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Random number generation in neglect patients reveals enhanced response stereotypy, but no neglect in number space
Neuropsychologia 47 (2009) 276–279
Contents lists available at ScienceDirect
Neuropsychologia journal homepage: www.elsevier.com/locate/neuropsychologia
Note
Random number generation in neglect patients reveals enhanced response stereotypy, but no neglect in number space Tobias Loetscher ∗ , Peter Brugger Department of Neurology, University Hospital Zurich, CH-8091 Zurich, Switzerland
a r t i c l e
i n f o
Article history: Received 9 May 2008 Received in revised form 19 July 2008 Accepted 6 August 2008 Available online 13 August 2008 Keywords: Spatial attention Numerical cognition Mental number line Working memory Order information Sequential memory
a b s t r a c t Based on interactions between number and space apparent from healthy subjects’ randomization attempts we expected random number generation (RNG) to be sensitive for the monitoring of unilateral spatial deficits. Specifically, we predicted patients with left-sided hemineglect to evidence “neglect in number space”, i.e. to produce a deficiency in the generation of small, “left-sided” numbers. In RNG of digits from 1 to 6, 19 patients with left-sided neglect generated sequences with a higher redundancy, but as many small numbers as did a matched control group. We discuss possible reasons for the absence of a smallnumber neglect and emphasize that the observed redundancy was not due to a counting bias, as known from other neurological patients, but to an unspecific imbalance in the use of response alternatives. We speculate that this may be the consequence of disrupted fronto-parietal functions normally serving in the sequential organization and manipulation of items in working memory. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction A variety of experimental approaches converge to the finding of a tight coupling between numerical and spatial operations (Hubbard, Piazza, Pinel, & Dehaene, 2005; Wood & Fischer, 2008). In general, we seem to implicitly associate small numbers with the left and larger number with the right side of space (but see also Galfano, Rusconi, & Umilta, 2006; Ristic, Wright, & Kingstone, 2006). A rather impressive demonstration of this association derives from clinical work with patients showing neglect for the left side of space (Zorzi, Priftis, & Umilta, 2002). When these patients had to name the midpoint of a given number interval, they deviated constantly towards larger numbers, i.e. to the right side on the mental number line (e.g. indicating “7” as the midpoint between the numbers “2” and “8”). This is strikingly similar to the bisection of lines in real space, where neglect patients also deviate to the right side. Such findings have led to the proposal that shifting the focus of attention along the mental number line is mediated by the same mechanisms that are involved in shifting attention in the outside world (Dehaene, Piazza, Pinel, & Cohen, 2003; Hubbard et al., 2005). However, we note that double dissociations between neglect on the mental number line and in real space have been reported (Doricchi, Guariglia, Gasparini, & Tomaiuolo, 2005).
∗ Corresponding author. Tel.: +41 1255 5579; fax: +41 1255 4429. E-mail address: [email protected] (T. Loetscher). 0028-3932/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.neuropsychologia.2008.08.005
We recently began to investigate spatial–numerical interactions in random number generation (RNG). In this paradigm, subjects are required to produce sequences of numbers without any underlying order. Marked deviations from true sequential randomness are the rule, and these are commonly ascribed to the limited capacity of working memory and frontal executive functions (Baddeley, 1998). In a series of retrospective analyses we established that healthy subjects, when generating long sequences of random numbers, produce significantly more small, “left-sided” numbers than expected by chance (Loetscher & Brugger, 2007b). Importantly, this smallnumber bias, interpreted as a pseudoneglect in number space, correlated with leftward biases in various spatial tasks. Well aware that these correlations not necessarily imply causation, we nevertheless speculated about a spatial component intrinsic to RNG (Loetscher & Brugger, 2007b). In a follow-up study we found further evidence for this assumption (Loetscher, Schwarz, Schubiger, & Brugger, 2008) when we demonstrated that turning one’s head to the left side of space enhanced one’s preference for small digits when mentally generating numbers “at random”. Thus, a mechanism known to allocate spatial attention in the outside world (head turning) seemed to influence random number choices systematically. Against this background it appears tempting to assume that patients with hemispatial neglect after a right parietal lesion might evidence a large-number bias in RNG, thereby “neglecting” numbers from the left side of the mental number line. The present experiment is an empirical attempt to test this assumption. We also planned to analyze neglect patient’s response stereotypy; to
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our knowledge, there is no published experiment on RNG in this special patient group. 2. Methods 2.1. Subjects Nineteen patients with left-sided spatial neglect (16 men; mean age = 62 years, S.D. = 12 years) participated in this study. Inclusion criteria were the presence of a right hemisphere lesion and symptoms of left-sided neglect in at least two paper and pencil tests. Demographic, clinical details and performances in the neglect tests can be found as supplementary material. Twenty-nine healthy subjects (well matched for age and years of education) served as control participants (12 men; mean age = 58 years, S.D. = 13 years). All subjects except one patient were right-handed. Testing was performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki. All subjects provided written informed consent prior to participation. 2.2. Randomization task The Mental Dice Task (MDT) requires subjects to generate 66 consecutive rolls of a die. Specifically, participants were instructed “to name the digits from 1 to 6 in a sequence as random as possible, i.e., as they might appear when rolling a real die over and over again” (Loetscher & Brugger, 2007b). Generation was paced to the beats of a metronome or rhythmic taps by the experimenter (1 Hz rhythm or, in exceptional cases, 0.5 Hz). 2.3. Measures The main dependent variable for the analysis of a spatial component in the MDT was the number of small digits (“1”, “2” and “3”). In addition, we determined the distribution of first-order differences (FODs), i.e. the arithmetic differences between each response and the immediately preceding number. These FODs vary between −5 (“6” is followed by “1”) and +5 (“1” is followed by “6”), a value of 0 indicating response repetition. In spatial terms, positive values reflect an ascending seriation, i.e. “rightward” orienting along the mental number line, whereas negative values indicate “leftward” shifts. The frequency of shifts between ascending and descending sequences is captured by the turning point index (TPI). Basically, the TPI compares the number of observed changes between ascending and descending sequences with the respective number of expected changes and is thus a measure of sequential response stereotypy. The sequence “1, 3, 4, 2, 1”, for example, has one turning point at response “4”. In spatial terms, the sequence changes at “4” from a rightward to a leftward orienting on the mental number line. Converted to percentages, TPI scores below 100 indicate fewer turning points than expected by chance. As global measures of randomness quality we calculated the redundancy score (R score) and the random number generation index (RNG index, Evans, 1978). The R score reflects deviations from the equiprobability of response alternatives and ranges from 0 (all alternatives generated equally frequently) to 100 (one single alternative provided on all trials). Similarly the RNG index assesses the degree of equiprobability of pairs of consecutive responses. It ranges from 0 (all 36 pairs in the case of randomization of six alternatives equally frequent) to 1 (perfect predictability of a digit from the preceding digit). Calculation of these randomness variables was achieved by a published computer program, freely downloadable at http://www.lancs.ac.uk/staff/towse/ rgcpage.html (Towse & Neil, 1998; see also for mathematical details on the measures used here).
3. Results In 66 rolls of a real die, 33 “small” and 33 “large” numbers are to be expected. The number of “small” digits generated did not differ between neglect patients (mean = 33.95, S.D. = 3.47) and controls (mean = 34.03, S.D. = 1.89; t(46) = .11, p > .91). While control participants’ small number bias was statistically significant (t(28) = 2.96, p < .01), the one by the patients was not (t(18) = 1.19, p > .24). The distribution of FODs is shown in Fig. 1. A multivariate analysis of variance (MANOVA) was conducted with group (patients and controls) as fixed factor and leftward and rightward responses (i.e. sum of negative FODs and sum of positive FODs) as dependent measures. It was not significant (Pillai’s Trace = .08, F(2, 45)=1.88, p > .16), indicating that the number of ascending and descending response pairings did not differ between the two participant groups. Turning behavior (TPI) did not differ between patients and controls either, i.e. there was a comparable number of changes
Fig. 1. Distribution of first-order differences (FODs) ± standard deviations.
from ascending to descending sequences and vice versa (patients’ mean = 84.1, S.D. = 13.2, controls’ mean = 84.7, S.D. = 12.1; t(46) = .15, p > .85). Since count-steps of ones and twos have previously been identified as a sensitive measures of stereotyped behavior in patient groups (e.g. Brugger, Monsch, Salmon, & Butters, 1996b; Spatt & Goldenberg, 1993) a multivariate analysis was run with group (patients and controls) as fixed factor and count-steps (ones and twos) as dependent measures. The MANOVA was again not significant (Pillai’s Trace = .04, F(2, 45)=.91, p > .41), indicating that neither counting steps in ones nor in twos differed between patients and healthy controls. The two groups differed significantly in both redundancy (R score; patients: mean = 1.50, S.D. = 1.2; controls: mean = .84, S.D. = .64; t(46) = 2.46, p < .02), and the frequency distribution of response pairs (RNG index; patients: mean = .44, S.D. = .047; controls: mean = .41, S.D. = .046, t(46) = 2.27; p < .03), with the patients being less random according to both these measures. There was no evidence for a relationship between the deviations in the line bisection task and the spatial parameters in RNG (p > .25 for all correlations). For a small subgroup of patients (n = 9), data on explicit numerical bisection (same bisection task as in Loetscher, Bockisch, & Brugger, 2008) were available. In this small subgroup the spatial parameters in the RNG task (i.e. small numbers) and number line bisection (i.e. deviation from true midpoint) were not related (Spearman’s rho = .06; p > .8). 4. Discussion Nineteen patients showing neglect for the left side of space and 29 healthy control subjects were administered a RNG task to investigate potentially asymmetric selections of small (“left-sided”) and large (“right-sided”) numbers. Our findings were clear-cut; the patients generated as many small numbers as did healthy control participants. The fact that the small-number bias in the controls (previously interpreted as a “pseudoneglect in number space”; Loetscher & Brugger, 2007b) was absent in the patient group is rather a consequence of the smaller number of participants than indicative of a genuine tendency to orient towards the right on a mental number line. Note that, numerically, patients still preferred smaller over larger digits. Other variables of RNG, potentially reflecting a spatial component, did also not convey any indication of the patients’ lateral deficit in real space to spread to the more abstract number space. Thus, the distribution of ascending and descending response pairs (Fig. 1) could theoretically be conceived of a distribution of rightward and leftward orienting tendencies. However, the 19 neglect patients generated as many backward (and forward) steps as did healthy controls. Still another possibility
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considered here was that patients exhibited a reluctance to switch between items relatively “left” and “right” in representational space (Loetscher & Brugger, 2007a). The relevant measure from the previous literature on RNG is the TPI, an indicator of the number of changes between ascending and descending sequences of consecutive numbers. Our analyses showed a symmetric “right-to-left” (and vice versa) shifting pattern in both patients and controls, and no significant differences between the two groups. We conclude that, contrary to our expectation, there is no slightest evidence for a spatial dimension of RNG that would reflect patients’ hemispatial deficit in visuo-motor space. The simplest explanation for this null-finding is that RNG does not involve a spatial component at all. This seems rather unlikely, however, because substantial evidence for numerical–spatial interactions in RNG has been accumulated with healthy subjects (Loetscher & Brugger, 2007b; Loetscher et al., 2008). We consider three alternative hypotheses instead: a) Implicit task demands: RNG does not require direct access and exploration of the mental number line. Previous work showed that patients with left-sided neglect evidenced a bias towards larger numbers particularly in paradigms, which require a conscious manipulation of numerical magnitudes. For instance, Zorzi et al. (2002) introduced a numerical bisection task in which the middle of a given numerical interval has to be estimated. Similarly, Vuilleumier, Ortigue and Brugger (2004) asked for explicit magnitude judgments of visually presented number pairs. In contrast, no neglect in number space was found in the SNARC paradigm (see Dehaene, Bossini, & Giraux, 1993), in which the ordinal arrangements of stimulus numbers are completely taskirrelevant (Priftis, Zorzi, Meneghello, Marenzi, & Umilta, 2006). As noted by these latter authors, one plausible explanation for a performance dissociation in these different tasks is in terms of different processing demands. Dissociations between explicit and implicit processing of left-sided visual information are well known in neglect patients (e.g., Marshall & Halligan, 1988). Priftis et al. (2006) demonstrated such a dissociation also in number space. Thus, the lack of a correlation among lateral deviations in mental number line and implicit RNG tasks in a subgroup of our patients fits well with the observation that neglect may affect explicit and implicit processing differently. Note however, that in healthy subjects, we found a weak correlation between these explicit and implicit numerical tasks (Loetscher & Brugger, 2007b, study 2). b) Number range: Perhaps the restriction to numbers 1–6 left us with too small a range to demonstrate spatial aspects of randomization. Neglect patients show a paradoxical leftward shift (“cross-over”) when bisecting short lines (Marshall & Halligan, 1989), as well as when bisecting small numerical intervals (i.e. a smaller-number shift, Zorzi et al., 2002). Thus, we cannot exclude the possibility that we missed a neglect in number space because of a “cross-over” effect along the number line. Future RNG studies should test this hypothesis by using larger number intervals. However, especially with larger number ranges one would have to consider that the representation of numbers may be compressed towards larger numbers (Dehaene & Mehler, 1992; Longo & Lourenco, 2007). A bisection of the interval at the arithmetic mean would thus not be spatially balanced. c) Linguistic factors: We have to keep in mind that number words do not occur with equal frequency in everyday language. For example, the word “two” is read and heard in English about ten times more often than the word “nine” (Dehaene & Mehler, 1992). Thus, two opposing tendencies may have produced an equiprobability of small and large numbers in neglect patients’ randomization attempts: a non-spatial bias towards more frequently occurring,
smaller numbers and a spatial orienting bias to the right side of number space. Note that, in contrast to neglect patients, in healthy subjects these linguistic and spatial (leftward) biases are additive, leading to a pronounced small-number preference. While the present RNG experiment produced no evidence for a neglect in number space, we found that patients displayed a more stereotyped, rule-governed randomization behavior than controls. Unlike the findings in other neurological patient groups, however, the observed stereotypies did not manifest themselves in more excessive counting in steps of one, but in an enhanced redundancy due to an unspecific imbalance in the use of single numbers (R score) and number pairs (RNG index). Pronounced counting is characteristic to patients with primarily frontal lobe dysfunction (Brugger et al., 1996b; Spatt & Goldenberg, 1993) and can be induced in healthy subjects by disruption of specifically left dorsolateral prefrontal cortex (Knoch, Brugger, & Regard, 2005). It is commonly interpreted as reflecting a failure to inhibit the most habitual way to sequentially arrange digits. Our findings may be relevant to the functional role of the right hemisphere for RNG. The only previous specific proposal of an involvement of the right hemisphere concerned the avoidance of repetitive responses (Brugger, Monsch, & Johnson, 1996a), but in the present study no differences emerged between patients and controls in the number of digit repetitions. Bilateral involvement was suggested by several neuroimaging studies. Thus, functional magnetic resonance imaging (Daniels, Witt, Wolff, Jansen, & Deuschl, 2003) and positron emission tomography (Jahanshahi, Dirnberger, Fuller, & Frith, 2000) studies have identified a widespread cortical network involving both hemispheres in RNG. Specifically, contrasted to simple counting, activations during RNG in a 1 Hz-rhythm have been reported bilaterally in the dorsolateral prefrontal and premotor cortices, the anterior cingulate region and the cerebellum. Inferior and superior parietal cortex was involved as well. This activation pattern is comparable to that described for a wide range of working memory tasks (Wager & Smith, 2003). Over and above their role as a buffer for verbal and spatial information, the posterior parietal lobes are about to be recognized as structures essential to the manipulation and organization of items in working memory (Champod & Petrides, 2007; Wendelken, Bunge, & Carter, 2008). It is thus tempting to assume that our neglect patients’ difficulties to spontaneously arrange numbers “randomly” was primarily based on disrupted fronto-parietal processes normally needed to handle sequential-organizational demands in working memory. Evidently, these hypothetical processes are not primarily concerned with the spatial characteristics of numerical information. To summarize, in a RNG task, neglect patients displayed a marked sequential non-randomness, but no neglect in number space. It will be interesting to learn about specific parietal lobe contributions to response randomization and to thus overcome the traditional research stereotype that had its focus perhaps too narrowly directed to the frontal lobes. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.neuropsychologia.2008.08.005. References Baddeley, A. (1998). Random generation and the executive control of working memory. Quarterly Journal of Experimental Psychology: Section A, 51(4), 819–852. Brugger, P., Monsch, A. U., & Johnson, S. A. (1996a). Repetitive behavior and repetition avoidance: The role of the right hemisphere. Journal of Psychiatry and Neuroscience, 21(1), 53–56.
T. Loetscher, P. Brugger / Neuropsychologia 47 (2009) 276–279 Brugger, P., Monsch, A. U., Salmon, D. P., & Butters, N. (1996b). Random number generation in dementia of the Alzheimer type: A test of frontal executive functions. Neuropsychologia, 34(2), 97–103. Champod, A. S., & Petrides, M. (2007). Dissociable roles of the posterior parietal and the prefrontal cortex in manipulation and monitoring processes. Proceedings of the National Academy of Sciences, 104(37), 14837–14842. Daniels, C., Witt, K., Wolff, S., Jansen, O., & Deuschl, G. (2003). Rate dependency of the human cortical network subserving executive functions during generation of random number series—a functional magnetic resonance imaging study. Neuroscience Letters, 345(1), 25–28. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology-General, 122(3), 371–396. Dehaene, S., & Mehler, J. (1992). Cross-linguistic regularities in the frequency of number words. Cognition, 43(1), 1–29. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3–6), 487–506. Doricchi, F., Guariglia, P., Gasparini, M., & Tomaiuolo, F. (2005). Dissociation between physical and mental number line bisection in right hemisphere brain damage. Nature Neuroscience, 8(12), 1663–1665. Evans, F. J. (1978). Monitoring attention deployment by random number generation—Index to measure subjective randomness. Bulletin of the Psychonomic Society, 12(1), 35–38. Galfano, G., Rusconi, E., & Umilta, C. (2006). Number magnitude orients attention, but not against one’s will. Psychonomic Bulletin and Review, 13(5), 869–874. Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6(6), 435–448. Jahanshahi, M., Dirnberger, G., Fuller, R., & Frith, C. D. (2000). The role of the dorsolateral prefrontal cortex in random number generation: A study with positron emission tomography. Neuroimage, 12(6), 713–725. Jahanshahi, M., Profice, P., Brown, R. G., Ridding, M. C., Dirnberger, G., & Rothwell, J. C. (1998). The effects of transcranial magnetic stimulation over the dorsolateral prefrontal cortex on suppression of habitual counting during random number generation. Brain, 121, 1533–1544. Knoch, D., Brugger, P., & Regard, M. (2005). Suppressing versus releasing a habit: Frequency-dependent effects of prefrontal transcranial magnetic stimulation. Cerebral Cortex, 15(7), 885–887. Loetscher, T., Bockisch, C., & Brugger, P. (2008). Looking for the answer: The mind’s eye in number space. Neuroscience, 151(3), 725–729.
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Loetscher, T., & Brugger, P. (2007a). A disengagement deficit in representational space. Neuropsychologia, 45(6), 1299–1304. Loetscher, T., & Brugger, P. (2007b). Exploring number space by random digit generation. Experimental. Brain Research, 180(4), 655–665. Loetscher, T., Schwarz, U., Schubiger, M., & Brugger, P. (2008). Head turns bias the brain’s internal random generator. Current Biology, 18(2), R60–R62. Longo, M. R., & Lourenco, S. F. (2007). Spatial attention and the mental number line: Evidence for characteristic biases and compression. Neuropsychologia, 45(7), 1400–1407. Marshall, J. C., & Halligan, P. W. (1988). Blindsight and insight in visuo-spatial neglect. Nature, 336(6201), 766–767. Marshall, J. C., & Halligan, P. W. (1989). When right goes left—an investigation of line bisection in a case of visual neglect. Cortex, 25(3), 503–515. Marshuetz, C. (2005). Order information in working memory: An integrative review of evidence from brain and behavior. Psychological Bulletin, 131(3), 323–339. Priftis, K., Zorzi, M., Meneghello, F., Marenzi, R., & Umilta, C. (2006). Explicit versus implicit processing of representational space in neglect: Dissociations in accessing the mental number line. Journal of Cognitive Neuroscience, 18(4), 680–688. Ristic, J., Wright, A., & Kingstone, A. (2006). The number line effect reflects top-down control. Psychonomic Bulletin and Review, 13, 862–868. Spatt, J., & Goldenberg, G. (1993). Components of random generation by normal subjects and patients with dysexecutive syndrome. Brain and Cognition, 23(2), 231–242. Towse, J. N., & Neil, D. (1998). Analyzing human random generation behavior: A review of methods used and a computer program for describing performance. Behavior Research Methods Instruments & Computers, 30(4), 583–591. Vuilleumier, P., Ortigue, S., & Brugger, P. (2004). The number space and neglect. Cortex, 40(2), 399–410. Wager, T. D., & Smith, E. E. (2003). Neuroimaging studies of working memory: A meta-analysis. Cognitive, Affective and Behavioral Neuroscience, 3(4), 255–274. Wendelken, C., Bunge, S. A., & Carter, C. S. (2008). Maintaining structured information: An investigation into functions of parietal and lateral prefrontal cortices. Neuropsychologia, 46(2), 665–678. Wood, G., & Fischer, M. H. (2008). Numbers, space, and action—From finger counting to the mental number line and beyond. Cortex, 44(4), 353–358. Zorzi, M., Priftis, K., & Umilta, C. (2002). Brain damage: Neglect disrupts the mental number line. Nature, 417(6885), 138–139.
Contents lists available at ScienceDirect
Neuropsychologia journal homepage: www.elsevier.com/locate/neuropsychologia
Note
Random number generation in neglect patients reveals enhanced response stereotypy, but no neglect in number space Tobias Loetscher ∗ , Peter Brugger Department of Neurology, University Hospital Zurich, CH-8091 Zurich, Switzerland
a r t i c l e
i n f o
Article history: Received 9 May 2008 Received in revised form 19 July 2008 Accepted 6 August 2008 Available online 13 August 2008 Keywords: Spatial attention Numerical cognition Mental number line Working memory Order information Sequential memory
a b s t r a c t Based on interactions between number and space apparent from healthy subjects’ randomization attempts we expected random number generation (RNG) to be sensitive for the monitoring of unilateral spatial deficits. Specifically, we predicted patients with left-sided hemineglect to evidence “neglect in number space”, i.e. to produce a deficiency in the generation of small, “left-sided” numbers. In RNG of digits from 1 to 6, 19 patients with left-sided neglect generated sequences with a higher redundancy, but as many small numbers as did a matched control group. We discuss possible reasons for the absence of a smallnumber neglect and emphasize that the observed redundancy was not due to a counting bias, as known from other neurological patients, but to an unspecific imbalance in the use of response alternatives. We speculate that this may be the consequence of disrupted fronto-parietal functions normally serving in the sequential organization and manipulation of items in working memory. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction A variety of experimental approaches converge to the finding of a tight coupling between numerical and spatial operations (Hubbard, Piazza, Pinel, & Dehaene, 2005; Wood & Fischer, 2008). In general, we seem to implicitly associate small numbers with the left and larger number with the right side of space (but see also Galfano, Rusconi, & Umilta, 2006; Ristic, Wright, & Kingstone, 2006). A rather impressive demonstration of this association derives from clinical work with patients showing neglect for the left side of space (Zorzi, Priftis, & Umilta, 2002). When these patients had to name the midpoint of a given number interval, they deviated constantly towards larger numbers, i.e. to the right side on the mental number line (e.g. indicating “7” as the midpoint between the numbers “2” and “8”). This is strikingly similar to the bisection of lines in real space, where neglect patients also deviate to the right side. Such findings have led to the proposal that shifting the focus of attention along the mental number line is mediated by the same mechanisms that are involved in shifting attention in the outside world (Dehaene, Piazza, Pinel, & Cohen, 2003; Hubbard et al., 2005). However, we note that double dissociations between neglect on the mental number line and in real space have been reported (Doricchi, Guariglia, Gasparini, & Tomaiuolo, 2005).
∗ Corresponding author. Tel.: +41 1255 5579; fax: +41 1255 4429. E-mail address: [email protected] (T. Loetscher). 0028-3932/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.neuropsychologia.2008.08.005
We recently began to investigate spatial–numerical interactions in random number generation (RNG). In this paradigm, subjects are required to produce sequences of numbers without any underlying order. Marked deviations from true sequential randomness are the rule, and these are commonly ascribed to the limited capacity of working memory and frontal executive functions (Baddeley, 1998). In a series of retrospective analyses we established that healthy subjects, when generating long sequences of random numbers, produce significantly more small, “left-sided” numbers than expected by chance (Loetscher & Brugger, 2007b). Importantly, this smallnumber bias, interpreted as a pseudoneglect in number space, correlated with leftward biases in various spatial tasks. Well aware that these correlations not necessarily imply causation, we nevertheless speculated about a spatial component intrinsic to RNG (Loetscher & Brugger, 2007b). In a follow-up study we found further evidence for this assumption (Loetscher, Schwarz, Schubiger, & Brugger, 2008) when we demonstrated that turning one’s head to the left side of space enhanced one’s preference for small digits when mentally generating numbers “at random”. Thus, a mechanism known to allocate spatial attention in the outside world (head turning) seemed to influence random number choices systematically. Against this background it appears tempting to assume that patients with hemispatial neglect after a right parietal lesion might evidence a large-number bias in RNG, thereby “neglecting” numbers from the left side of the mental number line. The present experiment is an empirical attempt to test this assumption. We also planned to analyze neglect patient’s response stereotypy; to
T. Loetscher, P. Brugger / Neuropsychologia 47 (2009) 276–279
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our knowledge, there is no published experiment on RNG in this special patient group. 2. Methods 2.1. Subjects Nineteen patients with left-sided spatial neglect (16 men; mean age = 62 years, S.D. = 12 years) participated in this study. Inclusion criteria were the presence of a right hemisphere lesion and symptoms of left-sided neglect in at least two paper and pencil tests. Demographic, clinical details and performances in the neglect tests can be found as supplementary material. Twenty-nine healthy subjects (well matched for age and years of education) served as control participants (12 men; mean age = 58 years, S.D. = 13 years). All subjects except one patient were right-handed. Testing was performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki. All subjects provided written informed consent prior to participation. 2.2. Randomization task The Mental Dice Task (MDT) requires subjects to generate 66 consecutive rolls of a die. Specifically, participants were instructed “to name the digits from 1 to 6 in a sequence as random as possible, i.e., as they might appear when rolling a real die over and over again” (Loetscher & Brugger, 2007b). Generation was paced to the beats of a metronome or rhythmic taps by the experimenter (1 Hz rhythm or, in exceptional cases, 0.5 Hz). 2.3. Measures The main dependent variable for the analysis of a spatial component in the MDT was the number of small digits (“1”, “2” and “3”). In addition, we determined the distribution of first-order differences (FODs), i.e. the arithmetic differences between each response and the immediately preceding number. These FODs vary between −5 (“6” is followed by “1”) and +5 (“1” is followed by “6”), a value of 0 indicating response repetition. In spatial terms, positive values reflect an ascending seriation, i.e. “rightward” orienting along the mental number line, whereas negative values indicate “leftward” shifts. The frequency of shifts between ascending and descending sequences is captured by the turning point index (TPI). Basically, the TPI compares the number of observed changes between ascending and descending sequences with the respective number of expected changes and is thus a measure of sequential response stereotypy. The sequence “1, 3, 4, 2, 1”, for example, has one turning point at response “4”. In spatial terms, the sequence changes at “4” from a rightward to a leftward orienting on the mental number line. Converted to percentages, TPI scores below 100 indicate fewer turning points than expected by chance. As global measures of randomness quality we calculated the redundancy score (R score) and the random number generation index (RNG index, Evans, 1978). The R score reflects deviations from the equiprobability of response alternatives and ranges from 0 (all alternatives generated equally frequently) to 100 (one single alternative provided on all trials). Similarly the RNG index assesses the degree of equiprobability of pairs of consecutive responses. It ranges from 0 (all 36 pairs in the case of randomization of six alternatives equally frequent) to 1 (perfect predictability of a digit from the preceding digit). Calculation of these randomness variables was achieved by a published computer program, freely downloadable at http://www.lancs.ac.uk/staff/towse/ rgcpage.html (Towse & Neil, 1998; see also for mathematical details on the measures used here).
3. Results In 66 rolls of a real die, 33 “small” and 33 “large” numbers are to be expected. The number of “small” digits generated did not differ between neglect patients (mean = 33.95, S.D. = 3.47) and controls (mean = 34.03, S.D. = 1.89; t(46) = .11, p > .91). While control participants’ small number bias was statistically significant (t(28) = 2.96, p < .01), the one by the patients was not (t(18) = 1.19, p > .24). The distribution of FODs is shown in Fig. 1. A multivariate analysis of variance (MANOVA) was conducted with group (patients and controls) as fixed factor and leftward and rightward responses (i.e. sum of negative FODs and sum of positive FODs) as dependent measures. It was not significant (Pillai’s Trace = .08, F(2, 45)=1.88, p > .16), indicating that the number of ascending and descending response pairings did not differ between the two participant groups. Turning behavior (TPI) did not differ between patients and controls either, i.e. there was a comparable number of changes
Fig. 1. Distribution of first-order differences (FODs) ± standard deviations.
from ascending to descending sequences and vice versa (patients’ mean = 84.1, S.D. = 13.2, controls’ mean = 84.7, S.D. = 12.1; t(46) = .15, p > .85). Since count-steps of ones and twos have previously been identified as a sensitive measures of stereotyped behavior in patient groups (e.g. Brugger, Monsch, Salmon, & Butters, 1996b; Spatt & Goldenberg, 1993) a multivariate analysis was run with group (patients and controls) as fixed factor and count-steps (ones and twos) as dependent measures. The MANOVA was again not significant (Pillai’s Trace = .04, F(2, 45)=.91, p > .41), indicating that neither counting steps in ones nor in twos differed between patients and healthy controls. The two groups differed significantly in both redundancy (R score; patients: mean = 1.50, S.D. = 1.2; controls: mean = .84, S.D. = .64; t(46) = 2.46, p < .02), and the frequency distribution of response pairs (RNG index; patients: mean = .44, S.D. = .047; controls: mean = .41, S.D. = .046, t(46) = 2.27; p < .03), with the patients being less random according to both these measures. There was no evidence for a relationship between the deviations in the line bisection task and the spatial parameters in RNG (p > .25 for all correlations). For a small subgroup of patients (n = 9), data on explicit numerical bisection (same bisection task as in Loetscher, Bockisch, & Brugger, 2008) were available. In this small subgroup the spatial parameters in the RNG task (i.e. small numbers) and number line bisection (i.e. deviation from true midpoint) were not related (Spearman’s rho = .06; p > .8). 4. Discussion Nineteen patients showing neglect for the left side of space and 29 healthy control subjects were administered a RNG task to investigate potentially asymmetric selections of small (“left-sided”) and large (“right-sided”) numbers. Our findings were clear-cut; the patients generated as many small numbers as did healthy control participants. The fact that the small-number bias in the controls (previously interpreted as a “pseudoneglect in number space”; Loetscher & Brugger, 2007b) was absent in the patient group is rather a consequence of the smaller number of participants than indicative of a genuine tendency to orient towards the right on a mental number line. Note that, numerically, patients still preferred smaller over larger digits. Other variables of RNG, potentially reflecting a spatial component, did also not convey any indication of the patients’ lateral deficit in real space to spread to the more abstract number space. Thus, the distribution of ascending and descending response pairs (Fig. 1) could theoretically be conceived of a distribution of rightward and leftward orienting tendencies. However, the 19 neglect patients generated as many backward (and forward) steps as did healthy controls. Still another possibility
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considered here was that patients exhibited a reluctance to switch between items relatively “left” and “right” in representational space (Loetscher & Brugger, 2007a). The relevant measure from the previous literature on RNG is the TPI, an indicator of the number of changes between ascending and descending sequences of consecutive numbers. Our analyses showed a symmetric “right-to-left” (and vice versa) shifting pattern in both patients and controls, and no significant differences between the two groups. We conclude that, contrary to our expectation, there is no slightest evidence for a spatial dimension of RNG that would reflect patients’ hemispatial deficit in visuo-motor space. The simplest explanation for this null-finding is that RNG does not involve a spatial component at all. This seems rather unlikely, however, because substantial evidence for numerical–spatial interactions in RNG has been accumulated with healthy subjects (Loetscher & Brugger, 2007b; Loetscher et al., 2008). We consider three alternative hypotheses instead: a) Implicit task demands: RNG does not require direct access and exploration of the mental number line. Previous work showed that patients with left-sided neglect evidenced a bias towards larger numbers particularly in paradigms, which require a conscious manipulation of numerical magnitudes. For instance, Zorzi et al. (2002) introduced a numerical bisection task in which the middle of a given numerical interval has to be estimated. Similarly, Vuilleumier, Ortigue and Brugger (2004) asked for explicit magnitude judgments of visually presented number pairs. In contrast, no neglect in number space was found in the SNARC paradigm (see Dehaene, Bossini, & Giraux, 1993), in which the ordinal arrangements of stimulus numbers are completely taskirrelevant (Priftis, Zorzi, Meneghello, Marenzi, & Umilta, 2006). As noted by these latter authors, one plausible explanation for a performance dissociation in these different tasks is in terms of different processing demands. Dissociations between explicit and implicit processing of left-sided visual information are well known in neglect patients (e.g., Marshall & Halligan, 1988). Priftis et al. (2006) demonstrated such a dissociation also in number space. Thus, the lack of a correlation among lateral deviations in mental number line and implicit RNG tasks in a subgroup of our patients fits well with the observation that neglect may affect explicit and implicit processing differently. Note however, that in healthy subjects, we found a weak correlation between these explicit and implicit numerical tasks (Loetscher & Brugger, 2007b, study 2). b) Number range: Perhaps the restriction to numbers 1–6 left us with too small a range to demonstrate spatial aspects of randomization. Neglect patients show a paradoxical leftward shift (“cross-over”) when bisecting short lines (Marshall & Halligan, 1989), as well as when bisecting small numerical intervals (i.e. a smaller-number shift, Zorzi et al., 2002). Thus, we cannot exclude the possibility that we missed a neglect in number space because of a “cross-over” effect along the number line. Future RNG studies should test this hypothesis by using larger number intervals. However, especially with larger number ranges one would have to consider that the representation of numbers may be compressed towards larger numbers (Dehaene & Mehler, 1992; Longo & Lourenco, 2007). A bisection of the interval at the arithmetic mean would thus not be spatially balanced. c) Linguistic factors: We have to keep in mind that number words do not occur with equal frequency in everyday language. For example, the word “two” is read and heard in English about ten times more often than the word “nine” (Dehaene & Mehler, 1992). Thus, two opposing tendencies may have produced an equiprobability of small and large numbers in neglect patients’ randomization attempts: a non-spatial bias towards more frequently occurring,
smaller numbers and a spatial orienting bias to the right side of number space. Note that, in contrast to neglect patients, in healthy subjects these linguistic and spatial (leftward) biases are additive, leading to a pronounced small-number preference. While the present RNG experiment produced no evidence for a neglect in number space, we found that patients displayed a more stereotyped, rule-governed randomization behavior than controls. Unlike the findings in other neurological patient groups, however, the observed stereotypies did not manifest themselves in more excessive counting in steps of one, but in an enhanced redundancy due to an unspecific imbalance in the use of single numbers (R score) and number pairs (RNG index). Pronounced counting is characteristic to patients with primarily frontal lobe dysfunction (Brugger et al., 1996b; Spatt & Goldenberg, 1993) and can be induced in healthy subjects by disruption of specifically left dorsolateral prefrontal cortex (Knoch, Brugger, & Regard, 2005). It is commonly interpreted as reflecting a failure to inhibit the most habitual way to sequentially arrange digits. Our findings may be relevant to the functional role of the right hemisphere for RNG. The only previous specific proposal of an involvement of the right hemisphere concerned the avoidance of repetitive responses (Brugger, Monsch, & Johnson, 1996a), but in the present study no differences emerged between patients and controls in the number of digit repetitions. Bilateral involvement was suggested by several neuroimaging studies. Thus, functional magnetic resonance imaging (Daniels, Witt, Wolff, Jansen, & Deuschl, 2003) and positron emission tomography (Jahanshahi, Dirnberger, Fuller, & Frith, 2000) studies have identified a widespread cortical network involving both hemispheres in RNG. Specifically, contrasted to simple counting, activations during RNG in a 1 Hz-rhythm have been reported bilaterally in the dorsolateral prefrontal and premotor cortices, the anterior cingulate region and the cerebellum. Inferior and superior parietal cortex was involved as well. This activation pattern is comparable to that described for a wide range of working memory tasks (Wager & Smith, 2003). Over and above their role as a buffer for verbal and spatial information, the posterior parietal lobes are about to be recognized as structures essential to the manipulation and organization of items in working memory (Champod & Petrides, 2007; Wendelken, Bunge, & Carter, 2008). It is thus tempting to assume that our neglect patients’ difficulties to spontaneously arrange numbers “randomly” was primarily based on disrupted fronto-parietal processes normally needed to handle sequential-organizational demands in working memory. Evidently, these hypothetical processes are not primarily concerned with the spatial characteristics of numerical information. To summarize, in a RNG task, neglect patients displayed a marked sequential non-randomness, but no neglect in number space. It will be interesting to learn about specific parietal lobe contributions to response randomization and to thus overcome the traditional research stereotype that had its focus perhaps too narrowly directed to the frontal lobes. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.neuropsychologia.2008.08.005. References Baddeley, A. (1998). Random generation and the executive control of working memory. Quarterly Journal of Experimental Psychology: Section A, 51(4), 819–852. Brugger, P., Monsch, A. U., & Johnson, S. A. (1996a). Repetitive behavior and repetition avoidance: The role of the right hemisphere. Journal of Psychiatry and Neuroscience, 21(1), 53–56.
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