Hemi-spatial neglect typically follows focal brain lesions in the parietal cortex or regions interconnected with the parietal cortex[2, 3]. Patients suffering from neglect usually fail to explore the side of space contralateral to their brain lesion site (; for a review see). These spatial manifestations of neglect can also be observed in patients’ everyday behaviour. For instance, patients with left-sided neglect may fail to eat food from the left side of their plate, ignore someone who approaches from their left side or fail to reach for an object on their left. When asked to bisect a physical line, neglect patients tend to indicate the midpoint of the horizontal line to be located to the right of the veridical center, as if they partially ignored the left part of the line. Thereby, line bisection is a simple and common clinical assessment tool for leftward hemi-spatial neglect, which can be used to quantify neglect severity.
In the current study we investigated the number magnitude representation in patients with unilateral leftward neglect. Examining neglect patients is informative about the neurocognition of numbers in general because of the very specific influence of neglect symptoms on the representation of number magnitude. Typically, neglect patients fail to explore stimuli in the contra-lesional (most often left) side of space (see above). Additionally, the metaphor of a left-to-right oriented mental number line (henceforth MNL) is a widely agreed upon conceptualization of human representation of number magnitude (cf.[8–10]). Combining these two approaches, there is evidence suggesting that number magnitude representation in neglect patients may be particularly problematic for the representation of relatively smaller numbers, corresponding to the left part of the MNL (e.g.,, see for a review, but see also for a review on dissociations between neglect in visual and numerical space). In the seminal study by Zorzi and colleagues, patients with leftward neglect were asked to indicate the midpoint of an orally presented number interval without performing mental calculations. The authors reported a significant rightward displacement of the midpoints reported by the neglect patients (e.g., reporting 7 to be the middle between 1 and 9). Thereby, the error pattern for mental number line bisection was virtually identical to the performance pattern, which neglect patients exhibit in physical line bisection (e.g.,). Additionally, as reported by Halligan and Marshall for the case of line bisection, this rightward displacement in number intervals was directly proportional to the width of the to-be-bisected interval: The wider the segment, the more pronounced the rightward displacement.
The displacement of the midpoint to the right exhibited by neglect patients in number interval bisection is usually interpreted to indicate that number magnitude is represented spatially along a mental number line mimicking physical attributes. From the similar effects of neglect on line bisection and bisection of numerical intervals Zorzi et al., concluded that physical and number space are isomorphic in the sense that they are organised according to similar Cartesian coordinates, which are explored by similar and interacting spatial processing mechanisms (such as the MNL) but which are nevertheless dissociable in principle (see). Yet, it is important to emphasize that the neglect patients exhibiting this form of space-based neglect had intact numerical and arithmetical skills. These original findings of Zorzi et al., were replicated and extended in a number of subsequent studies, employing number interval bisection tasks (e.g.,[16–19]) but also other tasks tapping on the magnitude representation of numbers (e.g.,[19–21]). Furthermore, the dissociation between impaired explicit processing and spared implicit processing of number magnitude information corroborates the notion that neglect produces a deficit in accessing an intact MNL, rather than a distortion in the representation of the MNL itself.
Taken together, there is converging evidence that neglect patients have specific problems to represent relatively smaller numbers corresponding to the left part of the MNL and in particular so when the task at hand requires explicit processing of number magnitude information. However, recent accounts have claimed that number-processing deficits in right-hemisphere-damaged patients may not be due to neglect, but rather due to a working memory deficit (e.g.,[22, 23]). In particular, it is argued that diverging performance in unilaterally brain damaged patients on tasks such as the number interval bisection task may not necessarily be the consequence of impairments in a spatial-attentional mechanism operating upon a mental number line (isomorphic to physical space), but may be due to an impaired working memory system with a position-specific deficiency. Alternatively, it has been put forward that the deficits in number processing may be due to the fact that the right hemisphere is specialised in processing "small" numbers.
However, there are at least two important limitations. First, previous studies have mostly focused on single-digit numbers or rather small numbers in the two-digit number range (see for a review). Therefore, knowledge about the way the representation of multi-digit numbers is affected in patients with neglect is still rather patchy. Second, the vast majority of previous results on the influence of neglect on number processing focused on the phenomenon of space-based conceptualizations of neglect, despite the fact that there is also empirical evidence suggesting other conceptualizations such as object-based symptoms in neglect. In particular, patients with object-based neglect tend to omit the left half of objects displayed across a scene[25–27], whereas patients with spatial neglect can produce an adequate representation of the right half of a scene presented whilst leaving out figures on the left side. Thereby, object-based neglect poses problems for the analysis of the parts of an object as they contribute to the recognition of the whole[28, 29], while space-based neglect is concerned with the position of an object as a whole, relative to a predefined spatial reference (e.g., the point of fixation, the subject’s midsagittal plane, or another object in the visual field). One of the first examples of object-based neglect was reported in a study of drawing by Gainotti and colleagues. The authors asked their patients to copy a model containing objects located on a horizontal axis (amongst others a house, a fence, trees). Some patients failed to draw the contralateral side of some of the objects from the model. But even though these patients omitted the left side of some objects they continued to reproduce other objects located even further to the left of the first drawing copies (see also[25, 31] for further cases).
Object based neglect in numerical cognition
As outlined above, neglect can also affect processing of the contra-lesional (mostly left) side of an individual object regardless of the spatial position of that object[29, 32, 33]. Therefore, the investigation of multi-digit number processing may be specifically informative concerning the evaluation of an influence of object-based neglect, which has not been evaluated so far in the context of numerical cognition. When conceptualizing a multi-digit number as a coherent object, applying the logic of object-based symptoms of neglect would suggest specific neglect of the left part of a multi-digit number. In case of a two-digit number like 73 this neglect should manifest itself in possibly impaired processing of the leftward tens digit (i.e. 7). Importantly, comparable manifestations of object-based neglect have already been reported for the case of reading. On the one hand, in line with the notion of space-based neglect, neglect patients tend to omit whole words located on the contra-lesional side of a sheet of paper. On the other hand, and of particular importance for the current study, these neglect patients also misread the beginnings of individual words because of omitting the first letters of a word (e.g., ‘word’ misread as ‘ord’,). Such an error pattern is in line with the notion of object-based neglect (e.g.,[29, 32]). Transferring this phenomenon to the case of multi-digit numbers, we assume that object-based aspects of neglect should result in deficient processing of leftward digits within a specific number, such as the tens digit in a two-digit number.
In turn, such deficient processing of, for instance, the tens digit might then lead to impairments in processing tens and units complying with the place-value structure of the Arabic number system because not all digits at all positions to be integrated may be accessed or activated equally well. Yet, it is important to note that this does not necessarily mean that the decade digit is not processed at all. Quite to the contrary, we suggest that processing of the tens digit is less precise rather than omitted completely.
First evidence for such a very specific impairment of place-value processing came from the study by Hoeckner et al.,. In a number bisection task involving two-digit numbers only administered to neglect patients and two control groups, participants were presented with number triplets and asked whether the central number was also the arithmetical middle of the interval (e.g., 21_25_29) or not (e.g., (21_27_29). Performance of neglect patients differed from that of controls in two important respects: Apart from replicating influences of space-based neglect, these patients showed specific difficulties processing place-value information. In particular, they were particularly impaired for triplets crossing a decade boundary (e.g., 35_38_41) compared to triplets within the same decade (e.g., 32_35_38). Importantly, the finding that neglect patients were particularly impaired for triplets crossing a decade boundary is in line with the notion of symptoms of object-based neglect mentioned above (e.g.,[29, 32]) in number processing. One might hypothesize that processing of the left side of fixated objects, for instance the decade digit in a two-digit number, was specifically impaired in neglect patients and that therefore changes in the decade number were particularly difficult to process.
The present study
Conceptualizing the observations of Hoeckner et al., within the framework of space- and object-based neglect is not the only possible account. For both, the hypothesis referring to object-based as well as to space-based neglect, there are counter-arguments. As regards object-based neglect, one might argue that in the study of Hoeckner et al., trials crossing a decade boundary were also generally much more difficult than trials not crossing that boundary, as documented by a high error rate of up to about 40% for the most difficult conditions (see also[36, 37] for the influence of decade crossing in healthy adults). Therefore, Hoeckner et al., may have observed a general difficulty effect.
As outlined above it was suggested recently that the apparent left-sided neglect in number bisection tasks may not necessarily reflect an isomorphism of physical and number space (as proposed by; see for a review) but may be driven by poor memorization of the initial items of the number sequences presented due to impaired working memory ([22, 23]; for a review see). For instance, van Dijck and colleagues used tasks indeed relying heavily on working memory such as indicating the number midway between two verbally presented numbers in a number bisection task to investigate influences of neglect on number processing. Such working memory processes may be even more important for bisection tasks involving more complex and more difficult two-digit numbers. Considering the issue of task difficulty, in the current study we pursued the issue of (impaired) representations of two-digit numbers in leftward neglect systematically employing the much easier number magnitude comparison task (cf.  with single-digit numbers), for which working memory influences should be less pronounced. If nevertheless differential effects can be found for neglect patients when performing such a task, this would also be informative with respect to the current debate on the origin of neglect-related deficits in number processing.
In our study a group of leftward neglect patients and two control groups (a patient control group with right-sided lesions but no neglect and a healthy control group) had to compare two-digit numbers to an internally memorized standard (i.e., 53 or 57). As we were particularly interested in effects of object-based neglect on place-value processing, the unit-decade compatibility effect, as observed in two-digit number magnitude comparison, seemed specifically informative. The unit-decade compatibility effect describes the observation that number pairs such as 32_47, in which separate comparisons of tens and units lead to the same decision (i.e., 3 < 4 and 2 < 7) RT and error rates are lower as compared to number pairs in which comparing tens and units separately yields incompatible decision biases (e.g., 37_52 for which 3 < 5 but 7 > 2) even though overall distance is held constant (i.e., 15 in both examples). As a consequence, the compatibility effect was interpreted to indicate separate and parallel processing of tens and units (e.g., [39–41]; for a review) and thereby an influence of the place-value structure of the Arabic number system. When processes of place-value integration are specifically impaired in neglect patients, especially with regard to the tens digits, a specific modulation of the compatibility effect by neglect may be expected. As the compatibility effect is generally more pronounced for large (e.g., 71_39, with 9 > 1 by 8) as compared to small unit distances (e.g., 75_46, with 6 > 5 by 1, cf.) two standards (i.e., 53 and 57) were used in the current study. This allowed for using an internal standard and unit distances up to 6 as compared to a standard of 55, where the maximum unit distance is 4 (see  for a more detailed discussion of this point). Thereby, we wanted to induce detection of the compatibility effect even in the case of small sample sizes and high variability of performance to be expected in neglect patients.
Objectives and hypotheses
In the study by Hoeckner et al., the number bisection task revealed strong differences between within- and between-decade items in neglect patients. From this pattern of results the authors concluded that neglect patients may have problems to mentally represent the decade digit’s magnitude, which are not due to perceptual impairments in neglect but due to impairments of the mental (spatial) representation of the presented two-digit numbers. Against this background the following more specific hypotheses can be derived.
First, associated with symptoms of object-based neglect, we hypothesized the unit-decade compatibility effect to be more pronounced in neglect patients as compared to controls. Because the tens digits of all two-digit numbers are located on the left, this compatibility effect could be due to possibly slower and less precise access to the magnitude of the tens’ digits in neglect patients. In turn, unimpaired access to the units should then exert a stronger influence on two-digit number processing, resulting in a more pronounced compatibility effect. Generally, the compatibility effect originates from interference between the (right-sided) irrelevant unit digit and the (left-sided) relevant decade digit, when these two digits are incompatible. Therefore, the stronger the focus on the irrelevant unit digit (for instance driven by neglecting the left-sided tens digit), the stronger the compatibility effect. Even more, in incompatible between-decade trials complete neglect of the tens digits would even generally lead to a wrong answer. Theoretically, such a finding would indicate that deficits in number processing and arithmetic may occur in neglect not only because the spatial representation of number magnitude along the MNL is impaired but also because the processing and integration of single digits of multi-digit numbers complying with the place-value structure of the Arabic number system may be impaired in neglect patients.
Second, our design, employing two separate internal standards, enabled us to contrast within-decade and between-decade comparisons even more directly than it had been possible in the study by Hoeckner et al.,. Comparisons with distances ±4, ±5, and ±6 were included for both between- and within-decade items (e.g., a distance of −5 corresponds to a between-decade comparison for the standard 53, i.e., 53_48, while it corresponds to a within-decade comparison for the standard 57, i.e., 57_52). Assuming the processing of the tens digits to be specifically impaired in neglect patients, we hypothesized that for the respective comparisons of within- and between-decade comparisons neglect patients should present with more difficulties for between- as compared to within-decade comparisons, because in the latter the tens digits are irrelevant. In summary, such a data pattern would provide first systematic evidence for the generalizability of the influence of object-based aspects of uni-lateral neglect to the processing of multi-digit numbers.
Despite these specific hypotheses we expected to replicate previous findings attributed to space-based symptoms of neglect, in particular, impaired processing of relatively smaller numbers associated with the left side of representational space. However, as the current manuscript focuses on the novel aspect of influences of object-based aspects of neglect on number processing, the interested reader is referred to Appendix A where the respective analyses and results are reported and discussed in detail.