Brain, Vol. 122, No. 1, 131-140,
January 1999
© 1999 Oxford University Press
Unilateral neglect and disambiguation of the Necker cube
1 Dipartimento di Psicologia, Università di Torino and 2 Divisione di Medicina Riabilitativa Ospedale Valduce, Lecco, Italy
Correspondence to:
Professor E. Bisiach, 22070 Lurago Marinone, Italy
| Abstract |
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Three groups of patients (right brain-damaged patients with or without left neglect, and left brain-damaged patients) and a group of healthy subjects, matched for age and educational level to the three groups of patients, were asked to report which of the two frontal surfaces of Necker cubes oriented in four different ways looked, at first sight, nearer to the viewer. The extent to which, and the way in which, disambiguation of the apparent perspective of Necker cubes occurred was found to vary across the four orientations and to be different in left-neglect patients compared with subjects of the other three groups. With normal subjects, the disambiguating factor is suggested to be a disposition to perceive the upper surface, which is nearly orthogonal to the frontal plane, as external to the cube. This would result from a navigation of the observer's spatial attention towards its target along a particular path that is altered in patients suffering from left neglect. It is suggested that comparison of the paths followed by the attentional vectors of normal subjects and left-neglect patients is potentially fruitful for a better understanding of the brain's normal mechanisms of spatial attention and of unresolved issues concerning the perception of the Necker cube.
unilateral neglect; Necker cube; spatial attention
C = control subjects; L = left brain-damaged patients; LANDMARK-M = manual version of the Landmark task; LANDMARK-V = verbal version of the Landmark task; R+ = right brain-damaged patients with left neglect; R- = right brain-damaged patients without neglect
| Introduction |
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Reversible-perspective cubes such as those shown in Fig. 1
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The first steps towards an understanding of the processes underlying the equivocal perception of the Necker cube were made by showing that perspective reversals of ambiguous figures are not prevented by the absence of retinal image motion (Pritchard, 1958
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As might easily be expected, the Necker cube has also been the subject of essays in artificial intelligence (e.g. Simon, 1967
Prompted by the findings of Kawabata and his co-workers, we decided to investigate whether and how pathological (as opposed to experimental) constraints imposed upon spatial attention affect the perception of Necker cubes. We therefore ran an experiment in which normal subjects and brain-damaged patients, including patients suffering from left visuospatial neglect, were asked for instant perceptual judgements about the perspective of Necker cubes such as those shown in
Fig.
1
. In agreement with Kawabata's results, a plausible expectation was that with cubes such as those shown in the first three columns of
Fig.
1
, where the two internal vertices are located on opposite sides with respect to the vertical midline, left-neglect patients would focus their attention on the right-side internal vertex; this being perceived as nearer than the opposite vertex. Consequently, one of the two conflicting perspectives would emerge while perception of the rival perspective would be inhibited. Another plausible expectation, however, was that neglect patients would more frequently judge the right frontal surface of these cubes as nearer, because of a perceptual bias by which the left frontal surface was seen as narrower (Milner and Harvey, 1995
) and therefore more distant. The results of the experiment did not match these expectations and therefore required an ad hoc interpretation that could not be grounded on earlier empirical knowledge about the perception of the Necker cube in normal subjects.
| Method |
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Subjects
Four groups of subjects participated in the experiment. One group (C) comprised 18 healthy subjects (10 male and 8 female) with mean age 53.78 years (SD = 14.19 years) and a mean educational level of 10.17 years (SD = 3.15 years). The other groups comprised 18 right brain-damaged patients with left neglect (R+), 13 right brain-damaged patients free from neglect (R) and 15 left brain-damaged patients (L) free from neglect with the exception of L patient 11. The four groups of subjects did not differ regarding age [F(3,60) = 1.31; P = 0.280] and educational level [F(3,60) = 1.95; P = 0.131]. Demographic and clinical data for brain-damaged patients are reported in Tables 13
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Contralesional neglect was defined on the basis of either (or both) of the following criteria: (i) mean error towards the ipsilesional side in bisecting a series of five 180-mm long and 1-mm thick black horizontal lines exceeding 10 mm (leftwards, i.e. right neglect) or 10 mm (rightwards, i.e. left neglect); (ii) left-side minus right-side omissions on the letter H cancellation task (Diller and Weinberg, 1977
The study was approved by the ethical committee of the Hospital Valduce; informed written consent was obtained from each subject in the presence of a witness.
Stimuli and procedures
The stimuli were 16 Necker cubes shown in four orientations (columns 14 in Fig. 1
). The outlines of the (relatively) frontal surfaces were one red and the other yellow in eight cubes; they were one red and the other one green (as shown in Fig. 1
) in the remainder. All other edges were black. All intersection points were white. The representation of each cube was printed at the centre of a vertical sheet of A4 white paper. The sides of the frontal surfaces measured 4 cm.
The entire set of cubes was shown 10 times, each item being presented at reading distance on a table in front of the subject; each time, the 16 cubes followed each other in a fixed random order. There were two blocked conditions following one another. In the first condition the subject had to name the colour outlining the surface that looked at first sight nearer; in the second condition the subject had to name the colour outlining the surface that looked at first sight farther away.
The R+ patients were also given two versions of the Milner Landmark Task (Milner et al., 1993
). In the verbal response version (LANDMARK-V), 180-mm horizontal lines composed of two segments of different length and printed in different colours (black on the left and red on the right, or vice versa) were shown. In a series of trials, patients had to name the colour of the shorter (or longer) segment. In the manual response version (LANDMARK-M), the lines were black and the subdivision between the left and right segments was marked by a short vertical bar superimposed on the horizontal line. In a series of trials, patients had to point at the shorter (or longer) segment. Two scores were computed from each version of the task: a perceptual bias score and a response bias score. From these scores, inferences can be drawn concerning the relative weights of two (putatively) different factors of neglect in individual patients. Procedural and computational details are reported elsewhere (Bisiach et al., 1998a
). It will be mentioned here only that each score may in principle range from 0 to 100. On the basis of cut-offs drawn from normative data, the following values have been held to indicate the presence of medium to severe perceptual bias and/or response bias (corresponding to left neglect) on the two versions of the task: perceptual bias (LANDMARK-V) > 60.21; response bias (LANDMARK-V) > 52.64; perceptual bias (LANDMARK-M) > 60.15; response bias (LANDMARK-M) > 51.74.
| Results |
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As is evident from Fig. 1
The combined percentages of `nearer' responses given by patients to left surfaces and `farther' responses given by them to right surfaces with cubes 1, 2 and 3, as well as the combined percentages of `nearer' responses given to upper surfaces and `farther' responses given to lower surfaces with cubes 4, are individually reported in
Tables
13![]()
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, in which the cumulative percentages concerning cubes 1, 2 and 3 are also listed. Group means, including the mean for group C, are reported in
Table
4
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All statistical analyses of the results were carried out following angular transformation of percentage values. An ANOVA (analysis of variance) with group as four-level between-subjects factor (C, R+, R and L) and type of cube as three-level within-subject factor (cubes 1, 2 and 3) showed a significant effect of group [F(3,60) = 5.43; P < 0.002]. Post hoc comparisons (Tukey test with significance level set at P < 0.05) showed that the behaviour of R+ patients was significantly different from that of the other three groups, among which no significant differences were found. On average, only in R+ patients was the percentage of identifications of the left surfaces of cubes 1, 2 and 3 as nearer (62.22) significantly different from 50% [t(17) = 3.58; P < 0.002]. The effect of type of cube was also significant [F(2,120) = 94.95; P < 0.0001], as was the group x type of cube interaction [F(6,120) = 3.90; P < 0.001]. One-way ANOVA showed a significant effect of group for cubes 1 [F(3,60) = 3.90; P < 0.02], cubes 2 [F(3,60) = 4.55; P < 0.01] and cubes 3 [F(3,60) = 4.43; P < 0.01]. In post hoc comparisons (Tukey test), significant differences were found between groups L and C and between groups L and R for cubes 1, between groups R+ and C and between groups R+ and L for cubes 2, between groups R+ and C and between groups R+ and R for cubes 3 (cf. means reported in Table 4
Regarding cubes 4, a one-way ANOVA did not show significant intergroup differences. The upper surface was more frequently perceived as nearer by C subjects [t(17) = 3.37; P < 0.005], R+ patients [t(17) = 8.38; P < 0.0001] and L patients [t(14) = 4.93; P < 0.0001], whereas no significant difference between the two competing perceptions was found in R patients.
No significant correlations were found in group R+ between the scores for the Necker cube task and the scores for the bisection and letter H cancellation tasks, or between the former and the length of illness. The frequency with which the left surface of cubes 3 was perceived as nearer by R+ patients was instead found to be positively correlated with perceptual bias (LANDMARK-M) [r(18) = 0.59; P < 0.01] and negatively correlated with response bias (LANDMARK-V) [r(18) = -0.53; P < 0.05]. The frequency with which these patients perceived the upper surface of cubes 4 as nearer was found to be positively correlated with response bias (LANDMARK-M) [r(18) = 0.47; P < 0.05]. The frequency with which R+ patients perceived the left surface of cubes 2 as nearer also showed a slight positive correlation with response bias (LANDMARK-M) approaching significance [r(18) = 0.43; P = 0.076].
Inspection of
Table
1
does not reveal any particular relationships between the behaviour of individual R+ patients on the Necker cube task and the intrahemispheric location of the lesion.
No correlation analysis was performed in order to investigate the relationships between collateral tasks (bisection, letter H cancellation and Landmark tests) because data about these relationships are already available from much larger groups of neglect patients (Bisiach et al., 1998a
, b
).
| Interpretation |
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The main results of the experiment may be summarized as follows (Table 4
Since these results do not confirm the expectations set out in the Introduction, an alternative, tentative interpretation is presented as follows.
Early perceptual processing of two-dimensional visual stimuli such as those shown in
Fig.
2
leads to a `cognitively impenetrable' (Pylyshyn, 1985
)i.e. incorrigibleinterpretation of each of those configurations as the perspective of a three-dimensional object. The binary perceptual reversibility of this perspective is itself, to a large extent, cognitively impenetrable: no matter how strenuously a normal observer endeavours to entertain only one of the two competing and mutually exclusive perspectives, the other one pops out at variable intervals and for variable time. However, our results demonstrate that the ambiguity of Necker cubes such as those shown in
Fig.
2
is not absolute, because at least in some instances (cubes 1, 2 and 4) the perceptual judgement of normal subjects is at first sight biased towards one of the two alternative interpretations. The literature on the Necker cube does not seem to offer any clue to the understanding of this bias, except for the results reported by Kawabata and his co-workers (see Introduction), which, as we shall presently see, give insufficient grounds for an exhaustive explanation of our results. If so, the only choice with which we are left is to start with a speculative working hypothesis and see how it fits the data. We may therefore tentatively assume that, once early perceptual processing presents the flat configuration of a Necker cube as a three-dimensional object compatible with either of two contrasting perspectives, the disambiguation is carried out by the dynamics of a complex attentional vector. Such a vector cannot be viewed as leading straight ahead from the observer to the visual stimulus as a whole, or to any of its details, because there seems to be no way in which the disambiguation could in this case be obtained under free viewing conditions (i.e. if not artificially constrained, e.g. by means such as those contrived by Kawabata and his co-workers). Consequently, we have to envisage an attentional vector impacting on the critical stimulus through a different route. Given that in our environment most (relatively small) three-dimensional objects, such as those (ambiguously) represented by Necker-cube drawings, lie on a supporting surface parallel to the ground and are most frequently viewed from above, we suggest that in the conditions of our experiment the final trajectory of the attentional vector of normal subjects (vector C in Fig. 2
) is perpendicularly directed from above towards the centre of the flat configuration after it has been interpreted as the centre of a (still ambiguous) cube. The first surface on which the vector impacts is therefore interpreted as a surface external to the cube.
On this hypothesis, we would indeed expect what is shown in
Fig.
2
. With cubes 1 and 2, the vector impacts on surface AEFB. This surface being interpreted as external, the left frontal surface ABCD of cube 1 is more frequently (83.47%) and the left frontal surface EFGH of cube 2 less frequently (22.50%) perceived as nearer to the observer. With cube 3, it impacts on the ambiguous intersection of surfaces ABFE and BFGC; the left frontal surface ABCD is therefore nearly as frequently perceived as nearer (46.25) or farther. With cube 4, it impacts on the upper frontal surface ABCD (standing out due to perceptual grouping on the basis of colour), which is therefore more frequently perceived as nearer (70.97%).
It is logically evident (and easy, though time-consuming, to check) that a rotation of the C vector in the frontal plane around the centre of cubes 14 would lead to predictions incompatible with the results of the experiment.
Regarding the left-neglect patients, their behaviour can be interpreted by assuming that, as a consequence of the brain lesion, the normal final trajectory of the attentional vector, though remaining in the frontal plane where the centre of the ambiguous cube is apparently located, is to some extent (by 45° in Fig. 2
) bent to the right. With cube 1, the R+ vector would thus impact on the edge connecting surfaces AEFB and BFGC; since these surfaces are interpreted by the brain as external to the cube, the left frontal surface ABCD is still more frequently perceived as nearer (76.53%). With cube 2 it would impact on the ambiguous intersection of surfaces AEFB and FBCG, so that the left frontal surface EFGH, which looked farther to normal subjects, is nearly as frequently perceived nearer (46.53%) as farther. With cube 3, it would impact on surface BFGC; this surface being interpreted as external, left-neglect patients (unlike normal subjects) perceive the left frontal surface ABCD more frequently as nearer (63.61). Finally, with cube 4, the R+ vector would impact, like the C vector, on the upper frontal surface ABCD, which is therefore also more frequently perceived as nearer (79.86%) by left-neglect patients.
With cubes 1 and 3, the findings of Kawabata could support an alternative and very plausible explanation of our patients' behaviour. If, due to contralesional visual neglect, attention had been focused on the internal vertex B, rather than having been divided between it and the internal vertex H, with cube 1, and on the internal vertex C, rather than having been divided between it and the internal vertex E, with cube 3, then the attended vertices (and therefore surfaces ABCD) would have appeared nearer to the viewer.
Figure
2
shows that this was indeed what we found. However, focusing attention on vertex G with cube 2 would have led to more frequent perception of left frontal surface EFGH as nearer, which was not the case. Furthermore, left neglect would not have biased attention on either internal vertex with cube 4, since both vertices F and D lie on the same vertical line; contrary to our results, no disambiguation would in this case be entailed. On the other hand, the alternative expectation of more frequent choices of right frontal surfaces as nearer, due to the perceptual bias mentioned in the introduction, is totally incompatible with the actual results obtained with each type of cube.
| Discussion |
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According to the above interpretation, the results of the experiment support the view that both in normal subjects and in left-neglect patients the dynamics of spatial attention required for the disambiguation of visual patterns such as the Necker cube should be conceived in terms of a mental analogue of three-dimensional navigation in real space rather than in terms of a beam of straight vectors leading directly from the viewer to configurational details. If the second alternative were the correct one, the two rival percepts of any variant of the Necker cube would (in the absence of any experimental constraints of the kind we mentioned in the introduction) be equiprobable.
What is suggested here is in some way similar to Gibson's (1966, p. 266286) concept of `affordance'. If, for example, the perspective of cube 1 in
Fig.
2
were slightly changed so that vertices B and H were made to coincide, the configuration would be much more likely to be perceived as the flat surface of a hexagon radially subdivided into six sectors. As soon as these vertices move away from one another, however, the configuration is perceived as a three-dimensional transparent object `inviting' potential actionssuch as inspecting, reaching and graspingthat are instrumental in normal perception, if the latter is held to occur `only when a viewer moves about in a natural environment' (Hochberg, 1986
, p. 290). Two incompatible sets of actions are thereby set in competition by the perspective ambiguity of the configuration. Which set wins depends on further attentional processes specifying how the observer would relate to the object if it were a three-dimensional object.
In interpreting the results of the experiment we have assumed that the attentional vector aimed at disambiguating the Necker cube reaches it from above in normal subjects, but obliquely from above and the right in left-neglect patients. We have also assumed that the way the Necker cube is perceived depends on which surface the vector first encounters and treats as external. Former interpretations have instead focused on the role of vertices (e.g. Simon, 1967
; Kawabata, 1986
; Feldman, 1985
; Rumelhart et al., 1986
; Enright, 1987
). However, there seems to be no parsimonious way in which our results might be accounted for by hypothesizing vertices or edges, rather than surfaces, as foci of (normal or pathological) spatial attention.
On our interpretation, the final trajectory of the R+ vector of spatial attention, rather than being perpendicular to the centre of a Necker cube, as in normal subjects, is bent towards the contralesional side. (It is worth noting that a slight, similar tendency regarding the vector of L patients is suggested by their perceiving the left surface of cubes 1 as nearer significantly less often than C subjects.) Why should this happen? It has been suggested that the metrics of space representation are contingent upon the equilibrium emerging in a widespread neuronal network of space-coding neurons (such as those investigated by, for example, Graziano and Gross, 1994
; Galletti et al., 1995
; Fogassi et al., 1996
) from a system of functional counterforts. Unilateral brain damage could result in a one-sided lack of counterpoise in such a system and lead to anisometry of the medium for space representation (Bisiach et al., 1998b
). This would imply a leftright skewness of the dynamics of spatial attention, giving rise to contralesional neglect and related phenomena. This might also be the reason why the attentional vector implied in the disambiguation of the Necker cube, being initially pushed rightwards in R+ patients (Kinsbourne, 1970
), must turn towards its target from right to left and impact (as shown in Fig. 2
) on the surface(s) approachable by its final trajectory and therefore more prone to being perceived as external to the apparently three-dimensional object. (We obviously refer to the dynamics of `covert' attention, since we expect that nothing would change in R+ patients following retinal fixation of the ambiguous figures by using, for example, the after-image technique.) As a consequence, the left frontal surfaces of cubes 1, 2 and 3, far from being neglected, are (cumulatively) more frequently perceived as nearer (or `showing') rather than farther (or `hidden').
It is hardly possible, so far, to make well-grounded conjectures about the precise nature of the brain dysfunction leading to the particular disambiguation of the Necker cube that has been found to be associated with right hemisphere damage giving rise to neglect. Nonetheless, it would be unjustified to totally disregard the clues offered, no matter how problematically, by the pattern of correlations between the behaviour of left-neglect patients on the Necker cube task and the Landmark tasks. Although fairly complicated, the results of our correlational analyses do indeed suggest that both perceptual and response-related biases could be responsible for the ways patients with left neglect, taken as a group, disambiguate variants of the Necker cube such as those used in the experiment. This is far from surprising, because the results of an extensive investigation of the Landmark task (Bisiach et al., 1998a
) show that perceptual and response biases, though double-dissociated, are not mutually exclusive and may co-determine the occurrence of neglect phenomena. Further work on single cases is, of course, required to clarify the respective contributions of perceptual and response biases in the disambiguation of the Necker cube by left-neglect patients; however, to the extent to which response-related factors may be held to reveal a premotor dysfunction, their implication in the mechanisms through which the Necker cube is disambiguated by left-neglect patients supports the view according to which these mechanisms are partly based (in these patients, as well as in normal subjects) on what the flat, but apparently three-dimensional, configuration affords for potential actions at the level of mental representation.
We were not able to find a satisfactory explanation for the lack of correlation between the results of the Necker cube task and the results of bisection and cancellation tasks, which is a vexing paradox since line bisection and letter H cancellation were in fact chosen as criteria for the selection of our group of neglect patients. More than ever, one is apparently forced to share Halligan and Marshall's (1992) scepticism regarding the `validity of any unitary concept' of unilateral visuospatial neglect. However, the different pattern of correlation of the results of bisection and cancellation tasks with perceptual and response bias scores on the verbal and manual versions of the Landmark test (Bisiach et al., 1998a
) suggests that the lack of correlation between the former and the Necker cube task is due to reciprocal masking of two different factors of unilateral neglect affecting the way patients perceive this ambiguous configuration: one more easily revealed by bisection errors, and the other by the inability to carry out the step-after-step visual scanning of a complex stimulus array.
| Conclusions |
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The results of the experiment were not as we expected on the basis of earlier empirical knowledge about the perception of the Necker cube by normal subjects and knowledge of the attentional disorders giving rise to unilateral visuospatial neglect. The interpretation we offer is tentative and invites further investigation, which may or may not support it. In any case, the results of the experiment are challenging vis à vis one of the oldestand still fascinatingissues of perceptual psychology. Furthermore, they strongly suggest that tasks of uni- or bidimensional perception such as those traditionally employed in the study of unilateral neglect should be supplemented by tasks based on the perception of (really or apparently) three-dimensional stimuli if we want to gain deeper insight into the attentional dynamics underlying this disorder and consequently into the dynamics of spatial attention in the normal brain.
| Acknowledgments |
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This paper has benefited greatly from constructive criticism of an earlier draft provided by two anonymous referees. The research was supported by CNR and MURST grants to the first author.
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Received August 3, 1998. Accepted August 24, 1998.
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