Accommodation and the Stiles–Crawford effect: theory and a ... · Accommodation and the...

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


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


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.


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.


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


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


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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