Evidence for Multiple Representations of Number in the Human Brain. Thesis. Presented in Partial Fulfillment of the Requirements for

Transcription

1 Evidence for Multiple Representations of Number in the Human Brain Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Frank Joseph Kanayet B. S. Graduate Program in Psychology The Ohio State University 2009 Thesis Committee: Professor John Opfer, Advisor Professor Simon Dennis Professor Wil Cunningham

2 Copyright by Frank Joseph Kanayet 2009

3 Abstract In the adult brain, the horizontal segment of the intraparietal sulcus (HIPS) is known to play a pivotal role in representing magnitudes of numerals. In children, however, prefrontal cortex may play a larger role than HIPS. This thesis tests a hypothesized link between developmental changes in locus of representation and the logarithmic-to-linear shift in representations of numerical magnitude. Participants were presented with number lines and asked to judge the accuracy of linear, logarithmic, or log-linear placements. Consistent with our hypothesis, event-related potentials (ERPs) revealed greatest parietal N1 amplitudes for linear trials and greatest frontal P3 amplitude for logarithmic trials. Additionally, the effects on linear trials were moderated by magnitude: parietal N1 amplitudes decreased with magnitude, while frontal P3 amplitudes increased with magnitude. Together, these results provide evidence for multiple representations of number and show that when linear representations are less entrenched, there is more involvement of the prefrontal cortex. ii

4 Dedicated to Catalina for being my companion and supporting me every step of the way My parents for giving me all the opportunities to grow as a person and pursue my dreams. Last but not least, to all the professors that played an instrumental role in motivating me to pursue a scientific career. iii

5 Acknowledgments First, I would like to thank my advisor John Opfer for all the discussions, thoughtful comments and support during the development and writing of this thesis. I am still amazed about all the transformations this project has been subjected to as a result of my meetings with John. I would also like to thank Wil Cunningham for opening the doors of his lab to me, and for providing his expertise in the conduction of brain imaging research to make this thesis a better product. In addition, I would like to express gratitude to Simon Dennis, Vladimir Sloutsky, and the rest of the members of the ERP reading group for providing a space to discuss issues related to ERP research that improved my understanding of this methodology and of the big picture implications of all the methodological decisions made in this thesis. Similarly, I am grateful to Jay Van Bavel for going with me through all the steps of the research, from plugging cables to data analysis. It would literally had been impossible to get to this point without the countless hours of troubleshooting that Jay and I spent in the ERP lab. I am also very grateful to Laura Nelson and Sima Finy for helping me with data collection and coding, and to Andrew Jahn, for helping with the data analysis. Finally, I would like to thank my parents and brother for being there for me at all times, and my girlfriend Catalina Torrente who motivates me, proofreads everything I write, listens to and discusses my ideas, and accompanies me in every single thing I do. iv

6 Vita February 4, Born Bogotá, Colombia B.S. Psychology, Universidad de los Andes B.S. Philosophy, Universidad de los Andes Presidential Fellowship, The Ohio State University to present Graduate Teaching Associate, Department of Psychology, The Ohio State University Fields of Study Major Field: Psychology Areas of Emphasis: Cognitive Psychology; Developmental Psychology v

7 Table of Contents P a g e Abstract...ii Dedication..iii Acknowledgments iv Vita...v List of Tables...viii List of Figures...ix Evidence for Multiple Representations of Number in the Human Brain..1 How is numeric magnitude represented in the brain?..3 Behavioral evidence from numerical estimation..10 Present study..13 Method.. 15 Participants.15 Task..16 Stimuli...16 Procedure and design ERP recording procedure. 17 Results Behavioral results..19 Electrophysiological results. 20 Discussion 26 vi

8 Future directions 29 Conclusion.. 31 References. 33 Appendix A: Tables 38 Appendix B: Figures vii

9 List of Tables Table Page 1 Table of ANOVA effects for accuracy and median reaction times viii

10 List of Figures Figure Page 1 Experimental procedure. Example of a) linear trials, b) logarithmic trials, and c) log-linear trials. ISI = Intertrial stimulus interval Montage and labels for electrodes used in the experiment. Top is front Area of circles represents the percentage of responses judged as correct by participants for linear trials (green), logarithmic trials (red), and log-linear trials blue) Mean d prime values for each numeral (+ SE). (a) and (b) represent significant differences at p < N1 component of ERP to placement of numbers on number lines. Figure shows the negative amplitude wave in the ms time window after a number was shown in either the linear, logarithmic, or log-linear position. The mean amplitude varies parametrically with the linearity of placement of the number. a) Average waveforms for representative posterior electrode P4 (right parietal) for linear trials (black), logarithmic trials (red) and log-linear trials (blue). b) Current source density topography for linear trials at 200 ms. c) Current source density topography for logarithmic trials at 200 ms 43 6 P3 component of ERP to placement of numbers on number lines. Figure shows the positive amplitude wave in the ms time window after a number was shown in either the linear, logarithmic, or log-linear position. The mean amplitude varies parametrically with the logarithmicity of placement of the number. a) Average waveforms for representative anterior electrode FC4 (right frontal) for linear trials (black), logarithmic trials (red) and log-linear trials (blue). b) Current source density topography for linear trials at 300 ms. c) Current source density topography for logarithmic trials at 300ms.44 ix

11 7 How the brain reacts to differences between the logarithmic and linear placements of numbers on number lines. Around 200 ms after presentation of a number line placement, a strong N1 component emerges in parietal cortex with the identification of the linear placement being correct. 100 ms later, in frontal cortex, a strong P3 component emerges with the recognition of the logarithmic placement being erroneous. a) Difference waveforms (logarithmic minus linear) for representative right posterior electrode P4 (red), and representative right frontal electrode (blue). b) Current source density topography for N1 (200 ms). c) Current source density topography for P3 (300 ms) 45 8 N1 component of ERP to placement of numbers on number lines for linear trials that were judged as correct by participants. Figure shows the negative amplitude wave in the ms time window after the placement of the hatch mark for small numbers (less than 500) and large numbers (greater than 500). a) Average waveforms for representative posterior electrode P4 (right parietal) for correct linear trials with small numbers after the presentation of a hatch mark (blue) and for large numbers after the presentation of a hatch mark (red). b) Current source density topography for small numbers at 200 ms. c) Current source density topography for large numbers at 200 ms.46 9 P3 component of ERP to placement of numbers on number lines for linear trials that were judged as correct by participants. Figure shows the positive amplitude wave in the ms time window after the placement of the hatch mark for small numbers (less than 500) and large numbers (greater than 500). a) Average waveforms for representative anterior electrode F7 (left frontal) for correct linear trials with small numbers after the presentation of a hatch mark (blue) and for large numbers after the presentation of a hatch mark (red). b) Current source density topography for small numbers at 300 ms. c) Current source density topography for large numbers at 300 ms N1 component of ERP to presentation of Arabic numerals. Figure shows the negative amplitude wave in the ms time window after the presentation of small numbers (less than 500) and large numbers (greater than 500). a) Average waveforms for representative posterior electrode P7 (left parietal) for trials with small numbers (blue) and for large numbers (red). b) Current source density topography for small numbers at 200 ms. c) Current source density topography for large numbers at 200 ms P3 component of ERP to presentation of Arabic numerals. Figure shows the positive amplitude wave in the ms time window after the presentation of small numbers x

12 (less than 500) and large numbers (greater than 500). a) Average waveforms for representative posterior electrode P7 (left parietal) for trials with small numbers (blue) and for large numbers (red). b) Current source density topography for small numbers at 350 ms. c) Current source density topography for large numbers at 350 ms...49 xi

13 Evidence for Multiple Representations of Number in the Human Brain Whether a pollster determining the sampling process for a presidential election poll, a parishioner telling the time by counting the tolls of a church bell, or a child figuring out how much candy she had received on this Halloween versus a previous one, mental representations of numerical magnitude are important for projecting the future, monitoring the present, and learning from the past. Moreover, this ability to code our experiences numerically must scale consistently regardless of the shape, size, sensory modality or context in which particular numeric magnitudes are presented. To understand how humans and other animals represent numerical magnitudes, neuroscientists have examined numeric coding in the brain. In this research, two prominent brain areas have been implicated: the prefrontal cortex and the horizontal segment of the intraparietal Sulcus (HIPS). Most studies have shown that HIPS plays a major role in numeric representation (Dehaene, Piazza, Pinel, Cohen, 2003 for a review). Thus, a dominant view regarding the role of HIPS is that magnitude is coded in this area as an abstract, notation-independent representation (Dehaene, Dehaene-Lambertz, & Cohen, 1998; Libertus, Woldroff & Brannon, 2007; Plodowski et al., 2003). However, studies that use different populations have found different results. For example, single cell recordings in monkeys (Nieder & Miller, 2004; Nieder & Merten, 2007) and fmri studies in children (Ansari et al. 2005; Ansari & Dhital, 2006) have shown stronger effects in the prefrontal cortex compared to HIPS. Additionally, fmri studies have shown greater activation of prefrontal cortex for tasks where children and adults are asked to count dots or sequences of taps (Cantlon, Brannon, Carter & Pelphrey, 2006; Venkatraman et al., 2005) compared to tasks that involve numerals. These findings 1

14 suggest that over the course of development, experience with numerical magnitudes may cause the prefrontal cortex to play a successively less prominent role (and HIPS a successively more prominent role) in numeric magnitude processing. To explain this developmental trend, we propose here a novel theory about how the brain develops a mature representation of numeric magnitude. The theory holds that: (1) at any given age, the brain represents numeric magnitudes using both a logarithmically-compressed code and a linear code, with the probability of a number being processed by the linear code increasing with age and experience; and (2) logarithmic-coding is predominantly processed in frontal areas, whereas linear coding is predominantly processed in parietal areas. The origin of this hypothesis stems from behavioral studies on the development of numeric representations (Booth & Siegler, 2006; Opfer & Siegler, 2007; Opfer & Thompson, 2008; Siegler & Opfer, 2003; Siegler, Thompson, & Opfer, in press; Thompson & Opfer, 2008; Thompson & Opfer, in press). In these studies, children and adults were asked to estimate the position of numbers on a blank line with the endpoints labeled 0 and 100, 0 and 1000, or 0 and This estimation task is particularly revealing about representations of numeric value because it transparently reflects the ratio characteristics of the number system. Overall, younger children s estimates typically follow Fechner s Law and increase logarithmically with actual value, whereas older children s estimates increase linearly. At any given age, however, individual children use both logarithmic and linear representations of number, depending on numerical context. That is, for very large numeric contexts (e.g., on and number lines), children s estimates increase logarithmically; however, the same children will use linear representations when estimating the magnitudes of numbers for small numeric contexts (e.g., on number lines). If our developmental theory is correct, it should be possible to identify two different patterns in the brain for 2

15 situations that are consistent with both the logarithmic and linear representations, thereby providing neural correlates for the logarithmic-to-linear shift hypothesis. Finally, the theory predicts that even adult brains are likely to use the logarithmic code for representing large numbers, much as children do for a wider range of numbers. To test our theory, we asked participants to judge whether a number had been accurately marked on the number line, and we evaluated the components of Event- Related Potentials (ERP) within 1s after participants were given a number to estimate and 1s after shown a target position for that number. By evaluating ERP components related to numeric estimation in the number line task, we were able to test several predictions derived from our developmental hypothesis. Specifically, we were able to provide a novel test of whether subjects expected positions on numbers on number lines to increase linearly, logarithmically, both, or neither with numeric value, and we were able to test whether these expectations were related to the topography of numerical processing, and to previous experience with the numbers to-be-estimated. In the following sections I will do a background review of some of the most significant findings that inform the work in this thesis. Initially I will review some of the major findings regarding the way numeric magnitude is represented in the brain. Next, I will review some behavioral data that reveals some clues about how numeric magnitude is represented in the brain both in adults and in children. Finally, I will end with a description of the studies presented here and of the general purpose they are designed to fulfill. How is numeric magnitude represented in the brain? With the improvement in neuroimaging techniques, there has been an increase in the number of studies that try to find neural correlates for the processing of numbers in the brain. Among the techniques that have yielded important findings are the functional magnetic resonance imaging (fmri) studies, single cell recording, and ERP. 3

16 As it would be expected of domains as complex as number and numerosity processing, many areas of the brain are involved. Therefore, an attempt to localize a single brain area as the place where everything is processed is very likely a major simplification. However, a robust finding in experiments that use fmri techniques is that the HIPS area plays a key role in numeric magnitude processing (Dehaene et al., 2003, for a review). Some of the tasks that have yielded high activation of HIPS are distance effects in comparisons of one-digit (Ansari et al. 2005; Chochon et al., 1999; Pinel et al., 2004; Venkatraman et al. 2005) and two-digit numbers (Pinel et al., 2001; Wood, Nuerk & Willmes, 2006), approximate addition and subtraction tasks (Chochon et al., 1999, Dehaene et al. 1999; Lee, 2000; Simon et al. 2002; Venkatraman et al. 2005), response to changes in numerosity in habituation tasks (Cantlon et al., 2006; Piazza et al., 2004), and recognition of meaning of signed numbers by hearing impaired adults (Masataka et al., 2005). It is worth noting that these studies use different tasks, stimuli, and populations, thus showing that the general importance of HIPS for magnitude processing. Moreover, Dehaene et al. (2003) propose that this area is number specific and it is the place for an abstract, notation-independent representation of numeric magnitude. In single cell recording studies with monkeys, Nieder & Miller (2004) found populations of neurons in the posterior parietal cortex that are selectively sensitive to different numerosities. Moreover, they found that the tuning curves of these populations of neurons are increasingly overlapping, providing a viable mechanism in the brain for the analog non-symbolic logarithmic representation discussed above. This finding of numerons in the posterior parietal cortex is consistent with the claim that HIPS is a number specific area. Another imaging technique that has produced interesting results regarding numerical representation in the brain is the ERP. Similar to what has been presented 4

17 above, studies using ERP in adults have found a notation-independent representation in the HIPS area (Libertus et al. 2007; Szucs & Csepe, 2005). Moreover, there are four major components that have been associated with magnitude processing (Deahene, 1996). The first ERP component that shows significant differences when comparing close numbers vs. far numbers is an occipito-temporal N1 component ( ms) related to number identification. The second component is the transition between the N1 and the P2p (second posterior positivity) in parieto-occipitotemporal electrodes. The third component corresponds to the amplitude of this second positivity ( ms). These last two components were related to magnitude comparison. Finally, the P3 ( ms) component was modulated by the distance effect and was related to the beginning of response preparation. A source localization analysis showed that the main location for processing this information was also located in the right IPS. Similarly, other studies have found effects in the N1 and P3 components that reflected processing of magnitude (Ullsperger, 1995) even when it is not taskrelevant (Schwarz & Heinze, 1998). Therefore, there is convergence in the findings obtained with fmri and ERP methodologies. In one of the very few ERP studies that has attempted to compare children and adults in a number comparison task, Temple and Posner (1998) found that 5-year-old children already present a distance effect for symbolic and non-symbolic magnitudes and that the effects concentrated around the same components and electrodes studied by Dehaene. However, in this study they did not use source localization analysis, therefore it is not easy to draw conclusions about the exact localization of the effects they found. Taken together, the evidence from ERP studies consistently shows that for adults and children, the N1, P2p and P3 components have a strong relation with the processing 5

18 of magnitudes using both numerals and numerosities. Additionally, the general finding is that these effects are stronger in temporo-parietal electrodes that are close to HIPS. The overwhelming amount of evidence that relates HIPS to the processing of numerosity is also consistent with the general properties of the IPS as a place critical for the processing of continuous magnitudes like luminosity and size (Pinel et al., 2004; Shuman & Kanwisher, 2004). Therefore, the recruitment of the areas that process analog magnitudes like space to process numeric magnitudes provides a simple interpretation for the findings reported above. However, it is clear that HIPS is not the only brain area that plays a central role in number processing. Another area that has been involved with several tasks of number processing is the prefrontal cortex. Generally speaking, the prefrontal cortex is more active in children than adults (Ansari et al., 2005; Ansari & Dhital, 2006; Cantlon et al., 2006, Kaufmann et al., 2006) and more active in studies that use numerosities like dots, sounds or sequences of taps, than in tasks that involve numerals (Cantlon et al., 2006, Kansaku et al., 2006; Venkatraman et al., 2005). The two critical points from these findings regarding the prefrontal cortex is the importance it plays in counting discrete numerosities and the fact that although it is still active in adults, it is more active in children. Similarly, in Nieder and Miller s (2004) single cell recording study, although there were numerons located in the posterior parietal cortex, the majority of the neurons were located in the prefrontal cortex. Because the largest concentration of these numerons was found in the prefrontal cortex, this study suggests that, at least in macaque monkeys, the processing of numerosities has a very important prefrontal component. As it was the case in the fmri studies reviewed in the previous paragraph, the studies that use numerosities, instead of numerals, tend to show more activation in the prefrontal cortex. 6

19 Recently, an ERP study with 3-month-old infants using a habituation paradigm found that number deviants generated activity in right inferior frontal areas (Izard, Dehaene-Lambertz & Dehaene, 2008). Therefore, from a developmental standpoint, it seems that there is converging evidence of a prefrontal to parietal shift in the processing of numeric magnitude that merits being explored further. Additionally, there are no ERP studies that have used a number line estimation task. As we foreshadowed in the beginning paragraphs, this task has shown important developmental differences, thus it is possible that by using this task we can identify some differences that are relevant to the issue at hand. Furthermore, because it is still an open question if the number line task will elicit the components normally related with magnitude processing, it is of interest to develop a characterization of the brain activity associated to this task and its relations to other number comparison tasks. Because the research on the neural correlates of number processing has been focused in posterior parietal regions, it is still not clear what is the precise role that is being played by the prefrontal cortex. One hypothesis that is consistent with what is known about the prefrontal cortex is that the increase in activation in studies that use numerosities instead of numerals, could signal domain-general attentional demands or processes of response selection that get active in the presence of more difficult tasks. Consistent with this hypothesis, it would be expected to find less activation in prefrontal cortex for numerals than for numerosities because we have experienced throughout our lives extensive training with numerals and its process could be more automatic. Similarly, this would explain the results in children because it is reasonable to assume that the tasks used are more difficult for children than for adults. Likewise, the single cell recording results could be explained from this perspective because the use of a delayed match to sample task typically demands resources from prefrontal cortex. Moreover, because most of the studies mentioned above that have found effects in the 7

20 prefrontal cortex were comparisons that used exclusively numerical stimuli, there is a confound between the numeric distance effect and task demands. Therefore, there is no way to test whether the results are due to attentional demands or number-specific processing. Despite these results, Cantlon et al. s (2006) study found greater activity in the prefrontal cortex in children for numerosity deviants vs. shape deviants in a habituation paradigm. Therefore, this finding rules out the response selection hypothesis because there was no response needed, and brings doubts about the attentional demands hypothesis because there should not be significant differences in attentional demands between the shape deviant and number deviant trials. A second possibility is that prefrontal activation is related to active maintenance (O Reilly & Munakata, 2000). If this were the case, it would make sense that studies that use numerosities show more activity in the prefrontal cortex than studies that use Arabic numerals. Similarly, children would show greater prefrontal activity because they would have more difficulty keeping the information active for enough time to perform the appropriate task. If this hypothesis is true, then there should not be a difference in activation of the prefrontal cortex in a number line task because it is not a task that requires major maintenance demands. Therefore, this study could provide some evidence for or against this hypothesis. Finally, as it was mentioned before, it is possible that the prefrontal activation is more specific to number and is the result of the amount of familiarity that the person has with the modality of the magnitudes, or the specific range of numbers. In this case, the prefrontal cortex would originally be critical for numeric representation and only through extensive learning this process would be automatized and transferred to a posterior cortex (HIPS). Therefore, if the logarithmic-to-linear shift hypothesis is true, adults linear representation should be more entrenched and therefore should produce 8

21 activation only in HIPS, especially with numerals. On the other hand, children would be just starting to learn this linear representation, and therefore their representation for Arabic numerals should not be completely entrenched in HIPS yet. Additionally, children should show more activation of prefrontal cortex than adults, both with numerals and numerosities. Rivera, Reiss, Eckert and Menon (2005) found that there is a developmental shift from prefrontal cortex to the Left IPS while performing arithmetic tasks. The authors found a negative correlation between age and prefrontal cortex activity and a positive correlation between age and Left IPS activity. Moreover, this activity was not related to task difficulty or gray matter density. Perhaps the most interesting aspect of this finding is that other studies have reported increased recruitment of the prefrontal cortex with age for skills such as word reading, response inhibition and working memory (Booth et al., 2003; Kwon, Reiss & Menon, 2002). Thus, consistent with the hypothesis proposed in this paper, these results suggest that the developmental pattern of change in the brain seems to be more related to the automatization of the representation of numbers than to other domain general processes associated to the prefrontal cortex. In conjunction with the results from fmri studies that use children, reviewed above, single cell recording studies bring support to a more numeric-specific role of the prefrontal cortex than just the domain-general process related to difficulty or attentional demands. Recently, Nieder and Merten (2007) extended their findings to numerosities up to 30, and again, in this study the neurons that coded for these numerosities were found in the prefrontal cortex. Similarly, Diester and Nieder (2007) taught monkeys to associate arbitrary shapes (numerals) to numerosities from 1 to 4 and after months of training recorded the activities of neurons in the prefrontal cortex and the IPS. The authors found that the proportion of association neurons that responded to both, sets of numerosities and 9

22 numerals was overwhelmingly higher for the prefrontal cortex in comparison with the IPS. These results support the view that the prefrontal cortex initially plays a major role in number processing and that the association of arbitrary shapes (e.g. Arabic numerals) is originally done there. Thus, these studies suggest that the involvement of the IPS as an abstract representation of numeric magnitude depends on learning and repeated exposure to the associations between the numerals and the magnitudes they represent. Therefore, evidence from single cell recordings appears to give support to a developmental interpretation for a prefrontal to HIPS shift of the abstract numeric magnitude representation. As it was briefly mentioned above, the developmental hypothesis outlined here was originated from results from behavioral experiments in cognitive development research. Therefore to fully understand this hypothesis it is important to summarize some of the behavioral signatures of number representation and number estimation that have been useful to understand the representation of numbers. In particular, studies on the development of number estimation have led to the proposal of a logarithmic to linear shift hypothesis of number representation that provides support for the frontal to parietal shift hypothesis argued in the preceding paragraphs. Behavioral evidence from numerical estimation: Several studies have found that although numerosity is a property of sets of discrete objects, animals and humans represent numerosity as a nonverbal analog magnitude that works in the same way as the representation of continuous quantities like length or brightness. This means that representations of sets follow Fechner s law. This property has been demonstrated by the presence of a size effect and a distance effect in number discrimination tasks (Dehaene, Dehaene-Lamberz & Cohen, 1998; Gallistell, & Gelman, 2000; Moyer & Landauer, 1967; Sekuler & Mierkewitz, 1977; Van Oeffelen & Vos, 1982). The distance effect refers to the fact that it is increasingly more difficult to 10

23 discriminate between two numbers if the distance between them decreases. Thus, it is harder to discriminate between 5 and 9 than between 5 and 15. The size effect means that even with equal distances, numbers are increasingly difficult to discriminate as the magnitude increases. As a result, it is harder to discriminate between 15 and 19 than it is to discriminate between 5 and 9. These two results have been found robustly in different species and both with non-symbolic numerosities and symbolic numerals (Moyer & Landauer, 1967; Van Oeffelen & Vos, 1982). Size and distance effects have two implications about the representation of numerical value. First, they suggest representations are noisy and inexact with a Gaussian distribution of activations around the representatives ( numerons ) of each numerosity. This leads small differences between numbers to be harder to discriminate than large differences (i.e., distance effects). Second, they suggest that the psychological distance between numbers decreases with magnitude, much like the compressed spacing of numerals on a logarithmic scale. This causes large numbers to be harder to discriminate than small numbers (i.e., size effects). In support of the log-gaussian hypothesis, young children s placement of symbolic numbers on number lines is highly logarithmic, especially for large and unfamiliar numeric ranges (Booth & Siegler, 2006; Opfer & Siegler, 2007; Opfer & Thompson, 2008; Siegler & Opfer, 2003; Siegler, Thompson, & Opfer, in press; Thompson & Opfer, 2008; Thompson & Opfer, in press). For example, when placing numbers on a number line, 2 nd graders estimates increase logarithmically with actual numerical value, whereas their placements of numbers on a number line are highly linear (Siegler & Opfer, 2003). Similarly, a test of people from the Mundurucu tribe an Amazonian tribe that has only a few words to denote numbers and that has very limited contact with Western societies and formal education (Dehaene, Izard, Pica & Spelke (2006) showed that their estimates for a 0-10 number line were logarithmic. 11

24 However, for a group of participants that had contact with formal education, their estimates in this range were linear (Dehaene, Izard, Spelke & Pica, 2008). This change in performance apparently arises from experience, and not just maturation. Evidence to support this claim comes from studies that show a logarithmic-to-linear shift after only providing feedback on a single trial for a number that has a value that differs quite strongly from that represented in a log-gaussian manner (Izard & Dehaene, 2008; Opfer & Siegler, 2007). Moreover, the developmental change in estimation performance does appear to occur at the level of the representation: training that induces linear numerical estimates transfers to other numerical judgment tasks (Opfer & Thompson, 2008), even when it entails a cost in accuracy (Opfer & DeVries, 2008; Thompson & Opfer, 2008). An account that seems to fit the data that has come from the number line task is that, initially, children s representation of numerals is logarithmically compressed, in the same way as the analog non-symbolic representation that we share with other animals. Later, with experience, children will encounter feedback about the inappropriateness of the logarithmic representation and will develop a linear representation on top of the original logarithmic one. However, this transition will only occur as a function of experience. Hence, because the frequency of smaller numbers in text and speech is disproportionally higher compared to larger numbers (Dehaene & Mehler, 1992), the linear representation will form earlier for smaller ranges of numbers. A possible consequence of this hypothesis for the neural representation of numbers is that experience leads the representation to change its location from prefrontal cortex to a more posterior site (i,e, HIPS). Evidence from perceptual learning has shown that the representation of complex conjunctive stimuli can change locus of representation to more basic levels of processing, both within the visual cortex (Mukai et al., 2007) and between the prefrontal cortex and visual cortex (Eriksson, Larsson, Nyberg, 2008). As a result, information changes from being processed serially and with 12

25 effort, to being processed in parallel and automatically. Therefore, for the case of numeric representation, the abstract representation that is needed for processing numeric magnitude regardless of shape, size, modality or context, could very likely be originally in prefrontal cortex, and then gradually shifted to HIPS based on the extensive amount of experience we have with tasks that require the extraction of magnitude. Present study The present study aims to use event-related potentials to provide a better temporal characterization of what happens in the brain during numerical processing, particularly number line estimation. At present, no previous studies have examined brain activity during number line estimation; consequently, it is unknown whether number line estimation relies on brain regions activated by the more commonly used number comparison tasks. For this purpose, I will present a range of Arabic numerals to participants and examine ERPs when hatch marks appear at various locations on a number line. The general expectation is that number line estimation requires access to representations of numerical magnitude, thereby requiring activation of the same cortical areas (in posterior parietal cortex) that are subject to distance and size effects in number comparison tasks. Additionally, this study examines whether numerical magnitudes are represented linearly, logarithmically, both, or neither. To examine representations of numerical magnitude, ERPs will be compared when locations of numbers on the number line appear in the linear (correct) position, the logarithmic position, and in a position midway between the linear and logarithmic position. Specifically, participants were asked to judge when a hatch mark on the number line was placed in the correct position for a given number. Components of event-related potentials to positions of numbers on number lines are potentially quite revealing about representations of numerical magnitude. Generally, 13

26 the greatest P3 amplitudes are elicited by targets that most violate subjects expectations. Thus, a higher P3 response to numbers marked in non-linear positions would indicate that the number was expected to appear in the linear position, whereas a higher P3 response to numbers marked in linear positions would indicate the number was expected to appear in a non-linear position. The N1 component of ERPs to positions of numbers on the number line is similarly revealing. Generally, the greatest N1 amplitudes are elicited when targets match the subject s orientation of attention. Thus, higher N1 responses to numbers marked in the non-linear position would indicate the numbers were expected to appear in the non-linear positions, whereas a higher N1 response to numbers marked in the linear position would indicate that they were expected to appear in the linear position. Previous behavioral studies have suggested that linear and logarithmic representations can co-exist and compete for use. Results from number line estimation tasks show that individual children know and use both linear and logarithmic representations of numerical magnitude; that with age and experience, children rely increasingly on correct, linear representations rather than intuitive, logarithmic ones; and that the same integers can elicit either a logarithmic or linear pattern of magnitude estimates, depending on the size of the numeric interval tested (Opfer & DeVries, 2008; Opfer & Siegler, 2007; Opfer & Thompson, 2008; Siegler & Opfer, 2003). Similar use of logarithmic representations have also been found among adult Amazonian indigene (Deahene et al., 2008; Pica et al., 2004), and logarithmic representations of numeric magnitude are also implied by adults numeric comparisons when pressured by time or economic competition (Moyer & Landauer, 1967; Furlong & Opfer, 2009; though see Cantlon et al., 2009, for an alternative interpretation). Thus, the design of our study allowed us to examine whether logarithmic representations also co-exist with linear representations among Western, college-educated students. Finally, this study can 14

27 provide information regarding the spatial topography of numerical representation. In particular, we were interested in exploring the possible roles that prefrontal cortex and HIPS play during number line estimation. 15

28 Method Participants Participants (N = 21, mean age = 20.5, 8 female) were recruited from an introductory psychology class and were awarded course credit for their participation in the experiment. Nineteen participants were right handed, and all had normal or corrected to normal vision. Task Participants were presented with a line flanked by a 0 and a 1000, an Arabic numeral, and were asked to determine if a hatch mark on the number line corresponded or not to the given numeral. For each of the numerals, the hatch mark appeared in three different positions. On the linear trials, the position of the hatch mark across the number line corresponded exactly to its linear position (e.g. 100 would appear at 10% of the line); on the logarithmic trials, the position of the mark was where log (n) would be (e.g. 100 would appear at 66% of the line); and in the log-linear trials, the position of the mark was half-way between the linear position and the logarithmic position. Stimuli Each problem presented a blank number line with a width of 255 pixels, labeled with 0 on the left end and 1000 on the right end. The numbers presented appeared on the top of the screen 192 pixels over the line (half point between the top of the screen and the number line). The numbers tested were 5, 78, 150, 606, 725 and 938. These numerosities were selected because they sample the whole length of the line and also maximize the discriminability between the linear and logarithmic representations. All 16

29 stimuli were presented in a dark and sound attenuated room using DirecRT (Jarvis, 2006). Procedure and design: Participants were instructed to identify if the position of the hatch mark on a number line corresponded to the numeral presented by pressing one key ( Q ) if the position of the hatch mark was correct, and by pressing another ( P ) if the position of the hatch mark was incorrect (keys were counterbalanced between participants). Additionally, they were instructed to answer as quickly and accurate as possible. Finally, participants were instructed to hold as still as possible, to avoid blinking, and to try to limit their blinks to the periods between trials (i.e. after responding). At the beginning of each trial, the number line with the marked end points appeared and a fixation was placed where the target numerals were going to be shown for a period of 1 second. Next, the stimulus (i.e. the numeral) replaced the fixation for another 1-second interval. After this period, the hatch mark was placed either in the linear, logarithmic or log-linear position. Once the hatch mark was in place, participants had to decide if the mark was correct or incorrect and press the appropriate key. A 2000 ms intertrial stimulus interval (ISI) was used (See Figure 1). Participants were tested on three different sets of trials and the design of the study was all within subjects. Thus, on each block, participants encountered each of the six numerals compared to three possible hatch mark positions (linear, logarithmic, loglinear). The experiment consisted of 16 blocks and presentation of the trials was randomized within each block. ERP recording procedure: After attaining informed consent from participants, a NuAmps quick cap with 32 Ag/AgCl electrodes (Compumedics Neuroscan, El Paso, TX, USA) was placed on their heads to record their brain activity (see Figure 2). Linked ears served as reference during 17

30 recording. Before the beginning of the experiment, impedances were held below 40 kω 1. The electroencephalogram (EEG) was amplified with an A/D conversion rate of 1000 and a gain of 250mV. Finally, a recording low-pass filter of 300Hz was used. Before analysis of the data, the raw EEG data were processed offline using BESA (Version 5.2). Raw data were re-referenced to an average of all electrodes to get a better approximation to the true voltage average over the whole head (Dien, 1998). Additionally, a digital 0.1 to 30Hz bandpass filter was used to eliminate noise from nonneural origin such as skin potentials, muscle contraction and electrical devices (Holinger et al., 2000; Luck, 2005). Also, using an artifact correction algorithm based on spatial components method (Berg & Scherg, 1994), the data were corrected to reduce ocular artifacts and blinks. After the data were corrected, an artifact rejection procedure (tailored to each individual) was conducted. In any case, trials that had ± 100µV, amplitude fluctuations from one point to the next greater than 11µV, or signal lower than.01 µv, were rejected. After artifact rejection, 7 participants who had less than 85% of the trials accepted were removed from further analyses. After artifact rejection, ERP epochs (-200ms to 1000ms) were created for the three trial types (i.e. linear, logarithmic and log-linear), for number size (i.e. small numbers and large numbers), and for hatch mark number size (i.e. hatch marks that corresponded to small numbers and large numbers). These epochs were selected to look at possible differences between linear and logarithmic representations of numbers, and to find the components relevant to the size effect, respectively. 1 Although the impedance threshold for accepting a participant was 40 kω, in reality most of the electrodes achieved impedances of 10 to 15 kω. 18

31 Results Behavioral results Number comparison is typically characterized by effects of distance and size on speed and accuracy of judgments (Moyer & Landauer, 1967). In this study we obtained similar results for judgments of number line placements (see Table 1). Consistent with distance effects, log-linear trials, which were closer to the correct (linear) placements than logarithmic ones, required more time to solve and resulted in the lowest accuracy rates. Consistent with size effects, judging the location of large numbers (i.e. larger than 500) on the number line required more time than judging the location of small numbers (i.e. smaller than 500), with accuracy also being lower for placement of large numbers compared to small numbers. Finally, there was evidence of interactive effects of size and trial type, with larger effects of trial type for small numbers (ω 2 =.54) than for large numbers (ω 2 =.32) on reaction times. This interaction is interesting because it suggests that representations of small numeric magnitudes are more strongly linear and nonlogarithmic than representations of large numeric magnitudes, leading to less discriminability between trial types for the large magnitudes. A potential problem with accuracy measures, such as those reported above is that they can fail to detect systematic response biases. To address this issue, d analyses were conducted for each number presented to assess the discriminability of the linear, logarithmic, and log-linear number line placements. Because performance of participants was near ceiling, hits and false alarm rates were corrected. Specifically, hit 19

32 rates were constructed by the formula (hits + 1)/(total trials + 2), and false alarm rates were constructed by the formula (false alarms + 1)/(total trials + 2). As predicted by the size effect, discriminability between the linear, and the logarithmic and log-linear trials declined with numeric size (see Figures 3-4). This result was confirmed by a one-way repeated measures ANOVA (F(5,100) = 36.83, p <.001, ω 2 = 0.59). An alternative explanation of this result is that it is due to the distance between the linear and logarithmic trials not being constant throughout the whole range of numbers (See Figure 3). Thus, it is possible that the reason why discrimination decreases for the numbers 725 and 938 is because the distances between the linear and logarithmic trials decrease too. To test this alternative hypothesis, I performed a planned comparison between two numbers that differ in size but that have the same distance between the linear and logarithmic hatch mark positions (5 and 725). As predicted by the size effect, even though the distance between the linear and logarithmic trials is equal for these two numbers, discriminability was significantly smaller for 725 (d = 2.25, SD = 0.78) than for 5(d = 3.04, SD = 0.56; p <.001). Electrophysiological results One of the main goals of this study was to understand the temporal characterization of the number line estimation task. Behavioral results showed that participants judged linear trials as correct whereas they judged logarithmic and loglinear trials as incorrect. Also, consistent with the distance effect, judgments for the loglinear trials were less accurate and took more time. Finally, consistent with the size effect, participants judgments for larger numerals were less accurate and took more time than their judgments for smaller numerals. However, behavioral measures only allow us to look at the end result of the processing. On the other hand, ERP measures allow a continuous inspection of processing in a millisecond-by-millisecond basis. Therefore, to gain insight into this pattern of results the use of EEG waveforms evoked 20

33 by the presentation of the hatch marks in the number line estimation task can be very useful. Two components that could be relevant for understanding these behavioral results are the N1 and P3 components. The N1 component has been found to be present in number identification (Dehaene, 1996) and therefore, could help explain the identification of linear trials as correct. On the other hand, the P3 component is related to surprise (Luck, 2005) and therefore could be involved in the rejection of logarithmic and log-linear trials. To understand the temporal characterization of the number line estimation task, average waveforms were computed for the three experimental trials (i.e. linear, logarithmic, log-linear). Additionally, these waveforms were averaged into four different electrode sites with the purpose of reducing the experiment wise error caused by computing multiple statistical comparisons. Because in previous studies there were no significant effects in the midline, these electrodes were not included in the analyses. The frontal left (FL) electrode site was computed by averaging the electrodes FP1, F3, F7, FC3, and FC7. The frontal right (FR) electrode site was computed by averaging the electrodes FP2, F4, F8, FC4, FC8. The parietal left (PL) electrode site was computed by averaging the electrodes CP3, TP7, P3, P7. The parietal right (PR) electrode site was computed by averaging the electrodes CP4, TP8, P4, P8 (See Figure 2). A visual inspection of these waveforms showed an effect of trial type at the N1, P3 and P1 peaks (see Figures 5-6). To calculate the N1 peak, mean amplitudes were computed over the ms time window. To calculate the P3 peak, mean amplitudes were computed for the ms time window. Although in previous experiments P2p and P3 components were discriminated, in the present study, there seems to be a strong overlap between these two peaks. Therefore, the time window mentioned above corresponds to the average of both P2p and P3 peaks. Finally, because there were no theoretical reasons to expect an effect at the P1 component, and because 21

34 there could be a high overlap between the P1 and N1 components (Luck, 2005), the P1 peak was not included in further analyses. The visual inspection of these waveforms is consistent with the main hypothesis from the study (see Figures 5-7). First linear trials generated a greater N1 peak than both the logarithmic and log-linear trials, especially in parietal electrode sites. Moreover, at frontal electrode sites, the logarithmic trials generated a greater P3 peak than the loglinear trials, and in turn, the log-linear trials generated a greater P3 peak than the linear trials. These effects suggest that even before the behavioral response is effectuated, there is a strong recognition of the linear placements of numbers followed by a signal of surprise related to the logarithmic and log-linear placements of numbers. To test for these effects statistically, a 3-way (trial type: linear, logarithmic, loglinear x electrode site: FL, FR, PL, PR x component: N1, P3) repeated-measures ANOVA was conducted on the mean amplitudes calculated for the N1 and P3 time windows. All reported p-values are Greenhouse-Geisser corrected for violations of sphericity assumptions. Results indicated a significant component x electrode interaction, (F(3,39) = 8.37, p <.001, η 2 = 0.39). This effect is largely due to a larger N1 component in parietal sites compared to frontal sites. Furthermore, as expected, a trial type x electrode x component interaction was significant (F(6,78) = 3.85, p =.033, η 2 = 0.23). A further exploration of this interaction determined that it was due to significant simple main effects of trial type at the N1 component for the PL (F(2,26) = 8.07, p =.003, ω 2 = 0.25) and PR (F(2,26) = 14.81, p <.001, ω 2 = 0.40) electrode sites, and to significant simple main effects of trial type at the P3 component for the FR (F(2,26) = 4.69, p =.048, ω 2 = 0.16) and PL (F(2,26) = 18.25, p <.001, ω 2 = 0.45) electrode sites. For the simple main effects at the N1 component (see Figure5), pairwise comparisons revealed that at the PL electrode site, there was a significant difference in mean amplitude between the linear trials (M = -0.11, SD = 0.81) and both the logarithm 22