1040-5488/15/9205-0544/0 VOL. 92, NO. 5, PP. 544Y550 OPTOMETRY AND VISION SCIENCE Copyright * 2015 American Academy of Optometry
ORIGINAL ARTICLE
Perceptual-Motor Computational Model of Anomalous Binocular Correspondence Clifton Schor* ABSTRACT Purpose. A head-centric disparity model of anomalous binocular correspondence (ABC) in strabismus provides a framework that captures several associated perceptual-motor characteristics that are unexplained by the retino-centric model (anomalous retinal correspondence) of Von Graefe and Burian. The head-centric model elaborates on the anomalous-projection model of Verhoeff and Brock, originally described by Wells, late in the 18th century, which proposes that three-dimensional space perception is based on information obtained separately from the two eyes in ABC, without binocular retinal correspondence. Binocular parallax angles formed by the two eyes’ monocular head-centric directions provide sufficient information to estimate distance but not enough to stimulate diplopia without a reference for zero disparity. Methods (Model Description). The retino-centric model computes binocular disparity from differences between retino-centric directions specified by the two eyes, with each eye’s direction referenced to its own primary visual direction. The head-centric analog to retinal disparity is binocular parallax that could provide distance information but not a stimulus for diplopia. Diplopia is computed from differences between binocular parallax angles subtended by object points and a reference for zero disparity, that is, the head-centric Horopter, which adjusts for viewing distance, independently of convergence of the eyes. Results (Clinical Observations). Several perceptual-motor phenomena associated with anomalous correspondence demonstrate two sensory fusion mechanisms in ABC that involve registered vergence signals and changes in the horopter, independent of vergence. Conclusions. In ABC, the subjective-squint angle is unaffected by registered vergence movements. Binocular sensory fusion is obtained via the head-centric model by adjusting the diameter of the head-centric horopter, independent of the vergence angle, from the fixation distance to the distance of another reference point. By altering the reference viewing distance for zero disparity, the sign and magnitude of disparity stimuli for fusion and diplopia are changed, thereby enabling the perception of a fused fixation target and the appreciation of physiological diplopia in strabismus. (Optom Vis Sci 2015;92:544Y550) Key Words: strabismus, anomalous correspondence, disparity, binocular parallax, stereopsis, binocular sensory fusion, diplopia, head-centric direction, retino-centric direction, suppression, perceived distance, eye position, horopter, disparity vergence
A
long-standing question in binocular vision is how do some people with manifest strabismus and anomalous binocular correspondence (ABC) perceive three-dimensional space in much the same way as people with aligned eyes and normal binocular vision? These strabismics perceive objects in the fixation plane as single without suppression, and they can appreciate stereoscopic depth of objects lying closer and farther from the fixation plane.1Y3 Under binocular viewing conditions, they perceive objects in accurate visual directions that lie in nonsuppressed regions of the deviating eye that correspond with monocular suppression zones of the
nondeviating eye as nonstrabismics do in the region of the blind spot. Usually, they have early-onset strabismus,4 and animal studies suggest that the strabismus disrupts the development of binocular vision with a lack of binocular cells in the primary visual cortex,5Y7 which suggests that they lack binocular retinal correspondence. The following is a synopsis of a detailed manuscript of the same title with a glossary of clinical terms that is available as Supplemental Digital Content online (available at http://links.lww.com/OPX/A209).
*OD, PhD, FAAO School of Optometry, University of California, Berkeley, Berkeley, California. Supplemental digital content is available for this article. Direct URL citations appear in the printed text and are provided in the HTML and PDF versions of this article on the journal’s Web site (www.optvissci.com).
Two general theories account for the constellation of characteristics that define abnormal binocular correspondence. The first theory, referred to as anomalous retinal correspondence (ARC) by Von Graefe and elaborated upon by Burian,4,8 proposes that there
Retino-Centric and Head-Centric Models of ABC
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Model of Anomalous Binocular CorrespondenceVSchor
is a shift in binocular retinal correspondence that is an adaptation to eliminate pathological diplopia of the fixation point. The shift is referred to as the angle of anomaly and it is quantified by the perceived disparity between visual directions of foveal images. In this retino-centric correspondence theory, binocular cells in the primary visual cortex code retinal image disparity but with a shift in the binocular mapping of the two retinal images that is frequently equal to the angle of strabismus. Sensory fusion occurs when objects are imaged on anatomically shifted corresponding retinal points, and convergence influences disparity by moving the retinal images onto noncorresponding retinal points. The second theory of anomalous correspondence, originated by Wells and elaborated upon by Verhoeff, proposes that there is a lack or absence of retinal correspondence, and binocular disparity is derived from visual directions sensed separately by the two eyes (?L and ?R) in head-centric coordinates.9Y12 Binocular sensory fusion occurs when the estimated binocular parallax angle subtended by an object (A¶ = ?L j ?R) is equal to a binocular parallax reference for zero disparity (AR) that is computed from monocular estimates of distance and direction of an attended fixation point. Fig. 1 is a block diagram that illustrates how the retino-centric and head-centric models differ by where absolute disparity is computed in the sequence of operations that compute head-centric direction from retinal and
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oculomotor signals. This difference in the models may be inconsequential for normal binocular vision, but it results in different predictions of how vergence eye position signals affect the perceived disparity between foveal afterimages in ABC (i.e., angle of anomaly).
Two Binocular Sensory Fusion Mechanisms for Head-Centric Correspondence In the head-centric correspondence theory, binocular sensory fusion is unperturbed by convergence, and fusion results from two sensory mechanisms. The first sensory mechanism for fusion responds to registered motor innervation signals for both version and vergence eye movements and is referred to by Hallden13 as the neural innervation mechanism for fusion. The second sensory mechanism for fusion responds to shifts in perceived fixation distance, independently of the angle of convergence, and it is referred to directly by Hallden and indirectly by Verhoeff as the diplopia mechanism.10,13 The neural innervation mechanism for sensory fusion was first proposed as a motor theory by Wells and later by Morgan and others.9,14Y16 The motor theory is based on direction constancy during version eye movements where, as proposed by Helmholtz,17 retinal image motion is combined with version eye position
FIGURE 1. Block diagram of retino-centric and head-centric models. Retino-centric and head-centric models differ by where absolute disparity is computed in the sequence of operations that compute head-centric direction from retinal and oculomotor signals. In the retino-centric model (left panel), absolute binocular disparity (CABS(RET)) is computed from the difference between right eye and left eye retino-centric directions (>L j >R), before the conversion to head-centric coordinates. Absolute retinal disparity is relative to a binocular parallax reference for zero disparity (AR) (i.e., the horopter). The absolute binocular retinal disparity is then combined with averaged eye-position signals (5B) to compute absolute head-centric disparity (CABS(H-C)). In the right panel, binocular parallax (AH-C) in the head-centric model is computed from the angle subtended by the intersection of the two monocular head-centric directions (?L j ?R) that are computed from the combination of each eye’s position (5) with its retino-centric (i.e., oculo-centric) direction (>). Binocular parallax in the absolute head-centric model (AH-C) is compared with a binocular parallax reference for zero disparity (AR) that is derived from monocular cues for distance (d ¶) and direction to compute absolute head-centric disparity (CABS(H-C)). Optometry and Vision Science, Vol. 92, No. 5, May 2015
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546 Model of Anomalous Binocular CorrespondenceVSchor
information to yield a stable percept of the world. However, changes in retinal image position caused by vergence movements normally cause changes in perceived direction so that when we voluntarily cross or converge our eyes, the fixation target doubles and two images appear to move apart. In anomalous correspondence, direction constancy also occurs with vergence movements that influence perceived direction in the same way as version movements so that the world remains stationary, independent of convergence. Vergence does not change perceived direction or binocular disparity of the fixation target. However, the motor theory is incomplete because it does not address its dependence on an absence of retinal correspondence. Vergence can only be associated with direction constancy by influencing perceived direction separately for the two eyes in head-centric coordinates. In contrast, in the retino-centric correspondence model, perceived direction and binocular disparity are influenced by vergence because disparity between the two retinal images is computed before eye position information can influence perceived direction to achieve direction constancy. Hallden’s diplopia mechanism for binocular sensory fusion in anomalous correspondence addresses the issue of sensory fusion of
disparate images without fusional convergence.13 Diplopia is based on estimates of absolute disparity that are computed from a reference distance for zero disparity. In normal retinal correspondence, the zero disparity reference is described by the horopter that identifies pairs of retinal points that are mapped onto binocular cells to represent zero disparity in the primary visual cortex, but without retinal correspondence, there is no anatomically fixed retinal correspondence system to define zero disparity. Yet, people lacking retinal correspondence can perceive the world with single binocular vision in the fixation plane and appreciate diplopia of objects located at more remote and proximal distances.
METHODS Computation of Absolute Binocular Disparity without Binocular Retinal Correspondence A major question is how is a binocular parallax reference for zero disparity (AR), which is used to compute absolute disparity, derived without binocular retinal correspondence? AR is computed
FIGURE 2. The left panel (retino-centric model) illustrates that the foveal target T, presented monocularly to the left eye with the right eye occluded, could originate from an object lying at one of many different viewing distances along the left visual axis; however, all of these possible object distances would lie in the same retino-centric direction (>L) relative to the visual axis of the left-fixating eye. Retino-centric directions are mapped onto the cyclopean eye and combined with averaged eye position (version position) to yield a single solution for binocular head-centric direction (EL; gray vertical dashed lineVgreen in the online version) from the cyclopean reference point (C). The right panel (head-centric model) illustrates that the foveal target T presented monocularly to the left eye with the right eye occluded during symmetrical convergence has a retino-centric direction of zero (>L = 0) and a monocular head-centric direction (?L) equal to the monocular vergence component (?L = 5L). The monocular head-centric vector (DL) (solid black diagonal vectors) is in reference to the nodal point and primary direction of gaze (solid black vertical line) and it is shown for three different estimates of d ¶. These correspond to three possible binocular head-centric directions (EL) (gray dashed vectorsVgreen in the online version) equal to the angles subtended at the cyclopean reference point (C) by three examples of monocularly estimated reference target distance (d ¶ ) or lengths of vector ?. A color version of this figure is available online at www.optvissci.com. Optometry and Vision Science, Vol. 92, No. 5, May 2015
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from head-centric directions that have two possible references. Head-centric directions such as ?L and ?R, shown in the right panel of Figs. 2 and 3, are referenced to the two eyes’ nodal points, that is, monocular head-centric directions. The angle formed by the intersection of ?L and ?R describes the estimated binocular parallax angle subtended by a target. The binocular parallax angle of the attended fixation target corresponds to AR and it determines which objects in the visual field appear single or double. Absolute disparity in head-centric coordinates (CABS(H-C)) is computed from the difference between the estimated binocular parallax angle subtended by a point at the two eyes (A¶)and AR, and this process is described as the relative head-centric model.
CABSðHL þ 5L and ?R ¼ >R þ 5R
ð3Þ
Head-centric directions such as EL, shown in the right panel of Fig. 2, are referenced to a single cyclopean eye lying on the interpupillary axis, that is, binocular head-centric directions. AR, expressed in radians, can be computed by triangulation from E, interpupillary distance (2a) and estimated viewing distance (d ¶) from equation 6.18 Absolute disparity is computed from the angle formed at the cyclopean eye between the two eyes’ binocular head-centric directions, and this is described as the absolute head-centric model.
CABSðH). In the absolute head-centric model, there are
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several solutions for E that depend on estimated target distance, location of the cyclopean reference point along the interpupillary axis, monocular eye position (5L and 5R), and retinal image location (>L and >R) (equation 5). The right panel in Fig. 2 illustrates that an estimate of monocular head-centric direction (?L) in reference to the nodal point of the left eye is independent of the target distance or vector length; that is, a single retinal image could be formed by points lying at many different viewing distances, yet all points lie in a single direction from the nodal point. However, the binocular head-centric directions (E) of points at the same distances are estimated from a binocular (cyclopean) reference point that is translated horizontally from the nodal point. Computation of egocentric direction (EL or ER) depends on monocular estimates of both distance (d ¶) and ?L or ?R (equation 5). Accordingly, in the absolute head-centric model, perceived distance influences binocular head-centric or egocentric direction estimated for each eye from the single cyclopean reference point under monocular and binocular conditions. In the retino-centric model, the two eyes are effectively combined into a single cyclopean eye whose reference nodal point for monocular head-centric direction coincides with the cyclopean reference point for egocentric direction, and binocular head-centric directions of the two eyes (EL and ER) and their absolute disparity (equation 1) are independent of perceived viewing distance. Thus, absolute disparity (CABS(H-C)) can be estimated, independent of vergence angle, either from binocular head-centric directions (EL and ER) and distance cues (d ¶) (equations 4 and 5), in reference to the cyclopean eye, or from monocular head-centric directions (?L and ?R) (equations 1 and 2), in reference to the two eyes’ nodal points. Squinters without retinal correspondence could shift their attention from one fixation distance to another and move AR to the new distance without changing convergence. Shifting the zero disparity reference distance (d ¶ ) and its binocular head-centric direction, without changing vergence angle, changes the binocular parallax reference for zero disparity (equation 6). This process is a mechanism that achieves binocular sensory fusion without changing convergence when there is no binocular retinal correspondence.
RESULTS Motor and Perceptual Phenomena That Characterize Anomalous Correspondence Several perceptual and motor phenomena characterize anomalous correspondence, and more of them are captured by the framework of the head-centric than the retino-centric model. Motor phenomena include harmonious anomalous correspondence, covariation of spontaneous variations in the objective angle of strabismus and the angle of anomaly, and slow anomalous vergence movements in response to base-out prisms.21,22 Perceptual phenomena include spontaneous changes in the angle of anomaly that are uncorrelated with changes in vergence angle, binocular sensory fusion of prism-induced disparity without vergence eye movements, and changes in the angle of anomaly in response to prism-induced disparity without vergence eye movements.23Y27 It is unlikely that any of the motor phenomena are consistent with the ARC model because adaptable shifts in retinal correspondence would be too slow and beyond a critical developmental period for binocular vision to account for rapid changes in retinal
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548 Model of Anomalous Binocular CorrespondenceVSchor
FIGURE 3. An example of a right esotropia of amplitude R, with harmonious anomalous correspondence, is illustrated for the retino-centric (left panel) and headcentric (right panel) models. In the left panel, a fixation target A is viewed at a remote distance, and T is a near target imaged on the two foveas. The left fovea corresponds with a nonfoveal nasal retinal locus in the right deviating eye (X), which is equal to the angle of anomaly (F) and R. The retino-centric geometric horopter is represented by the gray circle (green in the online version) that passes through the fixation target A. The near target T subtends a binocular parallax angle of A¶. The absolute disparity of target T (CABS(RET)) is computed from the difference between A¶ and AR. The right panel interprets the same example with the head-centric model. The gray circle (green in the online version) represents the head-centric geometric horopter, and its diameter is independent of vergence angle. AR is computed from monocular estimates of viewing distance (dA¶) and azimuth (EA = 0, not shown) (equation 6). The estimate of the binocular parallax angle (A¶) subtended by an individual point in the visual field (e.g., T at estimated distance dT¶ and azimuth ET¶) is computed from the angular difference between monocular head-centric directions (?L and ?R) (equation 2). Absolute binocular disparity of T (CABS(HC)) is computed from the difference between A¶ and AR. In this example, CABS(HC) of T is the angle of anomaly that is equal to R. The binocular parallax angle for the near target T (A¶) is greater than AR, and the target T subtends a crossed absolute head-centric disparity with the reference fixation target A (equation 1). If T became the fixation target by shifting attention without changing convergence, then the binocular parallax estimate of T would become AR and the binocular parallax estimate of A would be A¶ and the angle of anomaly would equal zero. The geometric horopter would be represented by the black circle and target A would subtend an uncrossed absolute disparity of R. A color version of this figure is available online at www.optvissci.com.
correspondence associated with vergence eye movements. As in normal retinal correspondence, binocular disparity is computed in ARC before eye position information can influence perceived direction or the disparity of foveal afterimages. In the retino-centric correspondence model, vergence eye movements would move retinal images of the fixation target away from corresponding retinal points and produce disparities that disrupt binocular sensory fusion; that is, vergence movements would produce diplopia, just as they do in NRC. Vergence movements interact differently with retinal image position and perceived direction in the absence of retinal correspondence. The registered monocular components of vergence interact
with retinal image position of each eye separately to produce direction constancy and single percepts of the fixation target during vergence-induced movement of the two retinal images. Consequently, the vergence signal that produces the strabismus causes the fixation target to remain single during retinal image movement and foveal afterimages to appear separated by the angle of anomaly that is equal to the objective angle of strabismus in a prevalent condition known as harmonious anomalous correspondence.21,22 Similarly spontaneous variations of the objective angle of strabismus covary with the angle of anomaly such that the fixation target continues to appear single in spite of changes in retinal image position caused by the vergence fluctuations.
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However, covariation is not perfect. Hallden13 noted that the amplitude of vergence change was not always equal to the changes in the angle of anomaly. He proposed that a diplopia mechanism stimulated a change in the angle of anomaly, independently of vergence, and resulted in perceived changes in disparity. The headcentric model interprets the source for spontaneous changes in disparity as spontaneous variations or noise of the estimated reference distance for zero disparity. The resulting disparity variations are independent of the noise variations of the objective angle of strabismus. The combination of these two motor and perceptual noise sources results in incomplete sensory fusion of the retinal disparity associated with vergence fluctuations. Another perceptual phenomenon that characterizes anomalous correspondence is the fusion of a prism-induced disparity of the fixation target that is independent of vergence eye movements. When the base-out prism is first introduced, the fixation target appears fused, and subsequently, there is a slow anomalous convergence response that can last from minutes to days to complete.23Y25 Fusion is experienced before, during, and after the vergence movements, without suppression of either eye. The sensory fusion (diplopia) mechanism achieves fusion by shifting the binocular parallax reference for zero disparity to equal the binocular parallax angle subtended by the fixation target viewed through the prism. During these changes in the attended fixation distance, there are changes in the empirically measured angle of anomaly such that the base-out prism causes foveal afterimages to appear to diverge or move in the direction of uncrossed disparity.25 Then, as slow anomalous convergence occurs, the afterimages are perceived to move slowly toward one another to reduce their uncrossed disparity.25 If diplopia is eliminated by shifts in the binocular parallax reference for zero disparity, then why is there a slow vergence response? It is possible that the anomalous vergence response may be an attempt to resolve a stimulus conflict to distance from motor cues of vergence and perceptual cues from binocular parallax. When the prism is first introduced, the binocular parallax subtended by the fixation target changes by the disparity of the prism. When the reference for zero disparity is shifted to the new binocular parallax angle without changing vergence, there is a cue conflict between these two distance cues. The slow vergence changes appear to be an attempt to match the vergence posture and perceptual cues to distance (i.e., estimated binocular parallax). However, it is also possible that the slow vergence changes may be a slow motor response that attempts to place the images of the fixation target on retinal locations that are habitually stimulated. Finally, there are changes in the angle of anomaly that depend on the complexity of the visual field and lighting. Anomalous correspondence tends to be harmonious when viewing natural scenes and unharmonious when viewing blank fields or in darkness. This may be related to the accuracy of distance estimates under these conditions, such as near biases of targets in darkness, known as the specific distance tendency, or perhaps a greater reliance on vergence angle as a distance cue in impoverished visual fields.28Y30
This project was supported by National Science Foundation grant BCS0715076 (Disparity Processing in Sensory and Motor Functions). The author thanks Dr. Pia Hoenig of the University of California at Berkeley, School of Optometry, for sharing her clinical observations of strabismus with anomalous correspondence; Dr. Martin Banks for his suggestions; Dr. Sangeetha Metapally for her comments on the paper organization; Dr. Nance Wilson for editing; Dr. Jonathan Horton for pointing out the historical reference to Wells 9; and the late Dr. Merton Flom for their conversations about his horopter research. Received August 25, 2014; accepted February 12, 2015.
Spatial Distortions of the Empirical Horopter in Head-Centric Correspondence
SUPPLEMENTAL DIGITAL CONTENT
There are several other phenomena associated with anomalous correspondence that require additional modeling. These include
A more detailed text of the same title with a glossary of clinical terms is available online at http://links.lww.com/OPX/A209.
shape differences between the empirical and theoretical-geometric horopter described by a central (foveal) concave notch and steep elliptical curvature deviation of the empirical from the circular geometric horopter.19,20 These spatial variations are likely to result from errors of estimated azimuth, that is, binocular head-centric direction (E), that are used to compute the binocular parallax reference for zero disparity (AR) (equation 6). The errors in perceived azimuth in the region of the notch could be caused by displacement of visual directions of objects imaged in suppression scotomas toward the fovea.31,32 The influence of errors of visual directions (E) on the curvature of the empirical horopter was first proposed by Ogle33 who described the differences in elliptical curvature of the normal empirical horopter from the circular geometric horopter as a consequence of monocular spatial distortions known as the Kundt and Munsterberg asymmetries.
CONCLUSIONS Anomalous binocular correspondence is modeled as a consequence of an absence of binocular retinal correspondence that allows monocular components of vergence eye position signals to interact with each eye separately to influence perceived direction and binocular disparity in head-centric coordinates as it does in lower vertebrates (Walls).34 Anomalous correspondence is not necessarily an adaptation of retinal correspondence to strabismus but rather a highly evolved form of binocular vision seen in lateral eyed animals. In some cases of strabismus, retinal correspondence may have been lost because of inhibition or suppression of binocular cells, and an effective treatment could be to break down the inhibition using antisuppression techniques that are intended to establish binocular triplopia and eventually normal correspondence between the two foveas. The binocular triplopia represents the coexistence of normal retinal and anomalous head-centric correspondence.35,36 If the eyes were physically aligned, either with surgery or orthoptics, then binocular inhibition could be eliminated to manifest an innate system of normal binocular retinal correspondence. Even without binocular retinal correspondence, voluntary foveal alignment of the two eyes would engage the motor innervation mechanism for fusion and result in harmonious anomalous correspondence, bifoveal fusion, and binocular correspondence would appear to be normal.
ACKNOWLEDGMENTS
Optometry and Vision Science, Vol. 92, No. 5, May 2015
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REFERENCES 1. Epstein DL, Tredici TJ. Microtropia (monofixation syndrome) in flying personnel. Am J Ophthalmol 1973;76:832Y41. 2. Henson DB, Williams DE. Depth perception in strabismus. Br J Ophthalmol 1980;64:349Y53. 3. Harwerth RS, Fredenburg PM. Binocular vision with primary microstrabismus. Invest Ophthalmol Vis Sci 2003;44:4293Y306. 4. Burian HM. Normal and anomalous correspondence. In: Allen JH, ed. Strabismus Ophthalmic Symposium II. St Louis, MO: C.V. Mosby; 1958:315. 5. Hubel DH, Wiesel TN. Binocular interaction in striate cortex of kittens reared with artificial squint. J Neurophysiol 1965;28:1041Y59. 6. Kumagami T, Zhang B, Smith EL, 3rd, Chino YM. Effect of onset age of strabismus on the binocular responses of neurons in the monkey visual cortex. Invest Ophthalmol Vis Sci 2000;41:948Y54. 7. Tychsen L, Wong AM, Burkhalter A. Paucity of horizontal connections for binocular vision in V1 of naturally strabismic macaques: cytochrome oxidase compartment specificity. J Comp Neurol 2004; 474:261Y75. 8. Graefe AV. Ueber doppelschen nach schieloperationen und incongruenz der netzhaute. In: Grefe AV, ed. Archiv fur Ophthalmologie. Berlin, Germany: Verlag von P. Jeanrenaud; 1854:82Y120. 9. Wells WC. Two Essays: One upon Single Vision with Two Eyes; The Other on Dew. Edinburgh, UK: A. Constable; 1818. 10. Verhoeff FH. Anomalous projection and other visual phenomena associated with strabismus. Arch Ophthalmol-Chic 1938;19:663Y99. 11. Brock FW. Anomalous projection in squint. Its cause and effect. New methods of correction. Report of cases. Am J Optom 1939; 16:201Y21. 12. Brock FW. Projection habits in alternate squints. Am J Optom 1940;17:193Y207. 13. Hallden U. Fusional phenomena in anomalous correspondence. Acta Ophthalmol Suppl 1952;37:1Y93. 14. Morgan MW. Anomalous correspondence interpreted as a motor phenomenon. Am J Optom Arch Am Acad Optom 1961;38:131Y48. 15. Boeder P. The ‘‘response shift’’. Doc Ophthalmol 1967;23:88Y100. 16. Kerr KE. Anomalous correspondenceVthe cause or consequence of strabismus? Optom Vis Sci 1998;75:17Y22. 17. Helmholtz HV. Handbook of Physiological Optics. [translation by Southall JPC from the 3rd German ed, Vol 1, of Handbuch der Physiologischen Optik. Hamberg: Voss; 1909]; New York, NY: Dover; 1962. 18. Erkelens CJ, van Ee R. A computational model of depth perception based on headcentric disparity. Vision Res 1998;38:2999Y3018. 19. Flom MC. Corresponding and disparate retinal points in normal and anomalous correspondence. Am J Optom Physiol Opt 1980;57:656Y65.
20. Boucher JA. Common visual direction horopters in exotropes with anomalous correspondence. Am J Optom Arch Am Acad Optom 1967;44:547Y72. 21. Bagolini. Sensorio-motorial anomalies in strabismus: (anomalous movements). Doc Ophthalmol 1976;41:23Y41. 22. Bagolini B. Sensory anomalies in strabismus. Br J Ophthalmol 1974;58:313Y8. 23. Alpern M, Hofstetter HW. The effect of prism on esotropia; a case report. Am J Optom Arch Am Acad Optom 1948;25:80Y91. 24. Burian HM. Fusional movements in permanent strabismusVa study of the roll of the central and peripheral retinal regions in the act of binocular vision in squint. Arch Ophthalmol-Chic 1941;26:626Y52. 25. Maraini G, Pasino L. Variations in the angle of anomaly and fusional movements in cases of small-angle convergent strabismus with harmonious anomalous retinal correspondence. Br J Ophthalmol 1964; 48:439Y43. 26. Bagolini B, Capobianco NM. Subjective space in comitant squint. Am J Ophthalmol 1965;59:430Y42. 27. Pasino L, Maraini G. Area of binocular vision in anomalous retinal correspondence. Br J Ophthalmol 1966;50:646Y50. 28. Brenner E, van Damme WJ. Judging distance from ocular convergence. Vision Res 1998;38:493Y8. 29. Owens DA, Leibowitz HW. Oculomotor adjustments in darkness and the specific distance tendency. Percept Psychophys 1976;20:2Y9. 30. Gogel WC, Tietz JD. Absolute motion parallax and the specific distance tendency. Percept Psychophys 1973;13:284Y92. 31. Jampolsky A. Characteristics of suppression in strabismus. AMA Arch Ophthalmol 1955;54:683Y96. 32. Bender MB. Disorders in Perception: With Particular Reference to the Phenomena of Extinction and Displacement. American Lecture Series: American Lectures in Neurology, Vol 120. Springfield, IL: Charles C. Thomas; 1952. 33. Ogle KN. Researches in Binocular Vision. Philadelphia, PA: Saunders; 1950. 34. Walls GL. The Vertebrate Eye and Its Adaptive Radiation. Bloomfield Hills, MI: Cranbrook Institute of Science; 1942. 35. Cass EE. Monocular diplopia occurring in cases of squint. Br J Ophthalmol 1941;25:565Y77. 36. Ramachandran VS, Cobb S, Levi L. Monocular double vision in strabismus. Neuroreport 1994;5:1418.
Clifton Schor School of Optometry University of California, Berkeley Berkeley, CA 94720 e-mail: [email protected]
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ORIGINAL ARTICLE
Perceptual-Motor Computational Model of Anomalous Binocular Correspondence Clifton Schor* ABSTRACT Purpose. A head-centric disparity model of anomalous binocular correspondence (ABC) in strabismus provides a framework that captures several associated perceptual-motor characteristics that are unexplained by the retino-centric model (anomalous retinal correspondence) of Von Graefe and Burian. The head-centric model elaborates on the anomalous-projection model of Verhoeff and Brock, originally described by Wells, late in the 18th century, which proposes that three-dimensional space perception is based on information obtained separately from the two eyes in ABC, without binocular retinal correspondence. Binocular parallax angles formed by the two eyes’ monocular head-centric directions provide sufficient information to estimate distance but not enough to stimulate diplopia without a reference for zero disparity. Methods (Model Description). The retino-centric model computes binocular disparity from differences between retino-centric directions specified by the two eyes, with each eye’s direction referenced to its own primary visual direction. The head-centric analog to retinal disparity is binocular parallax that could provide distance information but not a stimulus for diplopia. Diplopia is computed from differences between binocular parallax angles subtended by object points and a reference for zero disparity, that is, the head-centric Horopter, which adjusts for viewing distance, independently of convergence of the eyes. Results (Clinical Observations). Several perceptual-motor phenomena associated with anomalous correspondence demonstrate two sensory fusion mechanisms in ABC that involve registered vergence signals and changes in the horopter, independent of vergence. Conclusions. In ABC, the subjective-squint angle is unaffected by registered vergence movements. Binocular sensory fusion is obtained via the head-centric model by adjusting the diameter of the head-centric horopter, independent of the vergence angle, from the fixation distance to the distance of another reference point. By altering the reference viewing distance for zero disparity, the sign and magnitude of disparity stimuli for fusion and diplopia are changed, thereby enabling the perception of a fused fixation target and the appreciation of physiological diplopia in strabismus. (Optom Vis Sci 2015;92:544Y550) Key Words: strabismus, anomalous correspondence, disparity, binocular parallax, stereopsis, binocular sensory fusion, diplopia, head-centric direction, retino-centric direction, suppression, perceived distance, eye position, horopter, disparity vergence
A
long-standing question in binocular vision is how do some people with manifest strabismus and anomalous binocular correspondence (ABC) perceive three-dimensional space in much the same way as people with aligned eyes and normal binocular vision? These strabismics perceive objects in the fixation plane as single without suppression, and they can appreciate stereoscopic depth of objects lying closer and farther from the fixation plane.1Y3 Under binocular viewing conditions, they perceive objects in accurate visual directions that lie in nonsuppressed regions of the deviating eye that correspond with monocular suppression zones of the
nondeviating eye as nonstrabismics do in the region of the blind spot. Usually, they have early-onset strabismus,4 and animal studies suggest that the strabismus disrupts the development of binocular vision with a lack of binocular cells in the primary visual cortex,5Y7 which suggests that they lack binocular retinal correspondence. The following is a synopsis of a detailed manuscript of the same title with a glossary of clinical terms that is available as Supplemental Digital Content online (available at http://links.lww.com/OPX/A209).
*OD, PhD, FAAO School of Optometry, University of California, Berkeley, Berkeley, California. Supplemental digital content is available for this article. Direct URL citations appear in the printed text and are provided in the HTML and PDF versions of this article on the journal’s Web site (www.optvissci.com).
Two general theories account for the constellation of characteristics that define abnormal binocular correspondence. The first theory, referred to as anomalous retinal correspondence (ARC) by Von Graefe and elaborated upon by Burian,4,8 proposes that there
Retino-Centric and Head-Centric Models of ABC
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is a shift in binocular retinal correspondence that is an adaptation to eliminate pathological diplopia of the fixation point. The shift is referred to as the angle of anomaly and it is quantified by the perceived disparity between visual directions of foveal images. In this retino-centric correspondence theory, binocular cells in the primary visual cortex code retinal image disparity but with a shift in the binocular mapping of the two retinal images that is frequently equal to the angle of strabismus. Sensory fusion occurs when objects are imaged on anatomically shifted corresponding retinal points, and convergence influences disparity by moving the retinal images onto noncorresponding retinal points. The second theory of anomalous correspondence, originated by Wells and elaborated upon by Verhoeff, proposes that there is a lack or absence of retinal correspondence, and binocular disparity is derived from visual directions sensed separately by the two eyes (?L and ?R) in head-centric coordinates.9Y12 Binocular sensory fusion occurs when the estimated binocular parallax angle subtended by an object (A¶ = ?L j ?R) is equal to a binocular parallax reference for zero disparity (AR) that is computed from monocular estimates of distance and direction of an attended fixation point. Fig. 1 is a block diagram that illustrates how the retino-centric and head-centric models differ by where absolute disparity is computed in the sequence of operations that compute head-centric direction from retinal and
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oculomotor signals. This difference in the models may be inconsequential for normal binocular vision, but it results in different predictions of how vergence eye position signals affect the perceived disparity between foveal afterimages in ABC (i.e., angle of anomaly).
Two Binocular Sensory Fusion Mechanisms for Head-Centric Correspondence In the head-centric correspondence theory, binocular sensory fusion is unperturbed by convergence, and fusion results from two sensory mechanisms. The first sensory mechanism for fusion responds to registered motor innervation signals for both version and vergence eye movements and is referred to by Hallden13 as the neural innervation mechanism for fusion. The second sensory mechanism for fusion responds to shifts in perceived fixation distance, independently of the angle of convergence, and it is referred to directly by Hallden and indirectly by Verhoeff as the diplopia mechanism.10,13 The neural innervation mechanism for sensory fusion was first proposed as a motor theory by Wells and later by Morgan and others.9,14Y16 The motor theory is based on direction constancy during version eye movements where, as proposed by Helmholtz,17 retinal image motion is combined with version eye position
FIGURE 1. Block diagram of retino-centric and head-centric models. Retino-centric and head-centric models differ by where absolute disparity is computed in the sequence of operations that compute head-centric direction from retinal and oculomotor signals. In the retino-centric model (left panel), absolute binocular disparity (CABS(RET)) is computed from the difference between right eye and left eye retino-centric directions (>L j >R), before the conversion to head-centric coordinates. Absolute retinal disparity is relative to a binocular parallax reference for zero disparity (AR) (i.e., the horopter). The absolute binocular retinal disparity is then combined with averaged eye-position signals (5B) to compute absolute head-centric disparity (CABS(H-C)). In the right panel, binocular parallax (AH-C) in the head-centric model is computed from the angle subtended by the intersection of the two monocular head-centric directions (?L j ?R) that are computed from the combination of each eye’s position (5) with its retino-centric (i.e., oculo-centric) direction (>). Binocular parallax in the absolute head-centric model (AH-C) is compared with a binocular parallax reference for zero disparity (AR) that is derived from monocular cues for distance (d ¶) and direction to compute absolute head-centric disparity (CABS(H-C)). Optometry and Vision Science, Vol. 92, No. 5, May 2015
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546 Model of Anomalous Binocular CorrespondenceVSchor
information to yield a stable percept of the world. However, changes in retinal image position caused by vergence movements normally cause changes in perceived direction so that when we voluntarily cross or converge our eyes, the fixation target doubles and two images appear to move apart. In anomalous correspondence, direction constancy also occurs with vergence movements that influence perceived direction in the same way as version movements so that the world remains stationary, independent of convergence. Vergence does not change perceived direction or binocular disparity of the fixation target. However, the motor theory is incomplete because it does not address its dependence on an absence of retinal correspondence. Vergence can only be associated with direction constancy by influencing perceived direction separately for the two eyes in head-centric coordinates. In contrast, in the retino-centric correspondence model, perceived direction and binocular disparity are influenced by vergence because disparity between the two retinal images is computed before eye position information can influence perceived direction to achieve direction constancy. Hallden’s diplopia mechanism for binocular sensory fusion in anomalous correspondence addresses the issue of sensory fusion of
disparate images without fusional convergence.13 Diplopia is based on estimates of absolute disparity that are computed from a reference distance for zero disparity. In normal retinal correspondence, the zero disparity reference is described by the horopter that identifies pairs of retinal points that are mapped onto binocular cells to represent zero disparity in the primary visual cortex, but without retinal correspondence, there is no anatomically fixed retinal correspondence system to define zero disparity. Yet, people lacking retinal correspondence can perceive the world with single binocular vision in the fixation plane and appreciate diplopia of objects located at more remote and proximal distances.
METHODS Computation of Absolute Binocular Disparity without Binocular Retinal Correspondence A major question is how is a binocular parallax reference for zero disparity (AR), which is used to compute absolute disparity, derived without binocular retinal correspondence? AR is computed
FIGURE 2. The left panel (retino-centric model) illustrates that the foveal target T, presented monocularly to the left eye with the right eye occluded, could originate from an object lying at one of many different viewing distances along the left visual axis; however, all of these possible object distances would lie in the same retino-centric direction (>L) relative to the visual axis of the left-fixating eye. Retino-centric directions are mapped onto the cyclopean eye and combined with averaged eye position (version position) to yield a single solution for binocular head-centric direction (EL; gray vertical dashed lineVgreen in the online version) from the cyclopean reference point (C). The right panel (head-centric model) illustrates that the foveal target T presented monocularly to the left eye with the right eye occluded during symmetrical convergence has a retino-centric direction of zero (>L = 0) and a monocular head-centric direction (?L) equal to the monocular vergence component (?L = 5L). The monocular head-centric vector (DL) (solid black diagonal vectors) is in reference to the nodal point and primary direction of gaze (solid black vertical line) and it is shown for three different estimates of d ¶. These correspond to three possible binocular head-centric directions (EL) (gray dashed vectorsVgreen in the online version) equal to the angles subtended at the cyclopean reference point (C) by three examples of monocularly estimated reference target distance (d ¶ ) or lengths of vector ?. A color version of this figure is available online at www.optvissci.com. Optometry and Vision Science, Vol. 92, No. 5, May 2015
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from head-centric directions that have two possible references. Head-centric directions such as ?L and ?R, shown in the right panel of Figs. 2 and 3, are referenced to the two eyes’ nodal points, that is, monocular head-centric directions. The angle formed by the intersection of ?L and ?R describes the estimated binocular parallax angle subtended by a target. The binocular parallax angle of the attended fixation target corresponds to AR and it determines which objects in the visual field appear single or double. Absolute disparity in head-centric coordinates (CABS(H-C)) is computed from the difference between the estimated binocular parallax angle subtended by a point at the two eyes (A¶)and AR, and this process is described as the relative head-centric model.
CABSðHL þ 5L and ?R ¼ >R þ 5R
ð3Þ
Head-centric directions such as EL, shown in the right panel of Fig. 2, are referenced to a single cyclopean eye lying on the interpupillary axis, that is, binocular head-centric directions. AR, expressed in radians, can be computed by triangulation from E, interpupillary distance (2a) and estimated viewing distance (d ¶) from equation 6.18 Absolute disparity is computed from the angle formed at the cyclopean eye between the two eyes’ binocular head-centric directions, and this is described as the absolute head-centric model.
CABSðH). In the absolute head-centric model, there are
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several solutions for E that depend on estimated target distance, location of the cyclopean reference point along the interpupillary axis, monocular eye position (5L and 5R), and retinal image location (>L and >R) (equation 5). The right panel in Fig. 2 illustrates that an estimate of monocular head-centric direction (?L) in reference to the nodal point of the left eye is independent of the target distance or vector length; that is, a single retinal image could be formed by points lying at many different viewing distances, yet all points lie in a single direction from the nodal point. However, the binocular head-centric directions (E) of points at the same distances are estimated from a binocular (cyclopean) reference point that is translated horizontally from the nodal point. Computation of egocentric direction (EL or ER) depends on monocular estimates of both distance (d ¶) and ?L or ?R (equation 5). Accordingly, in the absolute head-centric model, perceived distance influences binocular head-centric or egocentric direction estimated for each eye from the single cyclopean reference point under monocular and binocular conditions. In the retino-centric model, the two eyes are effectively combined into a single cyclopean eye whose reference nodal point for monocular head-centric direction coincides with the cyclopean reference point for egocentric direction, and binocular head-centric directions of the two eyes (EL and ER) and their absolute disparity (equation 1) are independent of perceived viewing distance. Thus, absolute disparity (CABS(H-C)) can be estimated, independent of vergence angle, either from binocular head-centric directions (EL and ER) and distance cues (d ¶) (equations 4 and 5), in reference to the cyclopean eye, or from monocular head-centric directions (?L and ?R) (equations 1 and 2), in reference to the two eyes’ nodal points. Squinters without retinal correspondence could shift their attention from one fixation distance to another and move AR to the new distance without changing convergence. Shifting the zero disparity reference distance (d ¶ ) and its binocular head-centric direction, without changing vergence angle, changes the binocular parallax reference for zero disparity (equation 6). This process is a mechanism that achieves binocular sensory fusion without changing convergence when there is no binocular retinal correspondence.
RESULTS Motor and Perceptual Phenomena That Characterize Anomalous Correspondence Several perceptual and motor phenomena characterize anomalous correspondence, and more of them are captured by the framework of the head-centric than the retino-centric model. Motor phenomena include harmonious anomalous correspondence, covariation of spontaneous variations in the objective angle of strabismus and the angle of anomaly, and slow anomalous vergence movements in response to base-out prisms.21,22 Perceptual phenomena include spontaneous changes in the angle of anomaly that are uncorrelated with changes in vergence angle, binocular sensory fusion of prism-induced disparity without vergence eye movements, and changes in the angle of anomaly in response to prism-induced disparity without vergence eye movements.23Y27 It is unlikely that any of the motor phenomena are consistent with the ARC model because adaptable shifts in retinal correspondence would be too slow and beyond a critical developmental period for binocular vision to account for rapid changes in retinal
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548 Model of Anomalous Binocular CorrespondenceVSchor
FIGURE 3. An example of a right esotropia of amplitude R, with harmonious anomalous correspondence, is illustrated for the retino-centric (left panel) and headcentric (right panel) models. In the left panel, a fixation target A is viewed at a remote distance, and T is a near target imaged on the two foveas. The left fovea corresponds with a nonfoveal nasal retinal locus in the right deviating eye (X), which is equal to the angle of anomaly (F) and R. The retino-centric geometric horopter is represented by the gray circle (green in the online version) that passes through the fixation target A. The near target T subtends a binocular parallax angle of A¶. The absolute disparity of target T (CABS(RET)) is computed from the difference between A¶ and AR. The right panel interprets the same example with the head-centric model. The gray circle (green in the online version) represents the head-centric geometric horopter, and its diameter is independent of vergence angle. AR is computed from monocular estimates of viewing distance (dA¶) and azimuth (EA = 0, not shown) (equation 6). The estimate of the binocular parallax angle (A¶) subtended by an individual point in the visual field (e.g., T at estimated distance dT¶ and azimuth ET¶) is computed from the angular difference between monocular head-centric directions (?L and ?R) (equation 2). Absolute binocular disparity of T (CABS(HC)) is computed from the difference between A¶ and AR. In this example, CABS(HC) of T is the angle of anomaly that is equal to R. The binocular parallax angle for the near target T (A¶) is greater than AR, and the target T subtends a crossed absolute head-centric disparity with the reference fixation target A (equation 1). If T became the fixation target by shifting attention without changing convergence, then the binocular parallax estimate of T would become AR and the binocular parallax estimate of A would be A¶ and the angle of anomaly would equal zero. The geometric horopter would be represented by the black circle and target A would subtend an uncrossed absolute disparity of R. A color version of this figure is available online at www.optvissci.com.
correspondence associated with vergence eye movements. As in normal retinal correspondence, binocular disparity is computed in ARC before eye position information can influence perceived direction or the disparity of foveal afterimages. In the retino-centric correspondence model, vergence eye movements would move retinal images of the fixation target away from corresponding retinal points and produce disparities that disrupt binocular sensory fusion; that is, vergence movements would produce diplopia, just as they do in NRC. Vergence movements interact differently with retinal image position and perceived direction in the absence of retinal correspondence. The registered monocular components of vergence interact
with retinal image position of each eye separately to produce direction constancy and single percepts of the fixation target during vergence-induced movement of the two retinal images. Consequently, the vergence signal that produces the strabismus causes the fixation target to remain single during retinal image movement and foveal afterimages to appear separated by the angle of anomaly that is equal to the objective angle of strabismus in a prevalent condition known as harmonious anomalous correspondence.21,22 Similarly spontaneous variations of the objective angle of strabismus covary with the angle of anomaly such that the fixation target continues to appear single in spite of changes in retinal image position caused by the vergence fluctuations.
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However, covariation is not perfect. Hallden13 noted that the amplitude of vergence change was not always equal to the changes in the angle of anomaly. He proposed that a diplopia mechanism stimulated a change in the angle of anomaly, independently of vergence, and resulted in perceived changes in disparity. The headcentric model interprets the source for spontaneous changes in disparity as spontaneous variations or noise of the estimated reference distance for zero disparity. The resulting disparity variations are independent of the noise variations of the objective angle of strabismus. The combination of these two motor and perceptual noise sources results in incomplete sensory fusion of the retinal disparity associated with vergence fluctuations. Another perceptual phenomenon that characterizes anomalous correspondence is the fusion of a prism-induced disparity of the fixation target that is independent of vergence eye movements. When the base-out prism is first introduced, the fixation target appears fused, and subsequently, there is a slow anomalous convergence response that can last from minutes to days to complete.23Y25 Fusion is experienced before, during, and after the vergence movements, without suppression of either eye. The sensory fusion (diplopia) mechanism achieves fusion by shifting the binocular parallax reference for zero disparity to equal the binocular parallax angle subtended by the fixation target viewed through the prism. During these changes in the attended fixation distance, there are changes in the empirically measured angle of anomaly such that the base-out prism causes foveal afterimages to appear to diverge or move in the direction of uncrossed disparity.25 Then, as slow anomalous convergence occurs, the afterimages are perceived to move slowly toward one another to reduce their uncrossed disparity.25 If diplopia is eliminated by shifts in the binocular parallax reference for zero disparity, then why is there a slow vergence response? It is possible that the anomalous vergence response may be an attempt to resolve a stimulus conflict to distance from motor cues of vergence and perceptual cues from binocular parallax. When the prism is first introduced, the binocular parallax subtended by the fixation target changes by the disparity of the prism. When the reference for zero disparity is shifted to the new binocular parallax angle without changing vergence, there is a cue conflict between these two distance cues. The slow vergence changes appear to be an attempt to match the vergence posture and perceptual cues to distance (i.e., estimated binocular parallax). However, it is also possible that the slow vergence changes may be a slow motor response that attempts to place the images of the fixation target on retinal locations that are habitually stimulated. Finally, there are changes in the angle of anomaly that depend on the complexity of the visual field and lighting. Anomalous correspondence tends to be harmonious when viewing natural scenes and unharmonious when viewing blank fields or in darkness. This may be related to the accuracy of distance estimates under these conditions, such as near biases of targets in darkness, known as the specific distance tendency, or perhaps a greater reliance on vergence angle as a distance cue in impoverished visual fields.28Y30
This project was supported by National Science Foundation grant BCS0715076 (Disparity Processing in Sensory and Motor Functions). The author thanks Dr. Pia Hoenig of the University of California at Berkeley, School of Optometry, for sharing her clinical observations of strabismus with anomalous correspondence; Dr. Martin Banks for his suggestions; Dr. Sangeetha Metapally for her comments on the paper organization; Dr. Nance Wilson for editing; Dr. Jonathan Horton for pointing out the historical reference to Wells 9; and the late Dr. Merton Flom for their conversations about his horopter research. Received August 25, 2014; accepted February 12, 2015.
Spatial Distortions of the Empirical Horopter in Head-Centric Correspondence
SUPPLEMENTAL DIGITAL CONTENT
There are several other phenomena associated with anomalous correspondence that require additional modeling. These include
A more detailed text of the same title with a glossary of clinical terms is available online at http://links.lww.com/OPX/A209.
shape differences between the empirical and theoretical-geometric horopter described by a central (foveal) concave notch and steep elliptical curvature deviation of the empirical from the circular geometric horopter.19,20 These spatial variations are likely to result from errors of estimated azimuth, that is, binocular head-centric direction (E), that are used to compute the binocular parallax reference for zero disparity (AR) (equation 6). The errors in perceived azimuth in the region of the notch could be caused by displacement of visual directions of objects imaged in suppression scotomas toward the fovea.31,32 The influence of errors of visual directions (E) on the curvature of the empirical horopter was first proposed by Ogle33 who described the differences in elliptical curvature of the normal empirical horopter from the circular geometric horopter as a consequence of monocular spatial distortions known as the Kundt and Munsterberg asymmetries.
CONCLUSIONS Anomalous binocular correspondence is modeled as a consequence of an absence of binocular retinal correspondence that allows monocular components of vergence eye position signals to interact with each eye separately to influence perceived direction and binocular disparity in head-centric coordinates as it does in lower vertebrates (Walls).34 Anomalous correspondence is not necessarily an adaptation of retinal correspondence to strabismus but rather a highly evolved form of binocular vision seen in lateral eyed animals. In some cases of strabismus, retinal correspondence may have been lost because of inhibition or suppression of binocular cells, and an effective treatment could be to break down the inhibition using antisuppression techniques that are intended to establish binocular triplopia and eventually normal correspondence between the two foveas. The binocular triplopia represents the coexistence of normal retinal and anomalous head-centric correspondence.35,36 If the eyes were physically aligned, either with surgery or orthoptics, then binocular inhibition could be eliminated to manifest an innate system of normal binocular retinal correspondence. Even without binocular retinal correspondence, voluntary foveal alignment of the two eyes would engage the motor innervation mechanism for fusion and result in harmonious anomalous correspondence, bifoveal fusion, and binocular correspondence would appear to be normal.
ACKNOWLEDGMENTS
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Clifton Schor School of Optometry University of California, Berkeley Berkeley, CA 94720 e-mail: [email protected]
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