subjects continue to accommodate in the
absence of chromatic aberration and without feedback
from defocus blur (Kruger et al., 1997). There is also anec-
dotal evidence that some subjects who accommodate well in
the absence of chromatic aberration have highly decentred
Stiles±Crawford functions (Kruger et al., 1997). This lends
some support to Fincham's (1951) suggestion that the
Stiles±Crawford effect might mediate an accommodation
response to light vergence.
Stiles±Crawford effect
The Stiles±Crawford effect (Type I) results from the
®bre-optic or waveguide nature of retinal rods and cones
(Stiles and Crawford, 1933; Toraldo di Francia, 1949;
Enoch, 1960, 1961a,b, 1972; Snyder, 1966; Enoch and
Tobey, 1981; Enoch and Lakshminarayanan, 1991). Cones
gather light over relatively small acceptance angles and
light is most likely to be captured by the receptor if the
light is incident along the axis of the receptor (Bailey and
Heath, 1978; Enoch et al., 1978; Enoch and Lakshminar-
ayanan, 1991). In most eyes, the axes of retinal cone recep-
tors at all locations across the retina are aligned
approximately with the centre of the pupil (Laties and
Enoch, 1971; Enoch and Hope, 1972a,b; Enoch and Laksh-
minarayanan, 1991). As a result, light entering the eye near
the pupil centre is more effective for stimulating vision (and
thus appears brighter) than light entering close to the edge of
the pupil.
The Stiles±Crawford (S±C) effect is not homogeneous
across the retina. Both the location and the directionality
(or `peakedness') of the S±C function vary with retinal
location in several distinct patterns. The directionality of
the S±C effect is reduced at the centre of the fovea
(Westheimer, 1967), increases parafoveally (Vos and
Huigen, 1962), reaches a maximum between 0 and 28from the foveal centre (Enoch and Hope, 1973), and then
decreases toward the periphery (Enoch and Bedell, 1981).
The peak of the S±C function is usually located approxi-
mately 0.5 mm to the nasal side of the pupil centre, but the
horizontal decentration is sometimes much larger. Vertical
decentration of the S±C peak is also common (Stiles and
Crawford, 1933; Enoch, 1957; Applegate and Lakshminar-
ayanan, 1993; Gorrand and Delori, 1995). The sloping walls
of the foveal pit refract light away from the centre of the
fovea, and this alters the location of the peak of the S±C
function in a 28 annulus surrounding the foveal centre
(Williams, 1980). Within this annulus, the S±C function
peak for cones on the nasal and temporal sides of the foveal
pit tends towards the nasal and temporal sides of the pupil
respectively, and the peak for cones above and below the
fovea tends towards the superior and inferior parts of the
pupil respectively. Additional patterns are observed in
myopic eyes. In some myopic eyes, strain is exerted on
the retina originating at the temporal margin of the optic
nerve head. This strain produces systematic changes in the
orientation of foveal receptors across the horizontal meri-
dian of the nasal retina (Choi and Garner, 2000; Enoch et
al., 2000). Nasal decentration of the S±C peak is associated
with moderate and high myopia (Westheimer, 1968; Choi
and Garner, 2000; Enoch et al., 2000).
Accommodation produces small changes in the location
of the S±C function peak, most likely through a mechanism
of choroidal and retinal stretching (Enoch, 1973; Hollins,
1974). Accommodation tends to produce nasal shifts in the
location of the S±C function peak (Blank et al., 1975;
Enoch, 1975), which may be as large as 1 mm for marked
(9 D) accommodation (Enoch, 1975; Provine and Enoch,
1975). These small dynamic changes aside, the S±C func-
tion remains stable over the course of years in many indi-
viduals (Enoch and Lakshminarayanan, 1991; Rynders et
al., 1995).
The S±C function also differs between retinal rod and
cone receptors, with rods showing less directionality than
cones (Enoch and Lakshminarayanan, 1991). The present
study focuses on cone receptors and photopic vision because
rod receptors probably provide little or no contribution to
accommodation (Campbell, 1954; Johnson, 1976). There
are also differences in the S±C function between cone
classes: directionality is greatest in the S-cones, followed
by the M- and L-cones (Enoch and Stiles, 1961). Direction-
ality also varies with wavelength (Enoch and Stiles, 1961;
Enoch and Lakshminarayanan, 1991).
Fincham's model
Fincham's (1951) model of how the Stiles±Crawford
function could provide a stimulus to accommodation is illu-
strated in Figure 1(a)±(c). This model uses an indirect
method to determine light vergence using large numbers
of retinal cones. Light rays are shown coming to focus
behind and in front of the retina (hyperopic and myopic
defocus), and the out-of-focus image is positioned to the
right side of the foveal centre. Fincham assumed that the
axes of cones on either side of the foveal centre are aligned
with the centre of the globe (dashed lines). As a conse-
quence, cones on the right side of the fovea capture light
primarily from the left side of the pupil, and vice versa for
cones on the opposite side of the fovea (dashed lines). In the
case of hyperopic defocus, rays that form the left side of
the blurred retinal image (point spread-function) come from
the left side of the pupil, while rays that form the right side
of the point spread-function come from the right side of the
pupil. Conversely, in the case of myopic defocus, the limit-
ing rays cross before reaching the retina, and rays that form
the left side of the point spread-function come from the right
340 Ophthal. Physiol. Opt. 2001 21: No 5
side of the pupil. The point spread-functions (blur circles) on
the retina are shown schematically as Gaussian functions
which are similar [Figure 1(b)] for both conditions of defo-
cus. In an aberration-free optical system neither the size of
the spread-function (blur circle diameter) nor the intensity
distribution across the spread-function distinguish over-
from under-accommodation. Although the blur spread-func-
tions on the retina are symmetrical [Figure 1(b)], the peak of
the effective intensity distribution, after weighting for the S±
C effect, is skewed to the left in the case of under-accom-
modation, and to the right in the case of over-accommoda-
tion [Figure 1(c)]. In addition, the direction of asymmetry
changes for ®xation to the left or right side of the foveal
centre. Fincham suggested that the direction of asymmetry
speci®es focus behind or in front of the retina, and that small
eye movements (6±10 arc min) are used to move blurred
images from one side of the fovea to the other side in order
to use these effects to determine focus. Fincham's method
relies on the assumption that the axes of foveal cones are
aligned with the centre of the globe; rather than aligned
approximately with the centre of the pupil, as is now
generally accepted (Laties and Enoch, 1971; Enoch and
Hope, 1972a,b; Enoch and Lakshminarayanan, 1991).
Although Fincham's (1951) model is ¯awed due to this
assumption, it nevertheless provides a starting point for
further investigation.
Decentred Stiles±Crawford functions
It is important to recognise that it is common for the
peak of the S±C function to be decentred nasally in the
pupil, and that in some eyes the amount of decentration is
substantial (Westheimer, 1968; Applegate and Lakshmi-
narayanan, 1993). If the blur spread-function on the retina
is weighted by the activity of a large number of decentred
cones, then the asymmetry of the S±C function might help
distinguish under- from over-accommodation. The method
is illustrated in Figure 1(d)±(f) for an on-axis optical
system with all the cone receptors tilted toward the
nasal side of the pupil (N, dashed lines). To simplify the
description, angle alpha (between the optic and visual
axes) is not included. Light rays are shown travelling
from the edges of the pupil to focus behind or in front
of the retina. Rays from the edges of the pupil cross before
reaching the retina in the case of over-accommodation
(myopic defocus), and rays reach the retina uncrossed in
the case of under-accommodation (hyperopic defocus). In
the case of under-accommodation (hyperopic defocus),
light incident from the nasal side (N) of the pupil forms
the nasal side of the spread-function, and illuminates the
receptors along their axes. The temporal side of the point
spread-function is illuminated by light from the temporal
side of the pupil (T ) which reaches the retina at an angle
to the axes of the receptors. Conversely, in over-accom-
modation (myopic defocus) light from the nasal side of the
pupil illuminates the receptors on the temporal side of the
point spread-function along their axes. While the images
on the retina are symmetrical [Figure 1(e)], after weight-
ing for the Stiles±Crawford effect, the peak of the point
spread-function is skewed to the left in the case of under-
accommodation, and to the right in the case of over-
accommodation [Figure 1(f)]. Unlike Fincham's (1951)
method, the direction of asymmetry does not change for
®xation to the left or right side of the foveal centre
because all the foveal receptors are aligned with the
nasal edge of the pupil. It should be clear that in this
method the average S±C effect cannot provide a direc-
tional signal if all the cones are aligned precisely with
the centre of the pupil. In such a case, a symmetrical
point spread-function on the retina would remain symme-
trical after weighting for the S±C effect, both for under-
accommodation and over-accommodation.
There have been no previous investigations to determine
Accommodation and the Stiles±Crawford effect: theory and a case study: P. B. Kruger et al. 341
Figure 1. (a)±(c) Fincham's (1951) model for determiningtarget vergence. (d±f) Proposed method for determiningtarget vergence using a decentred Stiles±Crawford effect.
whether the S±C effect mediates the accommodative
response to light vergence, and the ®nding that subjects
accommodate in monochromatic light in the absence of
blur feedback (Kruger et al., 1997) provides incentive to
study the issue. In the present experiment we examined
the hypothesis that a decentred S±C function, averaged
over the central foveal region (4.128), mediates accom-
modation to changing light vergence. The two-dimen-
sional Stiles±Crawford function of a subject with a
markedly decentred S±C peak was measured using a
psychophysical technique, and specially designed apodis-
ing ®lters were imaged in the subject's eye to alter the
S±C effect while accommodation was monitored continu-
ously. Mean accommodative gain was not reduced signif-
icantly when the normal S±C function was neutralised or
reversed.
Methods
Subject
The subject was one of the investigators (NL) who volun-
teered to participate because detailed two-dimensional
measurements of his Stiles±Crawford function had been
made previously as part of a doctoral dissertation at Indiana
University (Rynders et al., 1997). Unfortunately, other
subjects from the original study were not available for addi-
tional testing, and the original instrumentation was no
longer available. Nevertheless, a case study on a subject
with a decentred S±C function would provide a starting
point for further investigation. The 30-year old subject
had no history of ocular disease, injury, strabismus or
amblyopia. He had worn glasses from 18 years of age,
and contact lenses for 5 months at 21 years of age. He
was in good general health and not taking any medications.
Subjective refraction and visual acuities were R.E. 24.50/
20.25 £ 50 (6/4.821) and L.E. 24.75 (6/4.8). Subjective
spectacle amplitudes of accommodation were normal
(R.E. 11.3 D; L.E. 9.5 D), and colour vision was normal
by Nagel Anomaloscope. All measurements of the S±C
effect and accommodation were made on the subject's
right eye. The subject gave informed consent, the accom-
modation study was approved by the Institutional Review
Board of the SUNY State College of Optometry, and the
procedures followed the tenets of the Declaration of
Helsinki.
Measuring and altering the Stiles±Crawford effect
Detailed psychophysical measures of the subject's
Stiles±Crawford function were made as part of an earlier
investigation that introduced a new method to measure and
neutralise the Stiles±Crawford effect (Rynders et al., 1997).
The apparatus and methods have been described in detail
(Rynders et al., 1995, 1997). In brief, the subject was posi-
tioned on a bite-plate and aligned on his achromatic axis
(Thibos et al., 1990). He then viewed a bipartite ®eld (4.128diameter) illuminated by monochromatic light (633 nm).
The subject's pupil was dilated by two drops of Paremyd
(hydroxyamphetamine hydrobromide 1% and tropicamide
0.25%; Allergan), separated by 5 min between drops. The
brightnesses of the two halves of the bipartite ®eld were
matched by the subject using an adjustable neutral density
wedge to alter the brightness of one side of the ®eld. The
light source for one half of the bipartite ®eld was a 1-mm
diameter pinhole illuminated by diffuse tungsten-halogen
light, and imaged in the subject's natural pupil as a 1-mm
arti®cial pupil. The other half of the bipartite ®eld was
illuminated by a computer-controlled video display viewed
through the subject's entire dilated pupil. The pinhole was
positioned randomly with an accuracy of 0.1 mm at one of
49 points in a 7 £ 7 mm square matrix that covered the
subject's dilated pupil. Three measures were made at each
of the 49 measurement positions within the pupil, except for
the centre point for which nine measurements were made.
The ®lter density for each pupil position was normalised,
and a least squares regression procedure was used to ®t the
form
SCF x; yÿ � � 10rx x2xc� �21ry y2yc� �2 �1�
to the data for points within a circle of radius r speci®ed by the
inequality x2 1 y2 # r2. Without the circle, SCF(x, y)� 0. In
Eq. (1), x represents the distance along the horizontal axis
(mm) where positive is nasal and negative is temporal; y repre-
sents the distance along the vertical axis (mm) where positive
is superior and negative is inferior; r x and r y are the rho values
in the x- and y-directions representing the directionality of the
S±C function; xc and yc are the horizontal and vertical positions
of the peak of the S±C function; r is the pupil radius (mm). The
zero point of the coordinate system is the achromatic axis of
the eye (Thibos et al., 1990). The values that describe the
Stiles±Crawford function [Eq. (1)] for the subject's right eye
are r x�20.052, r y�20.034, xc�11.24 and yc�20.93.
A three-dimensional plot of the S±C function is shown in
Figure 2(a). The decentred S±C function of the subject acts
to attenuate the light that has entered the eye from the
temporal and superior portions of the pupil. The peak of
the function is decentred 1.24 mm nasally and 0.93 mm
inferiorly from the achromatic axis of the eye. The decen-
tration of the peak is much larger than usual; most indivi-
duals have approximately 0.5 mm nasal decentration
(Applegate and Lakshminarayanan, 1993; Gorrand and
Delori, 1995).
A ®lter was designed that would alter the light entering
each point in the pupil to neutralise the subject's decentred
S±C function. A neutralising function (nSCF) is desired
such that
nSCF x; yÿ �
´SCF x; yÿ � � K
342 Ophthal. Physiol. Opt. 2001 21: No 5
within the circle of radius r. That is, the combination of the
neutralising ®lter and the eye's S±C function yields a ¯at
and uniform intensity transmittance for the pupil function
having the value of K. The neutralising function is then
given by
nSCF x; yÿ � � K=SCF x; y
ÿ � �2�within the circle of radius r. Here K is a constant of normal-
isation that corresponds to the minimum of the S±C func-
tion inside the circle of radius r [Eq. (1)], so that the constant
depends on the pupil diameter. For the subject's right eye
and a pupil radius of 2.5 mm, K� 0.1665. A three-dimen-
sional representation of the ®lter is shown in Figure 2(b).
The neutralising ®lter provides maximum transmission on
the temporal side of the pupil to compensate for the
subject's nasally decentred function.
Next, a ®lter was created to reverse the S±C function so
that the peak of the function for the subject's right eye
would be temporal and superior rather than its natural posi-
tion of nasal and inferior. The eye's natural S±C function is
given by Eq. (1), and the desired function after the reversing
®lter (rSCF) has been put in place is given by
SCF 0 x; yÿ � � 10rx x1xc� �21ry y1yc� �2
within the circle of radius r. Note that SCF 0 is obtained from
SCF simply by changing the signs of xc and yc. Now a
reversing ®lter (rSCF) is desired such that
SCF x; yÿ �
rSCF x; yÿ � � K 0´SCF 0 x; y
ÿ �where K 0 is a constant of normalisation, and so
rSCF x; yÿ � � K 0´SCF 0 x; y
ÿ �=SCF x; y
ÿ � �3�For the subject's right eye, K 0 � 0.19135, within the circle
of radius r� 2.5 mm. A three-dimensional representation of
the reversing ®lter is shown in Figure 2(c).
Finally, a ®lter was made to `double' the subject's S±C
function. The ®lter was made from the subject's normal S±
C function (SCF) shown in Figure 2(a). When imaged in the
subject's pupil, the doubling ®lter accentuates the subject's
normally decentred function and severely attenuates light
that enters the eye from the temporal side of the pupil.
The above functions were used to create apodising ®lters
(Metcalf, 1965) for imaging within the subject's natural
pupil. The use of such ®lters is a standard way to examine
the in¯uence of the S±C effect on visual performance
(Metcalf, 1965; Atchison et al., 1998; Zhang et al., 1999).
The apodising ®lters were grey-scale images of the func-
tions shown in Figure 2. Images were recorded on photo-
graphic transparency ®lm (Kodak Ektachrome ASA 100)
using a Montage ®lm-recorder. The gamma function of
the ®lm, ®lm-recorder and processing method was cali-
brated by generating 20 small test patches on the same
slide with nominal transmission values between 0 and
100%, in 5% increments. The actual transmission of each
patch was then measured in 633-nm monochromatic light,
and the resulting gamma function was used to produce the
®nal set of ®lters. Each ®lter was made four times larger
than the actual pupil size to allow for mini®cation imposed
Accommodation and the Stiles±Crawford effect: theory and a case study: P. B. Kruger et al. 343
Figure 2. (a) Normal S±C function (SCF) of the subject show-ing nasal (x� 1.24 mm) and inferior (y�20.93 mm) decen-tration of the peak; (b) ®lter (nSCF) designed to neutralise thesubject's S±C function; (c) ®lter (rSCF) designed to reversethe nasal and inferior decentration of the S±C function toprovide a temporal and superior decentration. In each graphthe z-axis represents relative luminous ef®ciency or transmit-tance, and the x- and y-axes represent horizontal and verticalpupil radius in millimetres. Maximum pupil diameter is 5 mm.
by the optical system between the ®lter plane and the
subject's pupil plane (Kruger et al., 1993).
Neutral density ®lters were used with each of the above
apodising ®lters to equate approximately the retinal illumi-
nances between conditions. Retinal illuminances (calcu-
lated to include the subject's normal S±C function) were
50 trolands in the normal condition and 91, 125 and 46
trolands in the neutralised, reversed and doubled conditions
respectively with the 3-mm arti®cial pupil in place (see
Section 2.4). (Retinal illuminances with the 5-mm pupil
varied when the natural pupil became smaller than 5 mm.)
Accommodation apparatus
Accommodation was monitored continuously (100 Hz)
by an infra-red optometer (Kruger, 1979) while the subject
viewed various targets in a Badal stimulus system (Kruger
et al., 1993). The limiting ®eld stop of the optometer was a
blurred circular aperture (15.2 D beyond the far point) that
subtended 11.58 at the eye.
In the main trials the target was a back-illuminated verti-
cal sine-wave grating (1 cycle per mm; 80% contrast; Sine
Patterns, Pen®eld, NY, USA) seen in non-Maxwellian view.
After spectacle magni®cation, this grating provided a spatial
frequency of 2.08 cycles per degree. Light from the metal
halide lamp of a high-luminance video projector (In Focus
Systems LitePro 620, Wilsonville, OR, USA) illuminated an
opal diffusing screen, and the sine-grating was positioned
¯at against the opposite side of the diffusing screen. The
video-projector was used simply as a source of diffused light
for illuminating the grating target. Light from the target was
®ltered by a 542-nm interference ®lter (12 nm bandwidth)
and an image of the monochromatic grating was viewed by
the subject in non-Maxwellian view in the Badal stimulus
system. Unlike Maxwellian view, where any irregularity in
the light source is imaged in the pupil plane, the present
method ensured a uniformly illuminated pupil (Westheimer,
1966).
Procedures
The subject was positioned in front of the instrument on a
dental bite plate and forehead rest while eye position was
monitored continuously by one of the investigators with an
infra-red camera and video display. Trial lenses were placed
in front of the right eye to correct the refractive error of the
eye (24.50/20.25 £ 50) and the left eye was patched. The
subject's eye was viewed on a video display and the ®rst
Purkinje image was used as a reference point for aligning
the eye. In a previous experiment using our apparatus, we
found that this method places the achromatic axis of the eye
(Thibos et al., 1990) very close to the optical axis of the
Badal system (n� 5, 20.07 to 10.43 mm). Thus although
the subject was not speci®cally aligned on the achromatic
axis, it is likely that the ®lters were aligned with a point
close to the achromatic axis of the eye.
The output of the infra-red optometer (Volts) was cali-
brated against the accommodative response (D) using a
method of bichromatic stigmatoscopy (Lee et al., 1999).
Following calibration, the subject's response to dioptric
blur was examined. In these trials the subject viewed a
Maltese cross (Kruger and Pola, 1986) seen in Maxwellian
view moving sinusoidally between 1 and 3 D at a temporal
frequency of 0.195 Hz under two conditions: (1) in white
light with normal longitudinal chromatic aberration intact;
and (2) in monochromatic light (550 nm, 10 nm bandwidth)
to eliminate longitudinal chromatic aberration. Each trial
lasted 40.96 s, and there were three trials of each condition
presented in random order. The trials were run under normal
closed-loop conditions using a 3-mm arti®cial pupil imaged
in the natural pupil plane by the optical system (Kruger et
al., 1993).
For the main experimental trials the target was a mono-
chromatic (542 nm, 12 nm bandwidth) green±black vertical
sine-wave grating (2.08 cycle per degree, 80% contrast)
seen in non-Maxwellian view, that moved sinusoidally
between 1 and 3 D at 0.195 Hz during trials lasting
40.96 s. The target was viewed through a 3-mm arti®cial
pupil imaged in the natural pupil plane. The subject was
instructed to concentrate his attention at the centre of the
target, and to keep the target clear. The subject was kept
completely unaware of the experimental condition that was
being presented. Ten trials were run for each of the follow-
ing four conditions: (1) normal S±C effect. No apodising
®lters were in place; (2) neutralised S±C effect. The nSCF
®lter was in place to neutralise the subject's S±C effect; (3)
reversed S±C effect. The rSCF ®lter was in place to reverse
the subject's S±C effect so that the S±C peak would be on
the opposite side of the pupil; and (4) `doubled' S±C effect.
The SCF ®lter was in place to accentuate the subject's
normal S±C effect.
Filters were carefully aligned on the optical axis of the
Badal optical system. Because a single subject design
(n� 1) was used, the order of presentation of the four condi-
tions was randomised within blocks without replacement
(Edgington, 1995). In another session, nine trials were run
for each condition in random order with a 5-mm arti®cial
pupil. During the trials with the 5-mm pupil, the examiner
estimated the minimum and maximum pupil diameter of the
eye on the video-monitor. The subject's pupil diameter
varied between 3 and 5.5 mm.
After the main sessions, accommodation was monitored
for 30 s without a target while the subject viewed a bright
empty ®eld (542 nm light) with a 0.75-mm arti®cial pupil
in place. Then the subject remained in the dark for 3 min
to allow accommodative adaptation effects to subside
(Rosen®eld et al., 1994), and a 30-s record of accommo-
dation was made in the dark to determine the dark focus
of accommodation.
344 Ophthal. Physiol. Opt. 2001 21: No 5
Analysis
Accommodation data from each trial were analysed using
a fast Fourier transform (FFT) and standard signal proces-
sing procedures (Lee et al., 1999) to determine gain and
phase of the response at the temporal frequency of target
motion. Several randomization tests were performed using a
method of random enumeration (n� 50,000; Manly, 1991).
A one-factor `within subjects' ANOVA by a randomization
procedure was used as an omnibus test for any overall differ-
ences between the four experimental conditions (Edgington,
1995). Then three comparisons were made using a two-
tailed repeated-measures t-test by a randomization proce-
dure (Edgington, 1995). Comparisons were made between
normal and neutralised conditions, normal and reversed
conditions, and normal and doubled conditions. These
powerful non-parametric procedures are valid for use in
single-subject designs (Edgington, 1995).
Results
In the preliminary trials, the subject's responses to diop-
tric blur were typical (Kruger et al., 1993, 1995, 1997). Gain
was 0.63 when measured in white light and was reduced to
0.28 in monochromatic light. Thus gain was reduced by
more than 50% when chromatic aberration was removed
by monochromatic light. Mean dark focus was 0.57 D,
and the accommodative level was 0.61 D while viewing a
bright empty ®eld.
The results of the main experiment are summarised in
Tables 1 and 2. Vector averaged gains and phase-lags
were similar for the four conditions (Table 1). For the
3-mm pupil, ANOVA by a randomization procedure
revealed a signi®cant overall effect of ®lter type
(ST2� 32.4, P� 0.017) that was due to a signi®cantly
lower gain in the doubled condition than in the normal
condition (Table 2). Contrary to our hypothesis, neutralis-
ing and reversing the S±C function had no signi®cant
effect on accommodative gain (Table 2).
For the 5-mm pupil, there was no overall effect of ®lter
type (ST2� 41.82, P� 0.10), although the differences
between the normal and neutralised conditions, and between
the normal and reversed conditions approached signi®cance
(Table 2). That is, neutralising or reversing the S±C func-
tion with a 5-mm pupil increased accommodative gain.
Overall, the results do not support the hypothesis that
neutralising or reversing the S±C effect with apodising
®lters should impair the accommodative response.
Discussion
For the present subject, the results do not support the
hypothesis that the average S±C effect, measured over an
extended area of the fovea (4.128), mediates a directional
signal for accommodation. For the 3-mm pupil, gains were
essentially the same in the normal, neutralised and reversed
conditions, but the doubling ®lter reduced gain signi®cantly.
Accommodation and the Stiles±Crawford effect: theory and a case study: P. B. Kruger et al. 345
Table 1. Vector-averaged gains and phase-lags for the four conditions at two pupil sizes (3 and 5 mm)
Condition Pupil size (mm)a Gain Phase (8)b
Normal 3 0.302 2 72Neutralised 3 0.309 2 65Reversed 3 0.300 2 71Doubled 3 0.205 2 78Normal 5 0.274 2 47Neutralised 5 0.393 2 51Reversed 5 0.397 2 50Doubled 5 0.345 2 55
aFor the 5-mm arti®cial pupil, the natural pupil varied in the range 3±5.5 mm.bNegative values indicate phase-lags.
Table 2. Results of the t-test by a randomization procedure for pair-wise comparisons at two pupil sizes (Nor, normal; Neu,neutralised; Rev, reversed; Dou, doubled)
Comparison Pupil size (mm)a Scalar difference in gain Pb
Neu±Nor 3 1 0.007 0.907Rev±Nor 3 2 0.002 0.914Dou±Nor 3 2 0.097 0.013Neu±Nor 5 1 0.119 0.086Rev±Nor 5 1 0.123 0.057Dou±Nor 5 1 0.071 0.259
aFor the 5-mm pupil, the actual pupil varied in the range 3±5.5 mm.bProbability values are for a two-tailed test.
The doubling ®lter provided a very narrow S±C function,
and effectively a smaller pupil. Thus, it is possible that the
reduced gain with the doubling ®lter was due to a larger
depth of focus from an effectively smaller pupil (Hennessy
et al., 1976; Ward and Charman, 1985). With a 5-mm arti-
®cial pupil in place, gain increased in the neutralised and
reversed conditions, although the increase did not reach
statistical signi®cance. An increase in gain in these condi-
tions contradicts the hypothesis that the normal S±C effect
mediates the response. The gain may be larger when the S±
C effect is neutralised if in the absence of the usual S±C
apodisation the effective pupil size is larger.
In the main trials, accommodation was driven by a sinu-
soidal stimulus motion. This approach is widely used and
allows the application of standard Fourier signal processing
techniques (Campbell and Westheimer, 1960; Carter, 1962;
Stark et al., 1965; Kruger and Pola, 1986; Kotulak et al.,
1995; Kruger et al., 1997). However, it could be argued that
the subject used voluntary accommodation, coupled with
the predictable nature of the sine motion (van der Wildt et
al., 1974), to mask poorer performance in the neutralised,
reversed and doubled conditions. While attention and
prediction appear to be important to normal accommodative
function (Hung and Ciuffreda, 1988; Francis et al., 1989;
Stark and Atchison, 1994), it seems unlikely that such
voluntary or predictive behaviour could completely mask
a poor response if the S±C function really does provide a
stimulus to accommodation. Many individuals have rudi-
mentary degrees of voluntary accommodation (Marg,
1951; Ciuffreda and Kruger, 1988; Jaschinski-Kruza,
1991), but the ability to replicate a particular accommoda-
tion response requires a great deal of training; even simply
to maintain a speci®ed steady response level (Randle, 1971).
Although the present subject clearly does not use the
average S±C effect to determine focus, the experiment
must be repeated on a larger group of subjects before the
hypothesised mechanism may be rejected. Broad individual
variation in accommodation responses to various aspects of
dioptric blur is well established and is a hallmark of accom-
modation (Fincham, 1951; Gwiazda et al., 1993, 1995;
Kruger et al., 1997). Thus, it would not be surprising to
®nd distinct differences among subjects regarding use of
the S±C effect.
Apodisation model of the Stiles±Crawford effect
It is important to consider whether the apodisation model
is a valid descriptor of the directionality of retinal receptors.
Apodising ®lters cannot correct for local variations in the
S±C function nor for small temporal variations in the S±C
function due to changing accommodation (see Introduc-
tion), and so the ®lters used in the present study are unlikely
to have been perfect. In addition, the subject's S±C effect
was measured with a target that subtended 4.128, while the
accommodation target subtended 11.58. Thus, the neutralis-
ing ®lter probably did not effectively neutralise the S±C
effect simultaneously at all points across the image of the
accommodation target. However, variations in the S±C
peak due to accommodation were probably negligible.
During the experiment the subject's peak-to-peak accom-
modative response did not exceed approximately 0.8 D
(gain < 0.4) and if we extrapolate from previous ®ndings
(Blank and Enoch, 1973; Provine and Enoch, 1975), this
would correspond to a nasal decentration of approximately
0.07 mm (from 1.24 to 1.31 mm for the present subject).
Despite the inadequacies of the apodisation model, it should
be noted that the inclusion of reversing and `doubling'
apodising ®lters in the present experimental design provided
a way to examine the effects of gross changes in the habitual
S±C function. These ®lters should have disrupted accom-
modation if the subject was using the S±C function as a
stimulus to accommodation. In future experiments, the
effects of local variations in the S±C function across the
fovea could be greatly reduced by using small centrally
®xated targets, provided that those targets are large enough
to provide an adequate stimulus to accommodation.
Stiles±Crawford directionality
In the present experiment a psychophysical technique
was used to measure the S±C function, and the apodising
®lters were based on psychophysical measures. However,
the S±C function has also been determined using the optical
methods of single-entry and multiple-entry re¯ectometry.
Optical and psychophysical methods give similar estimates
of the position of the S±C peak, but psychophysical
measures of S±C directionality (rho) are twice as broad as
single-entry re¯ectometry (Burns et al., 1995; Gorrand and
Delori, 1995, 1997; He et al., 1999; Marcos and Burns,
1999). Multiple entry-re¯ectometry gives results that are
closer to psychophysical measures of the S±C effect, but
directionality is still narrower using the re¯ectometric
method (Marcos and Burns, 1999). The discrepancy
between data from the two techniques is attributed to several
factors (Gorrand and Delori, 1997; Marcos et al., 1998;
Marcos and Burns, 1999). In the present experiment the
problem of differences between optical and psychometric
methods was mitigated to some extent by using several
apodisation models (neutralised, `doubled' and reversed)
to test our hypothesis. Even if the neutralised condition
did under-correct the S±C function, the effect of the
reversed condition was still considerable and should have
impaired accommodation if, as hypothesised, the average
S±C effect mediates the accommodative response.
Finally, in the present study, psychophysical measures
of the S±C function were made in monochromatic 633-
nm light but measures of accommodation were made in
542-nm light. This may have led to over-correction of the
S±C effect in the neutralised condition, because the S±C
function (rho) is larger for long-wavelength light than for
346 Ophthal. Physiol. Opt. 2001 21: No 5
mid-spectral light (Enoch and Bedell, 1981). Again, the
reversed condition still provided an appropriate test of the
hypothesis.
Directional stimuli for accommodation
The standard view of accommodation control has been
that blur from inaccurate focus provides the sole true stimu-
lus to re¯ex accommodation and is even-error in nature
(Stark and Takahashi, 1965; Ciuffreda, 1991). In this
model, any sources of directional information are relegated
to the inferior status of `cues' and are thought on a neural
level to provide information that is combined with that from
even-error blur to drive accommodation. In the absence of
anatomical or physiological data to either support or refute
this postulate, it has not been until recently that strong
empirical evidence against the necessity of even-error blur
has become available. For example, longitudinal chromatic
aberration has been found to drive accommodation in the
absence of blur feedback (Kruger et al., 1995; Lee et al.,
1999). In addition, subjects continue to accommodate in the
absence of both longitudinal chromatic aberration and blur
feedback (Kruger et al., 1997), and this supports the notion
of another (as yet unknown) odd-error stimulus with direc-
tional quality. These studies support a plural model of re¯ex
accommodation in which the visual system uses multiple
sources of information; including even-error blur (Phillips
and Stark, 1977). In the present experiment we considered
the S±C effect, averaged over the central ®eld of view, as a
possible directional stimulus to accommodation, but there
may be other candidates.
A directional signal might come from the monochromatic
aberrations of the eye, which are common at the fovea
(Howland and Howland, 1977; Campbell et al., 1990;
Artal et al., 1995; LoÂpez-Gil and Howland, 1999). Coma
produces an asymmetric point spread-function which alters
the spatial phase of grating images, especially at high spatial
frequencies (van Meeteren, 1974; Walsh and Charman,
1985, 1989). However, monochromatic aberrations may
not provide a reliable directional signal for accommodation
because they change with accommodation level (Ivanoff,
1956; Atchison et al., 1995; LoÂpez-Gil et al., 1998; He et
al., 2000).
Alternate methods of determining light vergence
As previously discussed, the apodising model in the
present study uses a S±C function that represents the aver-
age activity of a large number of directionally sensitive
foveal cones, most of which point approximately toward
the peak of the S±C function. Local variations in receptor
directional sensitivity (Westheimer, 1967) and receptor
orientation (O'Brien and Miller, 1953; Makous, 1968,
1977; Sansbury et al., 1974; Tobey et al., 1975; Bailey
and Heath, 1978; Williams, 1980) are ignored in the apodis-
ing model.
As an example of how local variations in receptor orien-
tation might provide a directional stimulus to accommoda-
tion, the annular variations noted by Williams (1980) are
considered. Williams used small targets (30 arc min) to
measure the position of the S±C peak for different locations
within the fovea. He concluded that the sloping walls of the
foveal pit refract light away from the centre of the fovea,
and this alters the S±C function across the central two
degrees of the fovea. The refraction of light simulates physi-
cal splaying of the foveal cones outward from the foveal
centre, so that the axes of the cones are aligned with the
edges of the pupil rather than the pupil centre. Conse-
quently, cones on the nasal and temporal sides of the foveal
pit sample light from the nasal and temporal sides of the
pupil, respectively, and cones above and below the fovea
sample light from the top and bottom of the pupil respec-
tively. In the experiment of Williams (1980), three of four
subjects showed this pattern of systematic local variations in
cone orientation, and one subject showed differences in
directional sensitivity for different foveal locations that
were not systematic.
Figure 3 illustrates the effect of such optically splayed
foveal cones on blurred image points positioned to one side
Accommodation and the Stiles±Crawford effect: theory and a case study: P. B. Kruger et al. 347
Figure 3. Effect of local annular variations in S±C functionaround the foveal centre (Williams, 1980).
of the fovea, for hyperopic and myopic focus. The defo-
cused point spread-function is positioned to the right side
of the foveal centre (Figure 3, top) and the left side of the
foveal centre (Figure 3, bottom) and the blur spread-func-
tion is sampled by a large number of similarly oriented
cones. The cones that sample light from the edges of the
pupil are located approximately 30 arc min from the centre
of the fovea where the slope of the foveal pit is largest
(Williams, 1980). When the blurred image is 30 arc min
to the right of the foveal centre, the peak of the effective
point spread-function (after weighting by the local S±C
effect) is decentred to the right in hyperopic defocus and
to the left in myopic defocus, and the direction of asymme-
try is reversed for blurred image points positioned approxi-
mately 30 arc min to the left of the foveal centre. These
changes in the position of the S±C peak produce local
changes in the spatial phase of grating targets after weight-
ing for the S±C function. The shift in spatial phase is
restricted to an annular region around the foveal centre
with a radius of approximately 30±60 arc min, and the
shift in phase is toward the centre of the fovea in myopic
defocus, and away from the foveal centre in hyperopic
defocus.
The methods of light vergence detection described so far
(Figures 1 and 3) use the average Stiles±Crawford effect
measured over patches of the fovea. However, the ®bre-
optic nature of cones might be used in other ways, by indi-
vidual cones or groups of cones, to determine the angle of
incidence or vergence of light. Light is propagated by retinal
cones as wave-guide modal patterns that result in a non-
uniform distribution of energy in receptors (Enoch, 1960,
1961a,b; Snyder, 1966; Enoch and Tobey, 1981; Enoch and
Lakshminarayanan, 1991). The modal patterns that are
propagated by the receptor change when the wavelength
or angle of incidence of the light changes (Enoch, 1960,
1961a,b; Snyder, 1966; Tannenbaum, 1975; Stacey and
Pask, 1994). Enoch (1961a,b) observed abrupt changes in
modal patterns when the angle of incidence of light was
changed by only a few degrees, and when the focal plane
of the illumination within the receptor was varied. Several
investigators have examined the role of wave-guide modes
in vision (Enoch, 1960, 1961a,b; Snyder, 1966; Snyder and
Hall, 1969; Tannenbaum, 1975; Lakshminarayanan and
Calvo, 1987; Stacey and Pask, 1994) but the possibility
that modal patterns might identify the sign of defocus has
not been considered.
Besides any role that wave-guide modes might play in
identifying the sign of defocus, several lines of evidence
suggest that individual cones have acceptance angles that
are narrower than the diameter of the pupil (Makous, 1968;
Sansbury et al., 1974; Tobey et al., 1975; Bailey and Heath,
1978). In addition, small groups of cones share common
alignment in the pupil, while neighbouring groups have
slightly different alignment (O'Brien and Miller, 1953;
Enoch, 1967; Ohzu and Enoch, 1972). Experiments by
Makous (1968, 1977) and Sansbury et al. (1974) suggest
that cones contain `channels' that capture light from oppo-
site sides of the pupil, and that separate channels reside in
the same cones. Although the idea is speculative, sampling
light simultaneously from opposite sides of the pupil as a
defocused edge moves across the retina is a basic method of
determining light vergence that follows the principles of
retinoscopy (Campbell et al., 1998). Again, individual
eyes might use diverse methods to determine the sign of
defocus. Ocular sensitivity to the vergence of light may be
important for re¯ex accommodation, and may have a role in
the long-term focussing process of emmetropization.
Acknowledgements
Supported by the National Eye Institute of the NIH (RO1-
EYO5901) and the Schnurmacher Institute for Vision
Research (97-98-088). We thank Maurice Rynders for
providing details of the S±C function obtained at the Indi-
ana School of Optometry, and David A. Atchison and Jay
M. Enoch for helpful comments on the manuscript.
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