Flora Schwartz
email: florasch@stanford.edu

Justine Epinat-Duclos

Jessica Léone

Alice Poisson

Jérôme Prado
email: jerome.prado@univ-lyon1.fr

Neural representations of transitive relations predict current and future math calculation skills in children

Keywords: math learning, transitive reasoning, arithmetic, neuromarker, fMRI

A large body of evidence suggests that math learning in children is built upon innate mechanisms for representing numerical quantities in the intraparietal sulcus (IPS). Learning math, however, is about more than processing quantitative information. It is also about understanding relations between quantities and making inferences based on these relations.

Consistent with this idea, recent behavioral studies suggest that the ability to process transitive relations (A > B, B > C, therefore A > C) may contribute to math skills in children.

Here we used fMRI coupled with a longitudinal design to determine whether the neural processing of transitive relations in children could predict their current and future math skills.

At baseline (T1), children (n=31) processed transitive relations in an MRI scanner. Math skills were measured at T1 and again 1.5 years later (T2). Using a machine learning approach with cross-validation, we found that activity associated with the representation of transitive relations in the IPS predicted math calculation skills at both T1 and T2. Our study highlights the potential of neurobiological measures of transitive reasoning for forecasting math skills in children, providing additional evidence for a link between this type of reasoning and math learning.

Introduction

The ability to understand and manipulate logical relations has long been thought to be associated with the acquisition of math skills in children. For instance, the pioneering developmental psychologist Jean Piaget famously proposed that the development of math cognition relies on the emergence of logical skills from childhood to adolescence [START_REF] Piaget | The child's conception of number[END_REF]. Educational policies have also often been influenced by the idea that learning math would contribute to the development of logical thinking in children, a claim whose premises can be found in the writings of philosophers such as Plato, Locke, and Bacon (Inglis and Attridge 2017). To date, however, evidence that math learning is related to logical reasoning remains scarce, most likely because research has largely focused on investigating to what extent math development relies on evolutionally old mechanisms for representing quantities [START_REF] Feigenson | Core systems of number[END_REF]).

Yet, mathematical development is about more than simply processing quantitative information. It is also about understanding relations between those quantities and making inferences based on these relations. For example, a type of logical relation that is prevalent in many math domains is that of transitivity. Transitivity is a property that arises from a set of items that can be ordered along a single continuum [START_REF] Wright | Reconceptualizing the transitive inference ability: A framework for existing and future research[END_REF]. A relation is said "transitive" when it allows reasoners to infer a relationship between two items (e.g., A > C) from two other overlapping pairs (e.g., A > B; B > C). The ability to recognize transitive relations and make associated inferences may contribute to the acquisition of many mathematical concepts. For example, transitive reasoning is fundamental to the acquisition of measurement skills in children [START_REF] Inhelder | The Growth of Logical Thinking from Childhood to Adolescence: An Essay on the Construction of Formal Operational Structures[END_REF][START_REF] Bryant | Spontaneous measurement by young children[END_REF][START_REF] Rabinowitz | Development of the middle concept[END_REF]. It also allows for the processing of ordinal and categorical information, which in turn contributes to domains such as arithmetic, algebra and geometry [START_REF] Bryant | Spontaneous measurement by young children[END_REF][START_REF] Rabinowitz | Development of the middle concept[END_REF][START_REF] Wright | Reconceptualizing the transitive inference ability: A framework for existing and future research[END_REF]).

Consistent with a role for transitive reasoning in math learning, a few studies have shown that the ability to understand transitive relations relate to math skills in children. For example, [START_REF] Morsanyi | The link between logic, mathematics and imagination: evidence from children with developmental dyscalculia and mathematically gifted children[END_REF] found that 10-year-olds with math learning disability (i.e., dyscalculia) performed significantly worse than typically developing controls in a transitive reasoning task. In contrast, children with high math ability performed significantly better than typically developing control on that same task. This is in line with a prior report showing that math performance is generally related to transitive (as well as conditional) reasoning ability in elementary school children [START_REF] Handley | Working memory, inhibitory control, and the development of children's reasoning[END_REF]).

What neural mechanisms may underlie the relationship between transitive reasoning and math learning in children? Two lines of evidence suggest that this relationship may be mediated by brain mechanisms involved in relational processing in the posterior parietal cortex [START_REF] Wendelken | Meta-analysis: how does posterior parietal cortex contribute to reasoning?[END_REF]. First, neuroimaging studies indicate that the processing of transitive relations consistently activates the intraparietal sulcus (IPS) (for a review, see [START_REF] Prado | The brain network for deductive reasoning: a quantitative meta-analysis of 28 neuroimaging studies[END_REF], a region that is also typically involved in tasks that require the manipulation of numbers [START_REF] Nieder | Representation of number in the brain[END_REF]. Arguably, both transitive reasoning and numerical cognition require individuals to process ordinal information. Therefore, this common reliance on IPS mechanisms may stem from the fact that this region is central to ordinal processing more generally, in line with studies showing that the IPS is also activated when participants represent learned series such as letters, days or months [START_REF] Fias | Processing of Abstract Ordinal Knowledge in the Horizontal Segment of the Intraparietal Sulcus[END_REF]). However, it is also possible that transitive relations and numbers are encoded in similar regions of the IPS because they both rely on representations that are inherently spatial and for which the posterior parietal cortex is key (Prado, Noveck et al. 2010). For example, numbers are often thought to be represented on a mental number line [START_REF] Hubbard | Interactions between number and space in parietal cortex[END_REF] in adults and children. This is notably suggested by the behavioral distance effect observed in number comparison tasks (i.e., reaction times decrease with the distance between numbers; [START_REF] Moyer | Time required for judgements of numerical inequality[END_REF], indicating that numbers that are close (e.g., 4 vs 5) are more difficult to distinguish than numbers that are far (e.g., 2 vs 8). This phenomenon has also been reported in transitive reasoning (i.e., reaction times decrease with the distance between items in a transitive ordering; Prado, Noveck et al. 2010), suggesting that transitive items are also arranged on a mental representation of space [START_REF] Alfred | Putting the pieces together: Generating a novel representational space through deductive reasoning[END_REF].

Furthermore, activity in same region of the IPS has been found to decrease with the distance between numbers [START_REF] Pinel | Modulation of parietal activation by semantic distance in a number comparison task[END_REF][START_REF] Pinel | Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments[END_REF][START_REF] Mussolin | Symbolic and nonsymbolic number comparison in children with and without dyscalculia[END_REF] or transitive items (Prado, Noveck et al. 2010), suggesting that transitive relations and numbers engage common spatial representations in the IPS.

A second line of evidence suggesting that the relationship between transitive reasoning and math learning may be mediated by the posterior parietal cortex is provided by a recent neuroimaging study. In that study, we contrasted performance and brain activity of typically developing children to children with math learning difficulties during a transitive reasoning task. While transitive relations were associated with IPS activity in typically developing children, that was not the case in children with math learning difficulty (Schwartz, Epinat-Duclos et al. 2018). Children with math learning difficulty also showed significantly less activity in the IPS than typically-developing children during the transitive reasoning task. Thus, the extent to which IPS mechanisms process transitive relations may be related to levels of math competence in children.

It is important to note that our previous study only provides indirect correlational evidence that math abilities may be associated with reasoning-related activity in the IPS.

However, it raises an intriguing possibility that may strengthen the claim that transitive reasoning plays an important role in math learning: current and future math skills in children might be predicted (at least to some extent) by the neural processing of transitive relations in the IPS. The goal of the present study was to formally test this hypothesis. FMRI activity of 31 typically-developing children from 9 to 13 was measured while they passively listened to transitive (e.g., A > B, B > C) and non-transitive (e.g., A > B, C > D) relations that were embedded in a coherent story that was designed to be as interactive (it took the form of a "choose your own adventure story") and as engaging as possible (Schwartz, Epinat-Duclos et al. 2018). This task allowed us to measure activity associated with transitive relation in a relatively ecological context (i.e., discourse comprehension). Arithmetic and math problemsolving abilities were measured for each child at the time of the fMRI session (T1), as well as 1.5 years later (T2). The predictive power of the neural representations of transitive relations in the IPS on current and future math skills was assessed using a machine learning approach with cross-validation [START_REF] Gabrieli | Prediction as a Humanitarian and Pragmatic Contribution from Human Cognitive Neuroscience[END_REF]. Specifically, we evaluated whether multivariate patterns of IPS activity associated with the processing of transitive relations (as compared to non-transitive relations) at T1 could accurately predict math scores of children at both T1 and T2.

Material and methods

Sample size justification

To our knowledge, this is the first study to examine the link between individual differences in math skills and the neural processing of transitive relations in typically developing children. For this reason, and because decoding accuracy in multivariate analyses may not reflect effect sizes [START_REF] Hebart | Deconstructing multivariate decoding for the study of brain function[END_REF], it is difficult to determine the optimal sample size for the multivariate analyses. However, we note that previous studies that have used multivariate analyses to predict math skills in children have used sample sizes ranging from n = 20 [START_REF] Evans | Brain structural integrity and intrinsic functional connectivity forecast 6 year longitudinal growth in children's numerical abilities[END_REF] to n = 28 [START_REF] Qin | Hippocampal-neocortical functional reorganization underlies children's cognitive development[END_REF]. Power calculations (performed with G*Power, https://stats.idre.ucla.edu/other/gpower/) indicate that such sample sizes would provide 80% power to detect a univariate brain-behavior correlation in the r=0.5 to r=0.6 range (with α = 0.05). Such a range is typically the one observed in most studies measuring univariate brain-behavior correlations (providing that regions in which correlations are observed are not selected from circular analyses; see Figure 5 in [START_REF] Vul | Puzzlingly high correlations in fMRI studies of emotion, personality, and social cognition[END_REF]. Therefore, we aimed to analyze a sample size of around 28 participants in the present study (similar to [START_REF] Qin | Hippocampal-neocortical functional reorganization underlies children's cognitive development[END_REF]. Because we expected to discard about 20% of participants due to technical, performance, or movement issues, we aimed to recruit around 35 participants.

Participants

Thirty-eight right-handed children from 8 to 13 were recruited using advertisements in schools, newspapers and social media. Seven of these participants were excluded from the analyses because of excessive head motion in at least 3 of the 4 fMRI runs (n=4), presence of a specific language impairment (n=1), and lack of behavioral data (n=2). Thus, 31 children were included in the analyses. Eighteen of these participants were already analyzed in Schwartz, Epinat-Duclos et al. (2018), in which only univariate analyses were used at T1 to compare brain activity associated with transitive reasoning between children with and without math learning difficulty. The mean age of the final sample was 10.97 years (SD = 1.37) at T1 and 12.57 years (SD = 1.39) at T2. All children were native French speakers and had no diagnosis of mental retardation or high intellectual potential. They also had no hearing deficit, no MRI contraindications and no history of neurological and psychiatric disorder. Parents gave their written informed consent and children gave their assent to participate in the experiment, which was approved by the local ethics committee (CPP Lyon Sud-Est II).

Families were paid 80 euros for their participation.

Testing

To ensure that participants had normal cognitive functioning and to provide a means of measuring math skill, children completed standardized tests at the time of scanning (T1) and after a follow-up period of between 1.45 and 1.79 years (T2) (mean = 1.60 years, SD = 0.07). First, the Nouvelle Echelle Metrique de l'Intelligence (NEMI-2) [START_REF] Cognet | NEMI-2[END_REF]) was used to obtain a measure of verbal intelligence (estimated using subtests of general knowledge, vocabulary and comparisons) and matrix reasoning at T1. Second, math skills were measured at both T1 and T2 using a French version of the Woodcock-Johnson Test of achievement (WJ III; [START_REF] Woodcock | Woodcock-Johnson III Tests of Achievement[END_REF]. To provide a comprehensive measure of math abilities in each individual, 3 subtests were used: Calculation, Math fluency, and Applied problems. The Calculation subtest is an untimed test in which participants perform math computations of increasing difficulty. The test begins with single-digit addition, subtraction and multiplication problems and progressively moves to double-digit problems (including division) and algebra, as well as logarithmic and trigonometric operations. Test administration is stopped after six consecutive errors or when the last item is reached (raw scores range from 0 to 45). In the Math fluency subtest, participants have 3 minutes to solve as many single-digit addition, subtraction and multiplication problems (raw scores range from 0 to 160). The Applied problems subtest measures the ability to analyze and solve math problems. While early items require application of basic numerical concepts (e.g., counting, performing simple addition and subtraction, reading clocks and coin values), most items require children to understand and analyze word problems. The test is untimed and testing stops after 6 consecutive errors or when the last item is reached (raw scores range from 0 to 63). Standardized IQ scores at T1 and raw math scores at T1 and T2 are shown in Table 1.

Task

Participants completed the task described in Schwartz, Epinat-Duclos et al. (2018). In the scanner, they listened to 4 stories that each included a series of 12 short scenarios (see Table 2). Each scenario ended up with a question that the child had to answer. Similar to a "choose your own story" structure, children were told to pay attention to every scenario because their responses to questions were critical for making progress in the story. Scenarios varied with respect to the type of relations involved. Twenty-four scenarios included linear-order relations such as "A is more than B", while 24 scenarios included set-inclusion relations such as "All As are Bs". In half (i.e., 12) of each type of scenario, the relations overlapped so that they could be integrated, and a conclusion could be inferred. For instance, the linear-order relations "White cows give more milk than black cows" and "Black cows give more milk than brown cows" in the bottom left cell of Table 2 should lead to the conclusion that "White cows give more milk than brown cows". Similarly, the set-inclusion relations "All old farms are made of stone" and "All farms that are made of stone are uphill" in the top left cell of Table 2 should lead to the conclusion that "All old farms are uphill". The question that followed scenarios with such transitive relations always tested whether children had inferred conclusions upon listening to the scenario (it was termed Reasoning question). The remaining scenarios included nontransitive relations, that is, relations that were not overlapping. Thus, no particular conclusion could be inferred upon listening to the scenario. For instance, the linear-order relations "The chocolate cake is baking faster than the apple pie" and "The strawberry pie is baking faster than the cheesecake" in the bottom right cell of Table 2 are non-transitive. Similarly, the setinclusion relations "All bedrooms with a red door are on the side of the chicken coop" and "All bedrooms with a green door are on the side of the barn" in the top right cell of Table 2 are nontransitive. In those cases, the question that followed simply tested whether children remembered some information that was explicitly given in the scenario and was termed Memory question (see Table 2).

Overall word count was controlled across the 4 types of scenarios. On average, word count was 69.1 for scenarios with transitive set-inclusion relations, 69.7 for scenarios with nontransitive set-inclusion relations, 73.6 for scenarios with transitive linear-order relations and 72.4 for scenarios with non-transitive linear-order relations. Because word count was not normally distributed, non-parametric testing was used to assess differences as a function of type of scenarios. A Kruskal-Wallis ANOVA indicated that word count did not differ between types of scenarios (H(3,48) = 4.14, p = 0.25).

The task was split into 4 runs that contained one story each. Each story contained 3 scenarios with transitive linear-order relations, 3 scenarios with non-transitive linear-order relations, 3 scenarios with transitive set-inclusion relations and 3 scenarios with nontransitive set-inclusion relations. Two wrap-up sentences concluded each story. Runs were randomized but scenarios were presented in a fixed order within a run. This was because scenarios were embedded in a coherent story. However, two scenarios of the same type were never following each other. Additionally, responses were counterbalanced across different variables. First, the order of the correct response was counterbalanced between the 4 types of stories across the whole experiment. Second, within a run, the order of the correct response was counterbalanced between scenarios with transitive and non-transitive relations and between scenarios involving linear-order relations and set-inclusion relations. Third, to prevent participants from developing expectations during the task and using heuristic strategies to respond to Reasoning questions, transitive relations were switched around in half of the problems. For example, linear-order relations could be presented in the order "A is faster than B and B is faster than C" in some scenarios and in the order "B is faster than C and A is faster than B" in other scenarios. Participants were instructed to press on a response button when they were ready to hear the next sentence. They were told to press on one of two other response buttons to choose between options in the question. Each story started with a red fixation cross at the center of the screen. The red cross turned orange after 6 s and green after 2 s. The green cross lasted 2 s and was immediately followed by the presentation of the first sentence of the first scenario. There was a 500 ms interval between each sentence in a scenario. A white fixation cross was displayed before and after each question for a random duration between 2 and 4 s. A fixation cross was also displayed for 20 s at the end of each story. See Figure 1 for a timeline of one scenario.

Experimental procedure

After standardized tests were administered at T1, children were familiarized with the fMRI environment in a mock scanner. They listened to a recording of the noises associated with all fMRI sequences. A motion tracker system (3D Guidance trak STAR, Ascension Technology Corporation) was used to measure head movements and provide on-line feedback to participants. Finally, children practiced the task in that mock scanner. Different stimuli were used in the practice and in the scanning sessions. Stimuli were generated using Presentation software (Neurobehavioral Systems, Albany, CA). Stories were spoken through headphones sentence by sentence. During each scenario, a black-and-white picture illustrating the setting was displayed on a computer screen that was viewed by the participants through a mirror attached to the head coil. Behavioral responses were recorded using MR-compatible keypads placed in the left and right hands. rapid scan-to-scan movement greater than 1.5 mm were identified using ArtRepair [START_REF] Mazaika | Methods and software for fMRI analysis for clinical subjects[END_REF]. Those were removed and replaced by the interpolation of the 2 nearest non-repaired volumes. Runs with more than 10% of repaired volumes were excluded from the analysis. Finally, functional images were normalized into the standard Montreal Neurological Institute (MNI) space.

fMRI: statistical modeling

Statistical modeling was performed in the context of the General Linear Model, using the exact same procedure as in Schwartz, Epinat-Duclos et al. (2018). Sentences of interest were those containing transitive and non-transitive relations (i.e., sentences 4 and 5). These sentences were modeled as epochs with onsets time-locked to the presentation of sentence 4 and with offsets time-locked to the end of sentence 6 (i.e., the wrap-up sentence). Other sentences (as well as the question) were not explicitly modeled and were considered background noise. Linear-order and set-inclusion relations were modeled within the same regressors. All epochs were convolved with a canonical hemodynamic response function (HRF). The time series data were high-pass filtered (1/128 Hz), and serial correlations were corrected using an autoregressive AR(1) model.

fMRI: ROIs definition

Regions of interest (ROIs) were regions in which greater activity was observed for transitive than non-transitive relations. The definition of these regions involved the following 3 steps. First, for each subject, brain activity associated with non-transitive relations was subtracted from brain activity associated with transitive relations across the whole-brain.

Second, these contrast images (one for each participant) were submitted to a second-level onesample t-test. Third, the resulting t-map was thresholded using the non-parametric permutationbased Threshold-Free Cluster Enhancement (TFCE) method [START_REF] Smith | Threshold-free cluster enhancement: addressing problems of smoothing, threshold dependence and localisation in cluster inference[END_REF], implemented in the TFCE Toolbox r164 (http://dbm.neuro.uni-jena.de/tfce/). As in Schwartz, Epinat-Duclos et al. (2018), we had an a priori hypothesis that transitive relations would be processed in the IPS. Therefore, clusters were considered significant at a Family-Wise Error-(FWE) corrected threshold of p < 0.05, either across the whole-brain or within an anatomical mask of the IPS (i.e., small volume correction) defined using the Anatomy Toolbox v2.2 [START_REF] Eickhoff | A New SPM Toolbox for Combining Probabilistic Cytoarchitectonic Maps and Functional Imaging Data[END_REF]. Following Schwartz, Epinat-Duclos et al. (2018), (Schwartz, Epinat-Duclos et al. 2018)the IPS mask consisted in voxels with at least 50% probability of belonging to one of the IPS subdivisions (hIP1, hIP2, and hIP3) as defined in the Anatomy Toolbox [START_REF] Choi | Cytoarchitectonic Identification and Probabilistic Mapping of Two Distinct Areas within the Anterior Ventral Bank of the Human Intraparietal Sulcus[END_REF], Scheperjans, Eickhoff et al. 2008, Scheperjans, Hermann et al. 2008).

fMRI: ROI analysis

Activity in the ROIs defined using the procedure above was analyzed using both univariate and multivariate methods. First, we calculated for each participant the average activity associated with the contrast of transitive versus non-transitive relations within an ROI by averaging the fMRI signal across all voxels within that ROI. This average activity was then correlated with Calculation, Math fluency, and Applied problems raw scores at T1 and T2 across participants.

Second, because such standard univariate regression analyses only show associations between two within-sample variables but do not have any predictive power [START_REF] Gabrieli | Prediction as a Humanitarian and Pragmatic Contribution from Human Cognitive Neuroscience[END_REF], we used a multivariate machine-learning approach with cross-validation to test whether brain activity associated with the contrast of transitive versus non-transitive relations could predict math scores of out-of-sample participants at both T1 and T2. This was done using the pipeline for pattern regression analyses provided by the Pattern Recognition for Neuroimaging Toolbox (PRoNTo) v2.0 implemented in Matlab (http://www.mlnl.cs.ucl.ac.uk/pronto/) [START_REF] Schrouff | PRoNTo: Pattern Recognition for Neuroimaging Toolbox[END_REF]. This pipeline involves a training phase and a test phase [START_REF] Portugal | Can emotional and behavioral dysregulation in youth be decoded from functional neuroimaging?[END_REF]. During the training phase, a pattern regression model is trained by associating labels with spatial patterns of activity from a "training set". During the test phase, the association between patterns and labels (learned in the training phase) is used to predict the label of novel (i.e., out-of-sample) patterns from a "test set" [START_REF] Schrouff | PRoNTo: Pattern Recognition for Neuroimaging Toolbox[END_REF]). In the present study, spatial patterns were voxel-byvoxel patterns of activity for the contrast of transitive versus non-transitive relations from each ROI. Labels were Calculation, Math fluency, and Applied problems raw scores at both T1 and T2. Following studies suggesting that Kernel Ridge Regression (KRR) (Shawe-Taylor and Cristianini 2004) is computationally effective and may give better prediction accuracy than other linear methods [START_REF] Chu | Kernel regression for fMRI pattern prediction[END_REF][START_REF] Mihalik | ABCD Neurocognitive Prediction Challenge 2019: Predicting individual fluid intelligence scores from structural MRI using probabilistic segmentation and kernel ridge regression[END_REF], we used KRR to predict math scores. Briefly, KRR estimates a relationship between samples by minimizing an error function that includes the sum of the squared differences between model predictions and regression targets as well as a model regularization term [START_REF] Taylor | Auditory prediction errors as individual biomarkers of schizophrenia[END_REF].

Partitioning between the training and the test sets was done using a leave-one-subject-out cross-validation method. This involves leaving one participant out for test and training the model on 30 (i.e., n -1) participants. The procedure was then repeated 31 times, such that each participant was left out once.

Prediction accuracy of the model was measured using the Pearson's correlation coefficient (r) and the mean squared error (MSE) of the relationship between actual and predicted scores. Positive r values indicate that the model is able to predict the outcome, while negative r values (or r values close to 0) indicate that the model fails to predict the outcome.

Statistical significance was calculated using permutation tests with 1,000 iterations. That is, the value for each metric (i.e., r and MSE) was compared to the results of 1,000 random permutations (representing the distribution of correlations corresponding to the null hypothesis). P values were calculated based on how many times the value obtained with random permutations was higher than the correlation value observed in the data (divided by 1,000).

Results

ROIs definition

ROIs involved in transitive reasoning were identified by contrasting activity associated with transitive relations to activity associated with non-transitive relations. As shown in Table 3 and Figure 2, this contrast revealed significant activity in two clusters of the left IPS and one cluster of the right IPS. These clusters defined the voxels that were used in the subsequent univariate and multivariate analyses (see Methods). Specifically, both left IPS clusters constituted the left IPS ROI (158 voxels) and the right IPS cluster constituted the right IPS ROI (154 voxels). No region outside of the IPS survived whole-brain correction for multiple comparisons.

Relationships between math scores, transitive reasoning performance, and univariate IPS activity

Transitive reasoning performance was measured using accuracy to reasoning questions during the fMRI task at T1. Accuracy to reasoning questions ranged from 0.5 to 1 (average = 0.75, SD = 0.15). Accuracy to memory questions ranged from 0.61 to 1 (average = 0.82, SD = 0.12). Table 4 contains the correlations among math scores (at T1 and T2), accuracy to reasoning and memory questions, and univariate activity associated with the contrast of transitive versus non-transitive relations in the left and right IPS ROIs (all correlations partialled out differences in age and verbal IQ between participants). Although there was no relationship between Calculation scores and accuracy to Reasoning questions, there was a significant relationship between Calculation scores and univariate activity in the left and right IPS, at both T1 and T2 (see Table 4). However, these brain-behavior correlations were specific to the Calculation subtest, as neither Math fluency nor Applied problem scores were correlated with univariate activity in the IPS (at T1 or T2). This was despite the fact that accuracy to reasoning questions did correlate with scores on these two subtests (but only at T1). Thus, univariate activity associated with the contrast of transitive versus non-transitive relations was specifically related to math calculation skills at the time of scanning and 1.5 years later.

Predictive power of multivariate activity associated with transitive reasoning

The analyses above, however, only reflect in-sample associations between brain activity during the processing of transitive relations and math abilities. As such, they do not inform on whether brain activity during the processing of transitive relations may predict the outcome of out-of-sample individuals [START_REF] Gabrieli | Prediction as a Humanitarian and Pragmatic Contribution from Human Cognitive Neuroscience[END_REF]. To address this question, we used regression-based multivariate analyses with cross-validation. As in univariate analyses, age and measures of verbal IQ (general knowledge, comparison, and vocabulary) were included as confounding variables. These analyses tested whether patterns of IPS activity associated with the contrast of transitive versus non-transitive relations could accurately predict current math score in out-of-sample individuals, over and above differences in age and verbal IQ (see Methods). Accuracy of the machine learning algorithm was assessed by examining the relationship between actual and predicted math scores at T1. As shown in Figure 3 (left), there was no relationship between actual and predicted Calculation scores in the left IPS at T1 (r = -0.07, p = 0.60; MSE = 38.05, p = 0.64) or T2 (r = -0.26, p = 0.87; MSE = 92.77, p = 0.89) . However, the relationship was reliable in the right IPS at both T1 (r = 0.60, p = 0.002; MSE = 11.87, p = 0.006) and T2 (r = 0.39, p = 0.04; MSE = 23.92, p = 0.01). Figure 4 shows the consistency between the actual and the predicted Calculation score for each participant in the left and right IPS at both T1 and T2. In contrast to Calculation scores, there was no reliable relationship between actual and predicted Math fluency scores at either T1 or T2 (all rs < 0.11, all ps > 0.32; all MSEs > 466.17, all ps > 0.30). Similarly, there was no reliable relationship between actual and predicted Applied problem scores at either T1 or T2 (all rs < 0.06, all ps > 0.39; all MSEs > 54.31, all ps > 0.36). Consistencies between actual and predicted Math fluency and Applied problem scores are shown in Supplementary Figure 1 and Supplementary Figure 2. Therefore, patterns of right IPS activity associated with transitive relations predicted math calculation skills (but not arithmetic fluency or problem-solving skills) at the time of scanning and 1.5 years later.

Discussion

Over the past decades, developmental research on math learning has largely focused on exploring the role of mechanisms for representing numerical quantities [START_REF] Ansari | Effects of development and enculturation on number representation in the brain[END_REF]). Yet, acquiring math skills also involves understanding relations between quantities and drawing inferences based on these relations [START_REF] Singley | Neurodevelopment of relational reasoning: Implications for mathematical pedagogy[END_REF]. In keeping with this idea, recent behavioral studies suggest that understanding relations of transitivity may significantly contribute to math learning in children [START_REF] Handley | Working memory, inhibitory control, and the development of children's reasoning[END_REF][START_REF] Morsanyi | The link between logic, mathematics and imagination: evidence from children with developmental dyscalculia and mathematically gifted children[END_REF][START_REF] Morsanyi | The link between math and logic in adolescence: The effect of argument form. Individual differences in judgment and decision making from a developmental context[END_REF][START_REF] Morsanyi | The link between deductive reasoning and mathematics[END_REF]). Here we tested whether the brain representations of transitive relations may predict current and future math skills in children. Specifically, fMRI activity was measured while children listened to transitive and non-transitive relations embedded in a story context. Calculation, single-digit arithmetic, and problem-solving skills were also measured at the time of scanning (T1) and 18 months later (T2). A machine learning approach with cross validation was used to test whether activity associated with the contrast of transitive versus non-transitive relations could predict math skills at both T1 and T2.

First, we found that transitive relations were associated with greater activity than nontransitive relations in the IPS. This is consistent with several previous neuroimaging studies that have also found that transitive reasoning mainly relies on regions in and around the IPS [START_REF] Goel | Functional neuroanatomy of three-term relational reasoning[END_REF][START_REF] Acuna | Frontal and parietal lobe activation during transitive inference in humans[END_REF][START_REF] Knauff | Spatial imagery in deductive reasoning: a functional MRI study[END_REF][START_REF] Knauff | Reasoning, models, and images: behavioral measures and cortical activity[END_REF][START_REF] Heckers | Hippocampal activation during transitive inference in humans[END_REF][START_REF] Fangmeier | FMRI evidence for a threestage model of deductive reasoning[END_REF][START_REF] Hinton | Neural correlates of relational reasoning and the symbolic distance effect: involvement of parietal cortex[END_REF], Prado, Noveck et al. 2010, Prado, Van Der Henst et al. 2010[START_REF] Prado | Fractionating the Neural Substrates of Transitive Reasoning: Task-Dependent Contributions of Spatial and Verbal Representations[END_REF]. For example, quantitative meta-analyses of the literature show that transitive reasoning consistently activates the left and right IPS across studies in adults (Prado, Chadha et al. 2011, Wertheim and[START_REF] Wertheim | The Neural Correlates of Relational Reasoning: A Metaanalysis of 47 Functional Magnetic Resonance Studies[END_REF]. A recent study in children also points to the involvement of the parietal cortex in transitive reasoning [START_REF] Mathieu | Distributed neural representations of logical arguments in school-age children[END_REF].

Thus, together with these previous studies, our results suggest that the IPS houses mechanisms that are important for representing and processing transitive relations. This may be because transitive relations are mapped onto spatial mental models [START_REF] Goodwin | Reasoning about relations[END_REF] that are encoded and represented in the IPS [START_REF] Alfred | Putting the pieces together: Generating a novel representational space through deductive reasoning[END_REF]).

Second, a major novel finding from the present study is that IPS activity associated with the contrast of transitive versus non-transitive relations was predictive of math skills at both T1 and T2. This predictive power, however, was limited to math calculation skills as measured by the Calculation subtest of the Woodcock-Johnson III battery. That is, it was not observed with the Math fluency subtest (reflecting arithmetic fluency) and the Applied problems subtest (reflecting broader problem-solving skills). Why would neural activity in the (right) IPS during the processing of transitive relations specifically predict current and future calculation skills? One possibility is that math calculation recruits mechanisms supporting the processing of transitive relations in the IPS because numerical operations are inherently relational (i.e., they involve relating quantities to each other). Thus, activity related to transitive reasoning in the IPS may reflect the way this region processes relational information in children [START_REF] Wendelken | Meta-analysis: how does posterior parietal cortex contribute to reasoning?[END_REF], which may in turn relate to their calculation skills. This possibility, however, is undermined by the fact that activity associated with transitive reasoning does not predict arithmetic fluency (as reflected by knowledge of single-digit problems in the Math fluency subtest). Even though they are largely thought to be retrieved from long-term memory rather than calculated [START_REF] Campbell | Cognitive arithmetic across cultures[END_REF], single-digit arithmetic problems are also arguably relational and should therefore be related to IPS activity is it reflects relational processing.

Another possibility, which may be more likely, is that both transitive reasoning and calculation skills may rely on a set of IPS mechanisms involved in ordering information in working-memory. The role of the IPS in processing numerical and non-numerical ordinal information has been suggested by several studies [START_REF] Marshuetz | Order information in working memory: an integrative review of evidence from brain and behavior[END_REF][START_REF] Majerus | The Left Intraparietal Sulcus and Verbal Short-Term Memory: Focus of Attention or Serial Order?[END_REF]. Children with math learning disability, who exhibit anatomical and functional impairment in the IPS, have also deficits when processing ordinal information [START_REF] Attout | Working memory deficits in developmental dyscalculia: The importance of serial order[END_REF]Majerus 2015, De Visscher, Szmalec et al. 2015). A prominent cognitive theory of human reasoning (i.e., the Mental Models theory) claims that integrating transitive relations relies on the representation and manipulation of ordered mental models in working memory [START_REF] Goodwin | Reasoning about relations[END_REF]. The ability to order numerical information is also fundamental to arithmetic skills [START_REF] Knops | Numerical ordering and symbolic arithmetic share frontal and parietal circuits in the right hemisphere[END_REF]Willmes 2014, Lyons, Price et al. 2014). Therefore, it is possible that a common reliance on ordinal working memory in the IPS underlies the relationship between the processing of transitive relations and arithmetic learning. Interestingly, this possibility would account for the fact that IPS activity associated with transitive reasoning is not associated with Math fluency scores. As mentioned above, single-digit arithmetic problems are typically thought to be either directly retrieved from long-term memory or automatized through procedural knowledge by the end of elementary school [START_REF] Thevenot | Ten-year-old children strategies in mental addition: A counting model account[END_REF]). Thus, these problems are likely to rely to a lesser extent on working memory resources and involve brain mechanisms supporting the retrieval of verbal information in the left temporo-parietal cortex, rather than right IPS mechanisms typically associated with working memory and more effortful calculation strategies [START_REF] Zamarian | Neuroscience of learning arithmetic-evidence from brain imaging studies[END_REF][START_REF] Prado | Distinct representations of subtraction and multiplication in the neural systems for numerosity and language[END_REF].

More generally, it is also noteworthy that the predictive relationship between neural activity associated with transitive reasoning and calculation skills was observed in the right rather than the left IPS. Both the right and left IPS are involved in math cognition [START_REF] Sokolowski | Common and distinct brain regions in both parietal and frontal cortex support symbolic and nonsymbolic number processing in humans: A functional neuroimaging meta-analysis[END_REF][START_REF] Arsalidou | Brain areas associated with numbers and calculations in children: Meta-analyses of fMRI studies[END_REF]), but a dissociation may emerge during development. It has notably been suggested that the neural mechanisms supporting math skills may develop earlier in the right than in the left IPS, notably because they are involved in non-symbolic processing [START_REF] Cantlon | Functional Imaging of Numerical Processing in Adults and 4-y-Old Children[END_REF][START_REF] Emerson | Continuity and change in children's longitudinal neural responses to numbers[END_REF][START_REF] Vogel | Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: an fMR-adaptation study[END_REF]. Thus, although transitive reasoning has typically been associated with bilateral IPS activity [START_REF] Goel | Explaining modulation of reasoning by belief[END_REF][START_REF] Knauff | Reasoning, models, and images: behavioral measures and cortical activity[END_REF][START_REF] Fangmeier | FMRI evidence for a threestage model of deductive reasoning[END_REF], Prado, Noveck et al. 2010[START_REF] Mathieu | Distributed neural representations of logical arguments in school-age children[END_REF], Schwartz, Epinat-Duclos et al. 2018), its interaction with math processing may be more readily observed in the right than left hemisphere in children.

It is interesting to note that we did not find any relationship between IPS activity associated with transitive reasoning and Applied problems scores. One possibility is that transitive reasoning does not critically support math problem-solving skills (as measured by the Applied problems subtest) in the developmental range considered. For instance, transitive reasoning may be particularly important earlier on when acquiring foundational math concepts such as measurement, serial-order and categorization [START_REF] Bryant | Spontaneous measurement by young children[END_REF][START_REF] Rabinowitz | Development of the middle concept[END_REF][START_REF] Wright | Reconceptualizing the transitive inference ability: A framework for existing and future research[END_REF]). Although this suggests that transitive reasoning and math problem-solving may be observed in children younger than the age range considered here, accuracy to reasoning questions was related to score on the Applied problems subtest in our sample (at least at the time of testing). Thus, it is likely that brain activity during transitive reasoning also relates to Applied problems scores on some levels, even though this relation may be observed in brain regions that were not identified here. For instance, the Applied problems subtest involves math word problems that may rely on brain regions involved in verbal processing and executive control in the prefrontal cortex [START_REF] Prabhakaran | Neural substrates of mathematical reasoning: a functional magnetic resonance imaging study of neocortical activation during performance of the necessary arithmetic operations test[END_REF][START_REF] Newman | The effect of problem structure on problem-solving: an fMRI study of word versus number problems[END_REF]. Unfortunately, we were not able to localize frontal brain regions involved in transitive reasoning at the whole-brain level. This might be investigated in future studies.

Finally, we found that both univariate and multivariate activity associated with the neural processing of transitive relations in the IPS predicted calculation skills at T1 and T2, even though there was no behavioral relationship between transitive reasoning performance and calculation skills (over and above differences in age and verbal IQ). As pointed out by [START_REF] Evans | Brain structural integrity and intrinsic functional connectivity forecast 6 year longitudinal growth in children's numerical abilities[END_REF], this may indicate that neurobiological measures are more sensitive than behavioral measurements in predicting learning outcomes. For example, activity in the right inferior frontal gyrus has been found to predict long-term reading gains in children with dyslexia even though no behavioral measures of reading ability could do so [START_REF] Hoeft | Neural systems predicting long-term outcome in dyslexia[END_REF]). In the domain of math cognition, previous neuroimaging studies have found that neither measures of executive functions nor measures of numerical abilities could accurately predict short-term [START_REF] Supekar | Neural predictors of individual differences in response to math tutoring in primary-grade school children[END_REF] or long-term [START_REF] Evans | Brain structural integrity and intrinsic functional connectivity forecast 6 year longitudinal growth in children's numerical abilities[END_REF] changes in math performance, whereas functional and anatomical brain measurements were predictive of learning. Therefore, our findings add to the growing evidence that brain measures have a unique potential for forecasting academic skills.

In summary, the present longitudinal study indicates that the neural processing of transitive relations may predict current and future calculation skills in children. These results add to growing evidence that math learning does not only rely on the ability to process numerical quantities, but also critically involves the ability to understand and process logical relations [START_REF] Singley | Neurodevelopment of relational reasoning: Implications for mathematical pedagogy[END_REF]. Because calculation skills were not related to transitive reasoning performance, our findings also suggest that fMRI measures may be more sensitive than behavioral measures in predicting academic skills. Overall, the present results call for future studies investigating the relationship between neurobiological measures of relational reasoning and different types of math learning in younger children as well as in children with math learning disability. Participants pressed on a button to indicate that they were ready to listen to the next sentence, which was spoken after a 500 ms delay (not shown). The scenario ended with a question (Q), which was also spoken through headphones. This question was preceded and followed by a jittered interval ranging from 2 to 4 seconds. The sentences of interest considered in the analyses were sentences 4 to 6 (in red).

  fMRI: data acquisitionImages were collected with a Siemens Prisma 3T MRI scanner (Siemens Healthcare, Erlangen, Germany) at the CERMEP Imagerie du vivant in Lyon, France. The BOLD signal was measured with a susceptibility weighted single-shot Echo-planar Imaging (EPI) sequence. Imaging parameters were as follows: TR = 2000 ms, TE = 24 ms, flip angle = 80°, matrix size = 128 × 120, field of view = 220 × 206 mm, voxel size = 1.72 x 1.72 mm, slice thickness = 3 mm (0.48 mm gap), number of slices = 32. A high-resolution T1-weighted whole-brain anatomical volume was also collected for each participant. Parameters were as follows: TR = 3500 ms, TE = 2.24 ms, flip angle = 8°, matrix size = 256 × 256, field of view = 224 × 224 mm, voxel size = 0.87 x 0.87 mm, slice thickness = 0.9 mm, number of slices = 192." fMRI: data preprocessing Images were analyzed with SPM12 (Welcome department of Cognitive Neurology, London, UK). The first 4 images of each run were discarded to allow for T1 equilibration effects. Functional images were corrected for slice acquisition delays and spatially realignedto the first image of the first run to correct for head movements. Realigned images were smoothed with a Gaussian filter (4 x 4 x 8 mm full-width at half maximum). As in our previous studies[START_REF] Schwartz | The neural development of conditional reasoning in children: Different mechanisms for assessing the logical validity and likelihood of conclusions[END_REF][START_REF] Mathieu | Hippocampal spatial mechanisms relate to the development of arithmetic symbol processing in children[END_REF], Schwartz, Epinat-Duclos et al. 2018, Schwartz, Epinat-Duclos et al. 2018), volumes showing

FiguresFigure 1 .

 1 Figures

Figure 2 .

 2 Figure 2. Brain regions involved in transitive reasoning across all participants. Clusters in red

Figure 3 .

 3 Figure 3. Results of multivariate analyses. (A) Pearson's correlation coefficient (r) for the

Figure 4 .

 4 Figure 4. Individual predictions of Calculation scores. (A) Line plots showing the consistency

""

  There are farms uphill and downhill." "All old farms are made of stone." "All farms that are made of stone are uphill." "You have to find an old farm." Reasoning question: "Are you going uphill (response 1) or downhill (response 2)?" 2. "You are going uphill and you find the old farm.(response 1)" / "You are going downhill and the farmers pick you up" (response 2) "The farmers invite you in." "You need to bring your bag to your bedroom on the 2nd floor." "All bedrooms with a red door are next to the chicken coop." "All bedrooms with a green door are next to the barn." "The farmers' house is very big." Memory question: "Are you taking your bag to the 3 rd floor (response 1) or to the 2 nd floor (You need to milk the cows giving the most milk." Reasoning question: "Are you milking the brown cows (response 1) or the white cows (response 2)?" 3. "You are going to the 2 nd floor and bring your bag in." (response 2) / "You are going to the 3 rd floor and the farmers tell you to go down to the 2 nd floor." (response 1) "The next morning, the farmer is baking pastries." "The farmer is asking you to take them out of the oven now." "The chocolate cake is baking faster than the apple pie." "The strawberry pie is baking faster than the cheesecake." "It is very hot in the kitchen." Memory question: "Are you taking the pastries out of the oven now (response 1) or later (response 2)?" Notes. Numbers 1 to 4 indicate the order of presentation within the experimental run.

Table 1 .

 1 Demographic information and measures of verbal IQ, reasoning (matrices) abilities and math skills.

	Tables
	Variable	Mean (SD)
	Demographic information
	Age at T1 (in years)	10.97 (1.37)
	Age at T2 (in years)	12.57 (1.39)
	Verbal IQ (NEMI-II)
	General knowledge 1	105.85 (20.4)
	Vocabulary 1	106.75 (18.45)
	Comparison 1	112.15 (21.75)
	Matrix reasoning (NEMI-II)
	Raven's matrices 1	105.85 (20.85)
	Math skills (WJ-III)
	Calculation (T1) 2	18.90 (3.82)
	Math fluency at (T1) 2	51.16 (17.43)
	Applied problems at (T1) 2	37.23 (5.33)
	Calculation (T2) 2	22.03 (5.18)
	Math fluency at (T2) 2	62.97 (16.36)
	Applied problems at (T2) 2	40.03 (6.04)

Table 2 . Examples of each type of scenario (translated from French) Transitive relations Non-transitive relations Set-inclusion relations 1

 2 . "You are going on vacation to the

Table 3 . Clusters activated in the contrast of transitive versus non-transitive relations across all participants

 3 Notes. BA: Brodmann area; MNI: Montreal Neurological Institute; SVC: Small Volume Correction; FWE-corr: Family-wise error corrected.

	MNI coordinates

Table 4 .

 4 Partial correlation matrix of math scores, performance on the transitive reasoning task, and univariate IPS activity, controlling for age and verbal IQ (General knowledge, Vocabulary, Comparison).

		1	2	3	4	5	6	7	8	9	10
	1-Calculation (T1)	-
	2-Math fluency (T1)	0.50 **	-
	3-Applied problems (T1)	0.47 **	0.76 ***	-
	4-Calculation (T2)	0.49 **	0.49 **	0.62 ***	-
	5-Math fluency (T2)	0.31 †	0.74 ***	0.72 ***	0.68 ***	-
	6-Applied problems (T2)	0.30	0.64 ***	0.81 ***	0.71 ***	0.73 ***	-
	7-Reasoning (accuracy)	0.27	0.38 *	0.42 *	0.12	0.17	0.14	-
	8-Memory (accuracy)	0.41 *	0.11	0.10	-0.01	0.02	-0.05 0.47 **	-
	9-Left IPS (beta)	0.41 *	-0.02	0.06	0.50 **	0.09	0.10 0.14	-0.01	-
	10-Right IPS (beta)	0.34 †	0.02	-0.09	0.41 *	0.04	-0.07 0.07	-0.12 0.84 *** -
	Notes. †, p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001

Acknowledgments: This research was supported by a grant from the Agence Nationale de la Recherche (ANR-14-CE30-0002) to J.P. We thank the Hospices Civils de Lyon for sponsoring the research, as well as Romain Mathieu, Auriane Couderc, Inès Daguet, and the MRI engineers (Franck Lamberton and Danielle Ibarrola) at the CERMEP-Lyon platform for their assistance in collecting the fMRI data.

Conflict of Interest:

The authors declare no competing financial interests.


Hu Li

Tianbo Duan

Soumyajyoti Haldar

Biplab Sanyal

Olle Eriksson

S Hassan

M Jafri

Samar Hajjar-Garreau

Laurent Simon

Klaus Leifer
email: klaus.leifer@angstrom.uu.se

Direct Writing of Lateral Fluorographene Nanopatterns with Tunable Bandgaps and Its Application in New Generation of Moiré Superlattice

A desirable picture of graphene as an atomic thin "canvas" is to freely draw semiconducting/insulating lateral nanopatterns directly on the semi-metallic graphene surface, which is one of the most looked-for goals for monolayer device fabrications. Here, we have demonstrated a reversible electron beam activated technique, which allows to directly write semiconducting/insulating fluorographene lateral nanopatterns with tunable bandgaps on the graphene surface and a resolution down to 9-15 nm. This approach overcomes the conventional limit of semiconducting C4F in the single-sided fluorination of supported graphene and achieves insulating C2F. Moreover, by applying this technique on bilayer graphene, we have shown a new type of rectangular moiré patterns arising from the generated C2F boat/graphene superlattice for the first time. This novel technique constitutes a new approach to fabricate

graphene-based flexible and transparent electronic nanodevices with the CxF channels utilized as semiconducting or insulating counterpart, and also opens a route towards the tailoring and engineering of electronic properties of such materials in addition to the dominating triangular moiré patterns from graphene/hBN system.

Ⅰ. INTRODUCTION

The inherent two-dimensional feature of graphene raises a concept as an atomically thin "canvas", where different functional groups could be deposited onto this canvas similar to an "ink" on the surface leading to distinct materials properties in different areas on the graphene surface [1][2][3][4][5] . As a promising next-generation material in electronics, an ideal picture of graphene is to freely draw the semiconducting/insulating patterns directly on conductive graphene surfaces with nanometer resolution. Fluorine shows superiorities as the "ink" towards this goal, owing to the efficiency in tuning the bandgap and towards achieving a chemically stable product [6][7][8][9][10] . A large variety of techniques has been reported to fluorinate graphene ranging from pure F2 exposure to fluorine plasma etching 11,12 . In many works, the synthesis of monolayer fluorographene and stoichiometric fluorographene (perfluorographane, CF) hitherto is carried out by exposing graphene to XeF2 gas 7,8,13 . As for the more applicable supported graphene (e.g. graphene on SiO2) where a physical support substrate avoids the mechanical collapsing, the product with the highest fluorine content is semiconducting C4F for the above mentioned plasma based and chemical fluorination pathways 7,14 , and it is still difficult to synthesize insulating fluorographene, e.g. C2F and CF, on a substrate directly. Theoretical studies have predicted that, by tuning the concentration of fluorine 15,16 , the electronic properties of the fluorographene such as the presence of a bandgap can be tailored to meet different requirements.

Experimentally, optical band gap measurements have proved the possibility of bandgap opening in the fluorographene 13 , yet the electronic band gap, important for most applications of fluorographene, has rarely been shown 17,[START_REF] Liang | MRS Proc[END_REF] .

In this work, we explored graphene fluorination processes, the physical property and structure changes in a rather broad parameter space, both theoretically and experimentally. For the purpose of a well-defined and local synthesis of fluorographene, we have developed a reversible and local electron beam activated fluorination technique, where the nano-sized electron beam induced deposition process cracks the XeF2 precursor gas and atomic fluorine is bound to the graphene surface covalently. This reversible technique is proved here to be very efficient. When it is applied to free-standing graphene, perfluorographane (CF) can be obtained after short synthesis time. Our X-ray photoelectron spectroscopy (XPS) studies of these fluorinated graphene surfaces suggest that for single-sided fluorination on the supported graphene, insulating C2F with a bandgap of ~5.8 eV is obtained which implies that our technique has overcome the conventional limit of semiconducting C4F in the fluorination of supported graphene and achieved the insulating fluorographene. The valence band spectra analysis of the fluorographene of different F-concentration shows that fluorine induces a discernible band opening in the graphene electronic structure and the bandgap of the fluorographene increases with the fluorine coverage. From the theoretical analysis of the fluorographene, we analyze at which ordering conditions a bandgap opening could appear. The ordering of graphene could also be analyzed from the fluorination of bilayer graphene using the same technique. As an impactful observation, a new type of rectangular moiré pattern is for the first time observed between the upmost fluorinated graphene and the bottom unfluorinated graphene layer. This rectangular shaped moiré patterns can be well explained from the atomically locked bilayer C2F boat/graphene superlattices. In turn, these structure analysis results are compared to the theoretical electronic structure analysis of fluorographene.

ⅠⅠ. RESULTS AND DISCUSSION

The schematic picture of the electron beam activated fluorination (EBAF) is shown in Fig. 1A.

The core idea of this method is the local creation of F radicals by the electron beam, resulting in the nano-synthesis of fluorographene directly on monolayer graphene under the nanometer sized electron beam. For the fluorographene synthesis, graphene was placed under the focused electron beam irradiation [START_REF] Li | [END_REF][20][21] and simultaneously, a flux of XeF2 molecules was injected through a gas injection system. Since the electrons in the initial beam is well above the critical energy to dissociate XeF2 22,23 , the decomposition of the XeF2 molecules was induced, leading to the formation of local F radicals and finally resulting in the graphene fluorination. Here, two types of graphene samples were used: free-standing graphene (FG) on transmission electron microscopy (TEM) grid and supported graphene (SG) on SiO2 substrate (Fig. 1B and1C). It is found that the typical saturation time for the fluorination of FG and SG were 10 and 5 minutes for a selected 100 µm × 100 µm area in measured characteristics, receptively.

The XPS spectra of pristine graphene, fluorinated SG and FG samples are compared in Fig. 2A.

In the fluorinated FG, the area under the XPS C 1s peak consists to ~90% of the C(-F) peak, and C(sp 2 or sp 3 ) peaks are absent, implying the formation of perfluorographane (CF). The remaining peak area, about 10% of the total peak area, corresponds well to the position of C(-F2 / -F3) peaks. Such double or triple functionalized carbon atoms could be present at the defect sites. In contrast to the fluorination process of FG, the good adhesion of monolayer graphene to the substrate will render diffusion of fluorine beneath the graphene sheet less probable, which results in a single-sided fluorination of SG 7,24,25 . From the XPS of fluorinated SG, the major signals come from the C(sp 3 ) and C(-F) bonds, whereas the C(sp 2 ) bond signal (located at 284.4 eV) disappears in the C 1s spectrum. This indicates the complete transformation of in-plane sp 2 carbon to sp 3 carbon and C(-F) bonds. It is also noticed that there are two types of C(-F) bonds (locating at ~288 eV and ~290 eV, respectively) observed in the XPS spectrum, both of which have been observed in previous experiments 16 . Similar to free-standing perfluorographane, for the SG we also observe ~10% C(-F2 and -F3), which can be attributed to the defect sites as well.

When comparing the atomic proportion of the C(-F) and C(sp 3 ) carbon, the ratio of one is obtained in the C(-F)/C(sp 3 ) of the SG sample, indicating the formation of C2F. This result is significantly different from the conventional single-sided fluorination where an F concentration corresponding to C4F is reached on supported graphene by using long time exposure of pure XeF2. This implies that by using this electron beam activated fluorination technique, a high F concentration can be obtained that goes beyond the conventional limit of fluorination. The comparison of the Raman spectra of the three samples is shown in Fig. 2B. The characteristic Raman signal of pristine graphene has disappeared in the case of saturated FG, which matches well with the reported perfluorographane. Raman spectrum relates mainly to the complete optical transparency of the material 13 . As for the SG, prominent D (at ~1350 cm -1 ) and G peaks (at ~1580 cm -1 ) were observed in the Raman spectrum, whereas the 2D peak (at ~2700 cm -1 ) was found to be negligible. This result is consistent with the disappearance of the XPS C(sp 2 ) bond signals analysis in the SG sample. This promising result of EBAF based local fluorination of the graphene raises a question about the obtained CxF structure, its electronic structure as well as the lateral resolution that the EBAF process can achieve. To further investigate the formation of the fluorographene as well as the bandgap opening effect, additional experiments were conducted on the SG samples by varying the fluorination time. Fig. 3A exhibits the plot of fluorine concentration as a function of fluorination time (more deconvoluted XPS spectra can be found from Fig. S1-S4). From the plot, it can be seen that for shorter times, the fluorine concentration increases with the process time and the saturation occurs at 5 minutes with an atomic F concentration of ~35% corresponding to C2F. Due to the sufficient energy and small probe size, the electron beam has been used to remove functionalities on graphene and achieved great success in the patterning of graphene with nanometer resolutions 10,26 . Since in our process, the primary and secondary electrons can both dissociate the formed C-F bonds 10 , an electron beam activated defluorination process will be triggered when the gas flux is stopped and only apply the electron irradiation (schematic illustration can be found in Fig. S5). Here, the electron beam used in the de-fluorination process is the same as that in the fluorination process, i.e. a vacuum condition of 10 -6 mbar, an electron accelerating voltage of 5 kV and beam current of 650 pA. The detailed relation between the atomic F concentration and de-fluorination time is shown in Fig. 3B with the starting material being saturated SG (C2F) with the F concentration of ~35%. Here, it can be seen that the de-fluorination process is less efficient and it takes ~240 minutes to totally recovery the sample. Therefore, the combination of the electron beam activated fluorination and de-fluorination techniques constitutes a reversible approach to precisely control the fluorine concentration from pristine graphene to C2F in the SG samples and back to the pristine graphene sample.

To investigate the influence of covalently bonded F atoms on the electronic structure of graphene, XPS valence band spectra were recorded on fluorinated SG samples with various F concentrations. The XPS valence band spectrum reflects the projected density of states (PDOS) below the Fermi level (defined as the binding energy of 0 eV of the measured material). Thus, XPS provides an efficient tool to understand the modifications in the electronic structure of graphene with covalently bonded F atoms. Fig. 4 shows the comparison of the valence band spectra of the fluorographene. In pristine graphene, the fitted line of the valence band spectrum in the low binding energy region intersects with the energy axis at the Fermi level and this agrees well with a vanishing band gap of graphene. Another observed feature of the pristine graphene is the prominent peak at the binding energy of ~8.5 eV. This energy corresponds well to the peak position of the delocalized π orbital electrons of graphene. In contrast to pristine graphene, for the fluorinated graphene, when fitting the slope of the low energy tail of the PDOS, the line intersects the energy axis at positive energy, which indicates a band opening. The PDOS intensity close to the Fermi level vanishes. The values of the valence band maximum (VBM) obtained from the experimental curves increases with the sequence of pristine graphene, 11%, 22%, 28% and 35% fluorographene as 0 eV, 0.9 eV, 1.4 eV, 1.8 eV and 2.9 eV, respectively.

As for the π state peak, the intensity gradually decreases with the atomic F concentration which can be related to the increased electron localization of the C-F bonds. The π state peak is thus expected to decrease in intensity with higher F concentration as observed. For the saturated SG (C2F) sample, the π peak disappears. This result coincides with the disappearance of the sp 2 bond component in the C1s peak observed in the XPS spectra. Theoretically, in the single-sided fluorinated graphene, the interaction between the p orbitals of F and the π orbitals of C producing sp 3 bonds will make the Fermi level not appear in the exact center of the bandgap.

The related maximum energy shift of the Fermi level is ~0.2 eV 7,27 . Therefore, it is possible to estimate the band gap of the fluorinated SG samples from the experimental VBMs. Thus, it can be deduced that the C2F two-dimensional (2D) material has a band gap of 5.8±0.2 eV, equal to diamond and that should be sufficiently insulating where insulating areas of graphene are needed in graphene-based devices. The C4F exhibits a smaller bandgap of 2.8±0.2 eV corresponding to a semiconducting 2D material. To better understand the relation between structure and electronic structure, ab initio density functional theory (DFT) calculations were performed on the single-sided fluorinated graphene (see the supporting information Fig. S7-S11). From these simulations, it is found that band gap of graphene increases as the percentage of fluorine coverage increases, which is consistent with our experimental observations. Moreover, we also observe that the band gap of the fluorinated graphene (CxF where x is 2, 4 and 8) opens only when the atomic sites of the F atoms are positioned as boat configurations, while the chair configurations cannot open a band gap and this is consistent with the report report 28 . Together with the observed bandgap opening, this implies that the F atoms in the saturated single side fluorographene in this work have the potential C2F boat structures.

A milestone in graphene research is the discovery of triangular moiré patterns in the graphene/hBN heterostructure superlattice, in which the electronic properties of graphene can be tailored through periodic moiré potentials and variety of significant phenomenon have been observed, e.g. the emergence of van Hove singularities 29 , Hofstadter's butterfly 30,31 , high-temperature quantum oscillations 32 , etc. Graphene derivatives, e.g. graphane (CH) and fluorographene (CF, C2F chair and C2F boat), due to their lattice constant being slightly different from the one of graphene, can also form moiré superlattices when being well aligned and on top of monolayer graphene (the simulated moiré patterns can be found in Fig. S12).

However, since after synthesis, the obtained graphene derivatives always show a rippled surface 25 , the transferred graphene derivative layer cannot atomically lock to the flat graphene layer to form the moiré superlattice. To address this issue, our strategy is to fluorinate the topmost layer in the exfoliated bilayer graphene directly and to achieve the C2F/graphene superlattice structure. First of all, to understand the stacking order, scanning tunneling microscopy imaging (Fig. 5A) was carried out on the highly oriented pyrolytic graphite (HOPG) from which the bilayer graphene is exfoliated, and the triangular lattice indicates the typical Bernal stacking order of the graphite. The EBAF technique was applied on the exfoliated bilayer graphene which was then imaged by using the AFM. It is noted in the AFM experiment that the patterns start to appear after a long time (8 hours) scanning in a comparably large scale as shown in Fig. 5B. When we performed fine scanning at a higher magnification as shown in Fig. 5C, the moiré pattern consisting of bright and dark rectangular patterns is clearly visible and the denoted unite cell shows distances of 13.8 nm and 6.8 nm in two directions perpendicular to each other. This generated moiré patterns agree very well with our simulated moiré patterns arising from C2F boat/graphene structure (see supporting information Fig. S12a).

They are clearly distinct from the triangular moiré patterns obtained in hBN/graphene system and also would be expected, following our moiré simulations (Fig. S12b), in C2F chair/graphene and CF/graphene bilayer. To the best of our knowledge, this is the first experimental observation of the rectangular moiré patterns arising from the C2F boat/graphene superlattice (the detailed structure can be found in Fig. S13), which means that a new type of periodic electrostatic moiré potential has been generated. This materials system has the potential to reveal novel interesting properties of 2D materials since the interaction between the two 2D materials layers will be very distinct from the hitherto dominating graphene/BN moiré materials.

The fluorine can be attached to the graphene either in the boat or in the chair configuration. We have simulated the structure of the one-sided saturated C2F for both, the chair and the boat configuration (see supporting information Table S1) and found that for the boat configuration, the total energy per fluorine atom is 1.5 eV lower than that for the chair configuration. Thus, the boat configuration is energetically more stable, which is in good agreement with our experimental analysis.

Owing to the intrinsic property of the focused electron beam, the here demonstrated electron beam activated synthesis method is also a controllable, local and free writing technique, and thus provides the possibility to fabricate lateral semiconducting/insulating nanopatterns directly on the conductive graphene surface. To demonstrate this, we further employed the technique to fabricate sharp C2F and C4F nanostrips directly on a graphene and measured the relative conductivity of the different layers by conductive AFM (Fig. 6A). It can be seen from the contrast in the image that the insulating C2F nanostrips has the lowest conductivity with the current close to zero nA and the graphene shows the highest conductivity, while the C4F nanostrips have an intermediate conductivity. To further demonstrate the resolution of this technique, we have used the AFM adhesion mapping. This technique is based on the fact that the adhesion forces between the AFM tip and graphene will change after fluorination, thus providing the contrast in the maps between graphene and the C2F layer.

SUPPLEMENTARY MATERIAL

See the supplementary material for details on the XPS spectra, DFT models, moiré pattern simulations, process parameters and field-effect transistor performances.

ACKNOWLEDGEMENT

We acknowledge the Swedish Research Council (Vetenskapsrådet) (621-2012-3679, 2016-05259), Knut and Alice Wallenberg Foundation (KAW Graphene) (Swedish Graphene Initiative) and Formas Foundation (2019-01538) for the financial support. We are also grateful to Swedish National Infrastructure for Computing (SNIC) and the PRACE-2IP project CHARTERED2 resource Salomon cluster based in Czech Republic at the IT4Innovations for computer support.

APPENDIX: EXPERIMENTAL

Graphene samples: Large scale monolayer graphene samples were synthesized by chemical vapor deposition technique and transferred to Cu TEM grid and SiO2 (285 nm)/Si substrate.

Prior to the experiments, the samples were immersed in chloroform for 24 h and then blow dried under N2 to remove the contaminations. Finally, the cleaned samples were checked by light optical microscope, XPS and Raman. Bilayer graphene are exfoliated from highly oriented pyrolytic graphite and transferred onto SiO2 substrate and characterized through optical microscope and Raman spectroscopy.

Graphene fluorination:

The local electron beam activated graphene fluorination and defluorination were conducted in a FEI Strata DB235 FIB/SEM at a vacuum condition of 10 -6 mbar under the electron acceleration voltage of 5 kV and beam current of 650 pA (more details on the electron dose can be found in Fig. S14). The XeF2 molecules were injected through an equipped gas injection system (GIS) inside FIB/SEM chamber with a partial pressure at the exit of the gas nozzle of 600 Pa 33 . The fluorination was carried out under the electron beam irradiation with simultaneous XeF2 molecule injection, while the de-fluorination was performed only under electron beam irradiation.

Characterizations: A Physical Electronics Quantum 2000 Scanning ESCA microprobe was used to conduct the X-ray photoelectron spectroscopy (XPS) experiments with a Al Kα X-ray (1486.7 eV) source. In the XPS measurements, due to the low conductivity of the fluorographene, a shift of the XPS spectra move to the higher energy region took place, thus an electron gun neutralizer was open to compensate this charging effect. The Raman measurements were performed by using a Renishaw inVia Raman microscope with a 532 nm wavelength laser. The AFM was carried out in a Bruker Multimode 8 AFM under the peakforce quantitative nanomechanical mapping mode and conductive AFM mode 9,[34][35][36][37][38] .

DFT calculation: Ab initio density functional theory (DFT) calculations are performed by using a plane wave based pseudo potential method (VASP) to study the single sided fluorination of monolayer graphene 39,40 . We have used the generalized gradient method of Perdew, Burke and Ernzerhof for the exchange-correlation functional along with PAW potential 41,42 . The structures were optimized using conjugate gradient method with forces calculated from Hellman-Feynman theorem. The energy and force thresholds are kept at 10 -5 eV and 10 -2 eV/Å, respectively. For geometry optimization, a 5 × 5 × 1 Gamma centered k grid was used. For total energies and electronic structures, we have used a 15 × 15 × 1 Gamma centered k grid.

The electronic structure of the CnF structures (n = 2, 4, 8, 16, 32) were investigated. These structures represent respectively 50%, 25%, 12.5%, 6.25% and 3.125% single sided fluorination coverage of monolayer graphene (see supporting information Fig. S7-S11).

  Fig. 6B shows a 15 nm C2F line directly fabricated on monolayer graphene, which can best indicate the resolution of the technique. It should also be mentioned that the interfacial width in Fig. 6A (90% to 10% criterion) is 9 nm.

  ⅠⅠⅠ. CONCLUSIONIn conclusion, electron beam activated fluorination is shown to be a highly efficient technique to obtain a local fluorination and de-fluorination of graphene. The structuring of CxF nanostructures is possible with high resolution of 9-15nm both with respect to chemical composition as well as spatial resolution. The saturated fluorination on the FG leads to a stoichiometric fluorographene of CF, while on the SG sample to an insulating C2F was approached which has overcome the conventional limit of semiconducting C4F. The concentration of the covalently bonded fluorine can be precisely controlled and results in a quantitative band gap engineering of graphene. Moreover, this local nano-synthesis method constitutes a novel 2D lithography technique to freely write semiconducting and insulating nanopatterns directly on conductive graphene surface with 15 nm resolution. These findings pave the way to fabricate graphene-based flexible and transparent electronic nanodevices.

FIG 1 .

 1 FIG 1. Schematic and graphene samples. (A). Illustration of the electron beam activated

FIG 2 .FIG 3 .

 23 FIG 2. Spectroscopic characterization of fluorographene. XPS C 1s (A) and Raman (B)

FIG 4 .

 4 FIG 4. Valence band spectra of single-side fluorographene. The Fermi level is located at the

R.R. Nair, W. Ren, R. Jalil, I. Riaz, V.G. Kravets, L. Britnell, P. Blake, F. Schedin, A.S.