The bisection point across variants of the task

Bisection tasks are used in research on normal space and time perception and to assess the perceptual distortions accompanying neurological disorders. Several variants of the bisection task are used, which often yield inconsistent results, prompting the question of which variant is most dependable and which results are to be trusted. We addressed this question using theoretical and experimental approaches. Theoretical performance in bisection tasks is derived from a general model of psychophysical performance that includes sensory components and decisional processes. The model predicts how performance should differ across variants of the task, even when the sensory component is fixed. To test these predictions, data were collected in a within-subjects study with several variants of a spatial bisection task, including a two-response variant in which observers indicated whether a line was transected to the right or left of the midpoint, a three-response variant (which included the additional option to respond “midpoint”), and a paired-comparison variant of the three-response format. The data supported the model predictions, revealing that estimated bisection points were least dependable with the two-response variant, because this format confounds perceptual and decisional influences. Only the three-response paired-comparison format can separate out these influences. Implications for research in basic and clinical fields are discussed.     

Errors towards the perpendicular in children's copies of angular figures: a test of the bisection interpretation

Children distort angular figures so that the constituent angles are nearer 90 degrees than they should be. This could be due to a perpendicular bias or a bisection bias, or to both. A study is reported which was designed to establish whether a perpendicular bias would appear independently of bisection. Twenty four-year-old children were tested on two types of angular figure: (i) a baseline with another line joining at the end at 45 degrees, 90 degrees, or 135 degrees and (ii) a baseline with another line joining at the middle at 45 degrees or 90 degrees. Perpendicular errors were obtained both for 'end' and for 'middle' figures, but overall more strongly for 'middle' figures. However, while 90 degrees 'middle' figures were copied more accurately than 45 degrees/135 degrees figures, this effect was only obtained for vertical and horizontal presentations of 'end' figures, and was reversed for oblique presentations. Also, for 'end' figures, directional errors varied with subtended-line orientation, whereas for 'middle' figures they varied with baseline orientation. It is concluded that although errors towards the perpendicular do occur with single-angle figures, angle equalisation may take place when there are two adjacent angles in the figure. One interpretation of the differing orientation effects is that in 'middle' figures strong internal relational forces produce a distortion that varies with the angle at which the figure is viewed, whereas in 'end' figures the absence of relational forces within the figure leads to a stronger influence from external cues.     

The bisection of a spatial interval by the pigeon

Pigeons pecked two response keys to move a white dot until it was equidistant from two other dots on a screen. Continuous records of dot position showed the effects of reinforcement and stimulus parameters upon the accuracy with which the dot was positioned. The method may prove useful for studying the perception of distance in non-human organisms.     

Cognition in the bisection task

Bisecting a visually presented stimulus is a sensitive test for attentional and motor biases in both healthy and brain-damaged participants. There are, in addition, perceptual effects on performance and several idiosyncratic biases. Experiments that manipulated the stimulus attributes have suggested that higher-level cognitive processes can also influence bisection accuracy. These observations encourage a wider diagnostic use of the bisection procedure.     

The bisection of a brightness interval by pigeons

Pigeons were trained on a discrete trials, successive discrimination procedure, in which the stimuli were two luminance values on the center key. Behavior was maintained by 25% reinforcement of correct responses on two side-keys. During occasional test trials the luminance of the center key was maintained at one of a number of values, intermediate to those of the two training stimuli, and a function relating the relative frequency of responses on the two side keys to stimulus intensity was obtained. The intersection of this function with the 50% line provided an estimate of the bisection point. Since no bisection point occurred below the geometric mean of the interval, the results were not consistent with a logarithmic scale of brightness but fitted the general mean theorem with an exponent of 0.24. With continued testing, the performance of individual subjects oscillated in an irregular manner about the mean bisection point. The relative stability of the test behavior and the absence of context effects indicated that the method was suitable as a general procedure for measuring stimulus distances.     

Geometric and perceptual effects of the location of the observer vantage point for linear-perspective images

New geometric analyses are presented of three impressive examples of the effects of location of the vantage point on virtual 3-D spaces conveyed by linear-perspective images. In the 'egocentric-road' effect, the perceived direction of the depicted road is always pointed towards the observer, for any position of the vantage point. It is shown that perspective images of real-observer-aimed roads are characterised by a specific, simple pattern of projected side lines. Given that pattern, the position of the observer, and certain assumptions and perspective arguments, the perceived direction of the virtual road towards the observer can be predicted. In the 'skewed balcony' and the 'collapsing ceiling' effects, the position of the vantage point affects the impression of alignment of the virtual architecture conveyed by large-scale illusionistic paintings and the real architecture surrounding them. It is shown that the dislocation of the vantage point away from the viewing position prescribed by the perspective construction induces a mismatch between the painted vanishing point of elements in the picture and the real vanishing point of corresponding elements of the actual architecture. This mismatch of vanishing points provides visual information that the elements of the two architectures are not mutually parallel.     

Human performance on the temporal bisection task

The perception and processing of temporal information are tasks the brain must continuously perform. These include measuring the duration of stimuli, storing duration information in memory, recalling such memories, and comparing two durations. How the brain accomplishes these tasks, however, is still open for debate. The temporal bisection task, which requires subjects to compare temporal stimuli to durations held in memory, is perfectly suited to address these questions. Here we perform a meta-analysis of human performance on the temporal bisection task collected from 148 experiments spread across 18 independent studies. With this expanded data set we are able to show that human performance on this task contains a number of significant peculiarities, which in total no single model yet proposed has been able to explain. Here we present a simple 2-step decision model that is capable of explaining all the idiosyncrasies seen in the data.     

Pseudoneglect for the bisection of mental number lines

Patients with unilateral neglect of the left side bisect physical lines to the right whereas individuals with an intact brain bisect lines slightly to the left (pseudoneglect). Similarly, for mental number lines, which are arranged in a left-to-right ascending sequence, neglect patients bisect to the right. This study determined whether individuals with an intact brain show pseudoneglect for mental number lines. In Experiment 1, participants were presented with visual number triplets (e.g., 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Despite changing the spatial configuration of the stimuli, or their temporal order, the numerical length on the left was consistently overestimated. The fact that the bias was unaffected by physical stimulus changes demonstrates that the bias is based on a mental representation. The leftward bias was also observed for sets of negative numbers (Experiment 2)—demonstrating not only that the number line extends into negative space but also that the bias is not the result of an arithmetic distortion caused by logarithmic scaling. The leftward bias could be caused by a rounding-down effect. Using numbers that were prone to large or small rounding-down errors, Experiment 3 showed no effect of rounding down. The task demands were changed in Experiment 4 so that participants determined whether the inner number was the true arithmetic centre or not. Participants mistook inner numbers shifted to the left to be the true numerical centre—reflecting leftward overestimation. The task was applied to 3 patients with right parietal damage with severe, moderate, or no spatial neglect (Experiment 5). A rightward bias was observed, which depended on the severity of neglect symptoms. Together, the data demonstrate a reliable and robust leftward bias for mental number line bisection, which reverses in clinical neglect. The bias mirrors pseudoneglect for physical lines and most likely reflects an expansion of the space occupied by lower numbers on the left side of the line and a contraction of space for higher numbers located on the right.     

On the theory of biased bisection operations and their inverses

A theory of fundamental measurement involving multiple operations is presented. The operations are bisection, and two inverses of the bisection operation called right and left displacement, analogous to “displacement” of an interval “up” or “down” a scale. Following Pfanzagl, “biased” operations are permitted, and linear, reflexive representations are presented for the biased operations. Results are obtained on the constructability of unbiased scaling operations, and several methods of constructing standard sequences of equally spaced stimuli using biased operations, are described. A striking property of biased linear representation models is derived, showing that if bias is present, indefinite iterations of one or the other of the two displacement operations will converge to asymptote. Preliminary data support this surprising convergence property.     

Pseudoneglect for the bisection of mental number lines

Patients with unilateral neglect of the left side bisect physical lines to the right whereas individuals with an intact brain bisect lines slightly to the left (pseudoneglect). Similarly, for mental number lines, which are arranged in a left-to-right ascending sequence, neglect patients bisect to the right. This study determined whether individuals with an intact brain show pseudoneglect for mental number lines. In Experiment 1, participants were presented with visual number triplets (e.g., 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Despite changing the spatial configuration of the stimuli, or their temporal order, the numerical length on the left was consistently overestimated. The fact that the bias was unaffected by physical stimulus changes demonstrates that the bias is based on a mental representation. The leftward bias was also observed for sets of negative numbers (Experiment 2)--demonstrating not only that the number line extends into negative space but also that the bias is not the result of an arithmetic distortion caused by logarithmic scaling. The leftward bias could be caused by a rounding-down effect. Using numbers that were prone to large or small rounding-down errors, Experiment 3 showed no effect of rounding down. The task demands were changed in Experiment 4 so that participants determined whether the inner number was the true arithmetic centre or not. Participants mistook inner numbers shifted to the left to be the true numerical centre--reflecting leftward overestimation. The task was applied to 3 patients with right parietal damage with severe, moderate, or no spatial neglect (Experiment 5). A rightward bias was observed, which depended on the severity of neglect symptoms. Together, the data demonstrate a reliable and robust leftward bias for mental number line bisection, which reverses in clinical neglect. The bias mirrors pseudoneglect for physical lines and most likely reflects an expansion of the space occupied by lower numbers on the left side of the line and a contraction of space for higher numbers located on the right.     

Invariance tests for bisection and fractionation scaling

If we are to accept the power function as the psychophysical law, then estimates of the exponent of a particular form of the power law should be independent of changes in basic independent variables. In the present study, various power law hypotheses were tested for the bisection and fractionation scaling of brightness. The results indicated that for bisection, estimates of exponents were dependent on the particular interval bisected, suggesting rejection of the simple power law for bisection. For fractionation, two additional forms of the power law were tested, each form involving a threshold parameter. One was the Φ-law (involving a translation on the intensity axis), and the other was the ψ-law (involving a translation on the psychological axis). The Φ-law provided a poor fit to the data, whereas the ψ-law appeared to fit well for at least one S. Analysis for individuals showed that for all five Ss, the variance due to standards was appreciably larger for the Φ-law.     

Vertical and bisection bias in active touch

We investigated the conditions that underlie the vertical and bisection illusion in touch, in order to understand the basis of their similarity to visual illusions, and the means of reducing the biases in length perception by active touch. Movement, speed, and spatial reference cues were tested. Movements in scanning L-shapes in ipsilateral and contralateral (across the body midline) table-top space produced significant underestimation of the vertical line with the right hand, but not with the left hand. Right-handed scanning of L-shapes showed no significant bias when the vertical line in the figure was aligned to the body midline, suggesting that spatial cues were involved. The vertical line was overestimated in inverted T-shapes, but underestimated in rotated T-shapes, implicating line bisection. Holding scanning latencies constant reduced the vertical error for inverted T-shapes, but could not explain the bisection bias. Sectioning biases were predicted by the location of junctions on sectioned lines, showing that junction points act as misleading anchor cues for movement extents. The illusion was significantly reduced when reference information was added by instructing subjects to relate two-handed scanning of the figure to an external frame and to body-centred cues. It is argued that disparities in spatial reference (anchor) cues for movement extents are involved in vertical and bisection biases in active touch. The hypothesis that length illusions depend on disparities in spatial reference information can also account for the similarity of the tactile to the visual horizontal-vertical illusion.     

The effects of stimulus size on bisection judgements in near and far space

Primate data suggest that near (peripersonal) and far (extrapersonal) space are coded within distinct representations. Support for this claim has been gained from human studies of line bisection, many of which have focused on neuropsychological, rather than normative, samples. One important aspect of these bisection studies has been to control for the changes in angular extent of stimuli that normally accompany changes in viewing distance. The control of angular information, however, requires alterations in the linear dimensions (actual stimulus size) of stimuli. We report two experiments in which normal subjects made manual bisection judgements on stimuli positioned in near or far space, and which were oriented in either the left-right (Experiment 1) or radial plane (Experiment 2). Both experiments were designed to enable the separable effects of linear and angular extent to be disentangled. Viewing distance effects were obtained when angular information was controlled, but many of these were dependent on changes in linear extent, and were only apparent at the individual subject level. Our data confirm that genuine near/far effects may be observed in normative bisection, but that many previous studies which appeared to support a near/far distinction in both normal and brain-damaged bisection behaviour may reflect a failure to control for changes in stimulus size.     

The modulation of haptic line bisection by a visual illusion and optokinetic stimulation

Research has shown that a variety of different sensory manipulations, including visual illusions, transcutaneous nerve stimulation, vestibular caloric stimulation, optokinetic stimulation, and prism adaptation, can all influence people's performance on spatial tasks such as line bisection. It has been suggested that these manipulations may act upon the 'higher-order' levels of representation used to code spatial information. We investigated whether we could influence haptic line bisection in normal participants crossmodally by varying the visual background that participants viewed. In experiment 1, participants haptically bisected wooden rods while looking at a variant of the Oppel - Kundt visual illusion. Haptic-bisection judgments were influenced by the orientation of the visual illusion (in line with previous unimodal visual findings). In experiment 2, haptic-bisection judgments were also influenced by the presence of a leftward or rightward moving visual background. In experiments 3 and 4, the position of the to-be-bisected stimuli was varied with respect to the participant's body midline. The results confirmed an effect of optokinetic stimulation, but not of the Oppel -Kundt illusion, on participants' tactile-bisection errors, suggesting that the two manipulations might differentially affect haptic processing. Taken together, these results suggest that the 'higher-order' levels of spatial representation upon which perceptual judgments and/or motor responses are made may have multisensory or amodal characteristics.     

Voluntary manual exploration and vision in rod bisection

Two experiments were carried out to investigate contributions of voluntary manual exploration and vision to rod-bisection performance when the rod was placed in different spatial positions. Analysis showed both voluntary manual exploration and vision contributed to the accuracy of rod-bisection performance in healthy adults (Exp. 1: 12 men, 13 women; Exp. 2: 10 men, 11 women). Advantages of voluntary manual exploration were related to rod position and initial search direction. Results are discussed with regard to ecological approach to action and perception, two cortical systems (ventral and dorsal streams) for perception and control of actions, and the hemispheric activation hypothesis.     

Frontal and posterior ERPs related to line bisection

The aim of this study was to investigate the dynamic nature of the cortical visuospatial attention processes during the line bisection test, which is sensitive to perceptual asymmetries. EEGs of 26 normal volunteers were recorded during the administration of a computerized line bisection test, which requires participants mark the midline of lines using a mouse. Two event-related potentials subsequent and time locked to the line presentations, namely, P300 and a positive slow wave, were obtained. Findings suggested that both potentials were related to the test performance, and the right hemisphere was more active. Analysis suggested a right parietotemporal and superior parietal locus for the P300 and right prefrontal activity for the positive slow wave. A dynamic asymmetrical activity was identified, such that after primary visual perception, spatial processing is then initiated in the right parietotemporal cortex and then proceeds to the right prefrontal cortex.     

Failure of additivity in bisection of length

Subjects were intructed to select one rod to lie halfway in length between two given rods. These bisection instructions imply an additive model in the subjective metric. However, the data were inherently nonadditive; the length of the bisector could be an increasing or a decreasing function of the length of one given rod, depending on the length of the other given rod. A convexity analysis and a nonmetric analysis both showed that no monotone transformation could make the data additive. The bisection problem is used to contrast the axiomatic and functional approaches to measurement theory.