Konrad Żolna
email: konrad.zolna@gmail.com

Thibault Asselborn

Caroline Jolly

Laurence Casteran

Marie-Ange Nguyen Morel

Wafa Johal

Pierre Dillenbourg

Konrad Żo Lna

Marie-Ange Nguyen-Morel

The Dynamics of Handwriting Improves the Automated Diagnosis of Dysgraphia

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Even in the era of digital learning, handwriting is still a critical element of early childhood education. Handwriting is a complex task involving cognitive, perceptual, attentional, linguistic, and fine motor skills [START_REF] Bourdin | Is graphic activity cognitively costly? a developmental approach[END_REF][START_REF] Feder | Handwriting development, competency, and intervention[END_REF][START_REF] Mccutchen | From novice to expert: Implications of language skills and writingrelevant knowledge for memory during the development of writing skill[END_REF]. In France, children learn to write cursively around the age of 5 (preschool) and complete their handwriting mastery around ten years later [START_REF] Accardo | Development, maturation and learning influence on handwriting kinematics[END_REF][START_REF] Ziviani | The development of graphomotor skills[END_REF]. During learning, handwriting evolves both at qualitative (letter shapes and legibility) and quantitative levels (speed) [START_REF] Accardo | Development, maturation and learning influence on handwriting kinematics[END_REF][START_REF] Blöte | A longitudinal study on the structure of handwriting[END_REF][START_REF] Rueckriegel | Influence of age and movement complexity on kinematic hand movement parameters in childhood and adolescence[END_REF]. In a constant search for efficiency, handwriting progressively becomes automated thanks to motor programs, which allow the motor control system to produce integrated movements. These programs appear around the age of 8, and it is generally admitted that automation is complete around the age of 14 [START_REF] Blöte | A longitudinal study on the structure of handwriting[END_REF][START_REF] Chartrel | The impact of spatio-temporal constraints on cursive letter handwriting in children[END_REF][START_REF] Feder | Handwriting development, competency, and intervention[END_REF][START_REF] Rueckriegel | Influence of age and movement complexity on kinematic hand movement parameters in childhood and adolescence[END_REF][START_REF] Vinter | Effects of different types of learning on handwriting movements in young children[END_REF][START_REF] Ziviani | The development of graphomotor skills[END_REF]. A significant breakthrough during recent decades in the study of handwriting came from the development 1 arXiv:submit/2728579 [cs.LG] 12 Jun 2019 of graphics tablets and dedicated software, which enables the high-frequency sampling and subsequent analysis of handwriting dynamics. These tools greatly contribute to our current knowledge of handwriting kinematics, the development of motor skills for planning and controlling handwriting movements [START_REF] Pontart | Influence of handwriting skills during spelling in primary and lower secondary grades[END_REF], and the understanding of neuromotor deficits in handwriting disorders. In the CoWriter project [START_REF] Jacq | Building successful long child-robot interactions in a learning context[END_REF][START_REF] Johal | Child-robot spatial arrangement in a learning by teaching activity[END_REF][START_REF] Lemaignan | Learning[END_REF], we aim to adapt handwriting training for children with difficulties by using a learning by teaching approach in which children teach a humanoid robot how to write on a graphics tablet.

Despite correct training, between 5% and 25% of children never master handwriting like their peers. With the increasing cognitive demand for school work, handwriting rapidly becomes a limiting task for these children who cannot handle simultaneous efforts such as handwriting, grammar, orthography, and composition. These students may then rapidly face more general learning difficulties. Thus, it is crucial to detect and remediate handwriting difficulties as soon as possible [START_REF] Christensen | The critical role handwriting plays in the ability to produce highquality written text[END_REF][START_REF] Feder | Handwriting development, competency, and intervention[END_REF].

The handwriting tests are not difficult; children are asked to copy a standard text, which is then evaluated based on a predefined set of features. In French-speaking countries, the BHK test (The Concise Assessment Scale for Childrens Handwriting) [START_REF] Charles | Échelle dévaluation rapide de lécriture chez lenfant (bhk)[END_REF][START_REF] Hamstra-Bletz | A longitudinal study on dysgraphic handwriting in primary school[END_REF][START_REF] Soppelsa | Evaluation de lécriture chez ladolescent. le bhk ado[END_REF] is widely used for diagnosing dysgraphia. This test is recognized by health insurance companies, which pay the costs of both diagnosis and treatment.

But when it comes to diagnosing dysgraphia with the help of the BHK test, a number of difficulties may arise. These are related to the amount of time required for scoring the tests, variability across evaluators, and most notably the time lag often a period of 6 months or more between initial concerns about a childs handwrit-ing and the opportunity to consult with an expert.

In this context, we develop a new method to allow rapid and accurate discrimination between typically-developing (TD) and dysgraphic children. The goal of our study is to create a diagnosis tool independent of human expertise to avoid any subjectivity. We use a simple writing task, the alphabet task, which was proven previously to predict children handwriting performance and to discriminate between proficient and non-proficient writers [START_REF] Berninger | Language by hand: A synthesis of a decade of research on handwriting[END_REF][START_REF] Berninger | Scientific practitioner: Theorybased diagnosis and remediation of writing disabilities[END_REF][START_REF] Graham | Which manuscript letters do primary grade children write legibly?[END_REF][START_REF] Rosenblum | Using the alphabet task to differentiate between proficient and nonproficient handwriters[END_REF]. We showed that the main advantage of this tabletbased test it that it exploits the dynamic features of handwriting. Two models were compared: a Convolutional Neural Network only analyzing the data as a static image (like the pen-paper tests used nowadays) with a Recurrent Neural Network using the dynamical aspect of the data. We showed that the latter outperformed the former.

Related work 2.1 Handwriting legibility analysis

Assessing the legibility or readability of handwriting is not a new challenge as studies relating to this topic exist since the beginning of the twentieth century. The first scaling method was developed by Thorndike [START_REF] Thorndike | Handwriting[END_REF] in 1910. This constituted a very important contribution "not only to the experimental pedagogy but to the entire movement for the scientific study of education" [START_REF] Ayres | A scale for measuring the quality of handwriting of school children[END_REF]. Thorndike compared his invention to the thermometer. "Just as it was impossible to measure temperature beyond the very hot, hot, warm, cool, etc., of subjective opinion, so it had been impossible to estimate the quality of handwriting except by such vague standards as one's personal opinion that given samples were very bad, bad, very good, etc." [START_REF] Ayres | A scale for measuring the quality of handwriting of school children[END_REF].

Until now, two approaches are used to evaluate handwriting. The first is a global holistic method that evaluates the handwriting quality as a whole, while the second measures it according to several predefined criteria [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF].

The global holistic approach provides an overall judgment on the quality of handwriting by comparing handwriting samples previously sorted depending on quality. As a typical example of this approach, Ayres developed in [START_REF] Ayres | A scale for measuring the quality of handwriting of school children[END_REF] a scale in which teachers can mark the legibility of a handwriting sample by comparing it to eight handwriting references of increasing quality included with the scale. Assessment of the writing quality relies on subjective judgment by the teacher. After that, several updated scales with more objectivity were developed [START_REF] Bezzi | A standardized manuscript scale for grades 1, 2, and 3[END_REF][START_REF] Freeman | A new handwriting scale[END_REF][START_REF] Herrick | The evaluation of legibility in handwriting[END_REF].

The second approach for handwriting legibility analysis is based on predefined criteria (e.g., letter form, letter size, spacing, and linestraightness). The judgment is then made by individually grading these criteria and summing the sub-scores. Among the large number of tests developed during the past 40 years using this approach, we can cite the concise Evaluation Scale for Children's Handwriting (BHK) test [START_REF] Hamstra-Bletz | Concise evaluation scale for childrens handwriting[END_REF] as the primary reference for the diagnosis of dysgraphia in Latin alphabet-based languages [START_REF] Charles | Échelle dévaluation rapide de lécriture chez lenfant (bhk)[END_REF][START_REF] Hamstra-Bletz | A longitudinal study on dysgraphic handwriting in primary school[END_REF][START_REF] Karlsdottir | Problems in developing functional handwriting[END_REF][START_REF] Overvelde | Handwriting development in grade 2 and grade 3 primary school children with normal, at risk, or dysgraphic characteristics[END_REF][START_REF] Soppelsa | Evaluation de lécriture chez ladolescent. le bhk ado[END_REF]. We can also cite more recent tests, such as the Evaluation Tool of Children's Handwriting (ETCH-C) [START_REF] Amundson | Evaluation Tool of Children's Handwriting: ETCH examiner's manual[END_REF] and the Hebrew Handwriting Evaluation (HHE) [START_REF] Erez | The hebrew handwriting evaluation[END_REF], which take into account other features of handwriting. In the ETCH-C test, for example, the pencil grasp, pencil pressure, and in-hand manipulation are additional criteria taken into account. In the HHE, the examiner is asked to observe the pencil and paper position, the body posture and stabilisation, and the fatigue. However, the observation of all criteria remain partly subjective as a human being makes them.

With the emergence of new tools (e.g., digital tablet), the addition of several variables (hidden so far) to the analysis of handwriting legibility is now possible. In particular, the analysis of dynamic features of handwriting allows a better characterisation of childhood handwriting difficulties [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF]. Several techniques have then been proposed to classify the handwriting legibility, including the dynamics of the process [START_REF] Danna | Signal-to-noise velocity peaks difference: A new method for evaluating the handwriting movement fluency in children with dysgraphia[END_REF][START_REF] Di Brina | Dynamic time warping: A new method in the study of poor handwriting[END_REF][START_REF] Rosenblum | Identifying developmental dysgraphia characteristics utilizing handwriting classification methods[END_REF], thus leading to better accuracy and less subjectivity.

Models used in handwriting analysis

Recently, much research has generated models for handwriting analysis using machine learning. Indeed, machine recognition of handwriting is used in various fields, such as reading postal addresses on envelopes, amounts written on bank checks, and signature verification. Models are divided into offline and online recognition. Offline recognition focuses on the image of the handwritten text, while in online recognition, the location of the pen-tip is recorded as a function of time [START_REF] Plamondon | Online and off-line handwriting recognition: a comprehensive survey[END_REF]. As the offline/online designation might not be clear, we change the terminology for the remainder of this paper to static/dynamic as we believe these are more relevant. Static systems are less accurate than dynamic ones due to the absence of temporal data (dynamics of writing), which contains information that may be relevant [START_REF] Plamondon | Online and off-line handwriting recognition: a comprehensive survey[END_REF] for the model. Only the image of the handwritten text is available to the model. MNIST [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] is the most widely used benchmark for isolated handwritten digit recog-nition. Many machine learning techniques have been used to classify the MNIST dataset from the multi-layer perceptron (MLP) [START_REF] Simard | Best practices for convolutional neural networks applied to visual document analysis[END_REF] to complex variants of support vector machines (SVM) [START_REF] Decoste | Training invariant support vector machines[END_REF]. However, CNN models seem to perform best by providing outstanding results on the MNIST dataset as well as on Latin letters and Chinese characters, for example, [START_REF] Ciregan | Multi-column deep neural networks for image classification[END_REF].

For dynamic recognition, the temporal information about handwriting is available to the model. Different architectures of models are then used for the classification. For example, a Hidden Markov Model (HMM) [START_REF] Marti | Using a statistical language model to improve the performance of an hmm-based cursive handwriting recognition system[END_REF] or a hybrid system gathering time-delay neural network and HMM [START_REF] Jaeger | Online handwriting recognition: the npen++ recognizer[END_REF] has been used in the past twenty years. Since then, RNNs have been used with great success surpassing all previous results in this field [START_REF] Graves | Unconstrained online handwriting recognition with recurrent neural networks[END_REF][START_REF] Graves | A novel connectionist system for unconstrained handwriting recognition[END_REF].

While the literature reports a large number of models to recognise digits, characters, words or sentences, very few works aimed to assess handwriting legibility for dysgraphia detection purposes. Rosenblum et al. [START_REF] Rosenblum | Identifying developmental dysgraphia characteristics utilizing handwriting classification methods[END_REF] used a SVM classifier on a set of manually extracted handwriting features to classify between dysgraphic and nondysgraphic Hebrew writing children. The SVM binary classification led to the accurate labeling of 89 out of 99 writing products (89.9% accuracy). In [START_REF] Asselborn | Automated human-level diagnosis of dysgraphia using a consumer tablet[END_REF], Asselborn et al. used a Random Forest classifier to correctly identify 98% of the 56 dysgraphic children in their database. Their method was taking in input the BHK test (latin alphabet).

With our work, we embrace the same goal and propose to create an easier test to effectively detect dysgraphia in a classroom context. Instead on focusing on the analysis of an handwritten text, task that appears to be challenging for a wide range of children, we developed a test working on individual letters (and numbers) that are taught earlier in school curriculum [START_REF] Asselborn | Bringing letters to life: handwriting with haptic-enabled tangible robots[END_REF][START_REF] Piasta | Developing early literacy skills: A meta-analysis of alphabet learning and instruction[END_REF][START_REF] Ritchey | The building blocks of writing: Learning to write letters and spell words[END_REF]. In addition, by characterizing difficulties independently for each letter, we envision to build a personalized training system that would generate words with letters specifically selected to suit the training needs of the child.

The next section presents our motivations and the machine learning tools used to build our models for the diagnosis.

Motivations and Technical Grounding

The overall goal of this project is to enable teachers to evaluate in a fine grain handwriting in order: (1) to detect children in high difficulties and at risk for dysgraphia, (2) to generate a handwriting profile for the child pointing gyphs that should be practiced more in later-on handwriting training. For this, we propose to build a system able to assess handwriting legibility automatically. We present below pros and cons of several methods and argument our choices. Artificial neural network (ANNs) are particularly useful in the situations where the relationship between inputs and outputs is complex and challenging to identify by a non-expert human observer. Practice has proven that artificial neural networks may be successfully employed to solve many of practical problems. They have been used in a variety of applications, for example, machine translation [START_REF] Cho | Learning phrase representations using rnn encoder-decoder for statistical machine translation[END_REF], image recognition [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], and even playing computer games on the level beyond human skills [START_REF] Mnih | Playing atari with deep reinforcement learning[END_REF].

In this work, we focus on two types of ANNs: LSTM recurrent neural networks (RNN) [START_REF] Hochreiter | Long short-term memory[END_REF] and convolutional neural network (CNN) [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. RNNs were introduced to analyse and interpret sequences of data. Hence, this approach meets our applicative setup where each glyph 1 may be understood as a trajectory or the sequence of consecutive positions of the pen tip on the paper. This approach focuses on the temporal aspect of data and considers the dynamics of writing. This characteristic is known, and RNN has been applied to handwriting tasks before [START_REF] Graves | Generating sequences with recurrent neural networks[END_REF]. For this reason, we believe that RNN can provide better results to our applicative problem compared to other methods using static images (including the original way of evaluating BHK tests).

CNNs are widely used tools for deep learning. They are known to be particularly well suited for applications with images as inputs, although they are now more frequently used in other applications involving text [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF] or signals [START_REF] Li | Automatic musical pattern feature extraction using convolutional neural network[END_REF] as inputs. CNNs have become a standard in text recognition giving outstanding results [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF][START_REF] Wang | End-to-end text recognition with convolutional neural networks[END_REF].

In addition to the different architectures of the network, it is important to notice that contrary to RNN, CNN takes the 2D image representing the glyph as input, meaning that the dynamic of the movement is lost. We present here a new promising method based on a neural network taking into account dynamic features of handwriting to discriminate between TD and dysgraphic children.

Methods

Participants

The present study was conducted in accordance with the Declaration of Helsinki, was approved by the University of Grenoble Alps' ethics committee (agreement n 2016-01-05-79). It was con-1 A glyph is a graphic symbol that provides the appearance or form for a character. A glyph can be an alphabetic or numeric. In this paper, we focus on lower case cursive alphabetical [a-z] and numerical [0-9] glyphs.

ducted with the understanding and written consent of each child's parents and the assent of each child and in accordance with the ethics convention between the academic organisation (LPNC-CNRS) and educational organisations. A total of 971 typically-developing children were recruited in 14 schools from various suburbs to ensure different socio-economic environments (TD dataset). 43 classes were included from preschool to fifth-grade. None of the TD children included in the study presented known learning problems or neuromotor disorders. Twenty-four dysgraphic children recruited at the Learning Disorders Center of Grenoble hospital (Centre Référent des Troubles du Langage et des Apprentissages, CHU Grenoble) were also included in the study (D dataset) and were all diagnosed as dysgraphic based on their BHK scores.

Data collection

Children were asked to write cursively, without a time limit, the 26 letters of the alphabet in lower case, as well as 10 digits, randomly dictated. Two procedures following different dictations were performed, the first in the middle of the school year (January-February) and the second at the end of the school year (May-June). We checked that the dictation order did not affect children performances (data not shown). Dictations were performed on a sheet of paper placed on a Wacom Intuos 4 A5 USB graphic tablet (sampling frequency = 200 Hz; spatial resolution = 0.25 mm). The sheet of paper was used to place the children in their usual handwriting conditions as asking them to write directly on the tactile surface might be different due to a different friction coefficient [START_REF] Alamargot | Does handwriting on a tablet screen affect students graphomotor execution? a comparison between grades two and nine[END_REF]. All tracks were monitored using the Scribble software developed in the LPNC laboratory [START_REF] Jolly | Analysis of cursive letters, syllables, and words handwriting in a french second-grade child with developmental coordination disorder and comparison with typically developing children[END_REF][START_REF] Jolly | A oneyear survey of cursive letter handwriting in a french second-grade child with developmental coordination disorder[END_REF] As mentioned above, handwriting is a crucial skill to acquire and paramount for many school activities. Hence, children are taught handwriting in a standardised way from the beginning of their school education. As a result, the vast majority of data collected is inadequate compared to with what was taught in school (resulting in an over-representation of "good" examples). In other words, the class of glyphs that we are particularly interested in ("bad" examples) are underrepresented. In addition, we have a significant variety of instances that belong to the underrepresented class (illegible glyphs) as compared to a relatively small variance in the overrepresented class-there are only a few ways to write a letter correctly, while there are many ways to do it incorrectly. Hence, the discrimination between typicallydeveloped and dysgraphic children seems to be challenging as the class of illegible glyphs we want to describe is underrepresented and varies a lot. We argue that using a black box to solve this problem would require a considerable amount of data, which we find infeasible. We are then left to work in a regime of underrepresented positive cases meaning that a more sophisticated solution must be developed. We applied an idea of transferring learning to address this challenge.

Transfer learning

In machine learning, transfer learning is an approach where a new task is solved through the transfer of knowledge from a related proxy problem that is simpler to solve [START_REF] Torrey | Transfer learning. Handbook of Research on Machine Learning Applications and Trends: Algorithms[END_REF].

A small group of shapes are fixed to be letters or digits, and people are trained to write and read these symbols. The glyphs written by dysgraphic children tend to be harder to decode as they are not written conventionally. This is why we believe that the problem of identifying dysgraphic children may be resolved by using an approach that mimics humans. Hence, we decided to train a recognising model to classify glyphs and use its prediction to discriminate dysgraphic children. We assume that if the model fails to predict which glyph is the given one, then the writer has a higher probability to be dysgraphic. Our recognising model is trained to discriminate glyphs, in other words, the model assesses legibility of a given glyph. It is important to notice that this approach (using the predictions of the recognising model by a high-level diagnosing model to classify a writer) is plausible only if the recognising model performs well while presented legible glyphs. In other words, the diagnosing model based on a poorly performing recognising model would wrongly label all children dysgraphic. Hence, we have to make sure that the recognising model performs well enough when glyphs written by non-dysgraphic children are provided as the input. Since we are in the regime of scarcity of annotated data for dysgraphic children (as mentioned in Section 4.3.1), we decided to use a straightforward diagnosing model (taking an average of given predictions) built on top of an advanced recognizing model (that may leverage the massive data we have for non-dysgraphic children).

Assessing glyphs' legibility

We divided our data into three disjoint sets:

• Training set -80% of TD dataset.

• Validation set -20% of TD dataset.

• Dysgraphic set -consists of all glyphs written by dysgraphic children (D dataset).

Hence, the recognising model is trained on examples which are believed to be more legible.

The model's input consists of a single glyph, and there is a class for every glyph. Depending on the model used, the inputs may be the trajectories of a pen (point-by-point for the RNN case) or the static images (final, visual results for the CNN). This means the model treats the problem as a multi-class classification balanced by the design of the data collecting procedure (each child has to write down every 36 glyphs). In real-life applications, a child would be asked to write down a given glyph and, even if the ground truth is known, we let the model predict the glyph's label. Afterwards, the discrepancy between the ground truth and the model's prediction is measured. Assuming that the model discriminates the glyphs properly with high accuracy, high discrepancy means that glyphs are not legible and the child is likely to be dysgraphic. Therefore, while each child writes down all glyphs, the model predicts in parallel the probability for every glyph to be the requested one. These glyphlevel scores are averaged per child, and this value is understood as a statistic for evaluating writing proficiency, and we call this statistic D (dysgraphic statistic).

Taking an unweighted average is a simple, arbitrary choice and more sophisticated methods may be invented as can be seen in Section 5.3. However, we found this approach works sufficiently well.

Final dysgraphic prediction

The lower the value of D statistic, the more likely the child will be dysgraphic. However, to solve our main problem, we have to answer the following question:

Having a value of D statistic for the given child, is the writer dysgraphic?

In other words, we have to find the threshold (the critical value) of D statistic that divides the children into two groups (dysgraphic and nondysgraphic children). In the original BHK test, children are considered dysgraphic if they obtain a score beyond two standard deviations from the normative group. In practice, it was found that 8.6% of the population is dysgraphic [START_REF] Charles | Échelle dévaluation rapide de lécriture chez lenfant (bhk)[END_REF]. We, therefore, decided to use the same approach to compute the threshold in a way that the 8.6% of the children who obtained the lowest values of D statistic will be considered dysgraphic by our diagnosing model. Of course, this threshold value depends on the recognising model. The main property of this approach for fixing the threshold is that, on average, 8.6% of children are labelled dysgraphic, precisely like the original BHK test.

Discrimination procedure

To summarise, the discrimination procedure for a given child consists of the following steps:

1. The child is asked to write down the 36 glyphs on the same digital tablet.

2. The recognising model, which is trained to discriminate glyphs, is evaluated for all drawings and a single value (the probability that the glyph is the one requested) for each is obtained. A higher value represents, the greater legibility of the glyph.

3. All 36 scores are averaged giving the D statistic.

4. The value of the D statistic describes the child's handwriting proficiency. It is then compared with a threshold calculated in such a way that, on average, 8.6% of children score below. A child with a score below this threshold is labelled dysgraphic.

Results

As mentioned in Section 4, we trained the model to calculate the D statistic for a given child. This value is compared with the threshold found previously. If the value of D statistic is below the threshold, then the child will be labelled "dysgraphic."

In this section we address the following question: Do 8.6% of all children suspected by our prototype represent the group of children truly diagnosed as dysgraphic by the original BHK test?

We show in this section that our model provides very promising results when it uses dynamics of writing as an input.

Recurrent neural network

We present results obtained using recurrent neural networks which, as pointed out in Subsection 3, utilises the temporal aspect of the data to take into account the dynamics of writing. We argue that this temporal information provides valuable insights so that our model is advantageous over other methods only using static images (including the original BHK test).

A k-fold cross-validation [START_REF] Friedman | The elements of statistical learning[END_REF] (with k = 5) produced the graph presented in Figure 1. Details about the model, the learning procedure, and the training procedure may be found in the Appendix. In Figure 1, the x-axis are quantiles and By definition, 8.6% of validation results are below the threshold, which separates the children into dysgraphic and non-dysgraphic groups. It means that in the perfect case all dysgraphic children should score below the threshold. The most important finding is that only two dysgraphic children are not below the threshold, and hence more than 90% of dysgraphic children are correctly diagnosed. This means that even the prototype of the digital BHK test (trained only on single glyphs) that takes into account the dynamics of writing would identify nearly all children labelled by the original BHK test.

This result is impressive as without any further research, we can prepare a model capable of identifying more than 90% of all dysgraphic children without the laborious review by a human expert. Additionally, this digital approach for labelling children is highly objective. Hence, this work is a milestone in achieving our longterm mission of creating a digital BHK test.

Convolutional neural network

As explained in Subsection 3, CNN is a category of neural networks shown to be very effective in tasks, such as image recognition and classification. They work only with static 2D data, so, contrary to the recurrent neural network (see Subsection 5.1), the dynamics of handwriting (i.e., the timeframe) cannot be taken into account. Only the final trace will be used by the model to predict the label of the data.

Figure 2 follows the same procedure as used to generate Figure 1 with the only difference being that CNN is the recognising model instead of RNN. It can be seen that even if the score of dysgraphic children is significantly lower than the one not diagnosed (which does not guarantee all these children are non-dysgraphic), we failed to obtain a clear separation of 90% of the subjects as found with the RNN. As can be seen in Figure , only 25-30% of the dysgraphic are below the threshold line suggesting that the model can successfully identify approximately a quarter of the dysgraphic children.

The results presented in this section suggest the superiority of the RNN over CNN for this task. We believe this difference in efficiency is due to the incorporation of the movement's dynamics for the RNN. This assumption will be further explored in Section 5.4. The estimated threshold is included, and each point represents D statistic for one child (averaged on each fold, the error bar represents the standard deviation). By definition, 8.6% of the validation results are below the threshold, which divides the children into dysgraphic and nondysgraphic groups such that, in the perfect case, all dysgraphic children should score below the threshold. In our case, the model is not very efficient as less than 30% of dysgraphic children are correctly diagnosed.

Discriminative glyphs

The method used to calculate the D statistic for children may be slightly modified to obtain a similar score for glyphs. We can calculate glyphlevel (not child-level) D statistic separately for the dysgraphic children (D dataset) and for the others (TD dataset) (validation part only) to identify the most discriminative glyphs. These glyphs are particularly hard to write for dysgraphic children compared to the general population making them possibly the most helpful for identifying dysgraphic children.

In Figure 3, the D statistics (averaged for each student and averaged on the five folds, and the standard deviation were not presented for clarity) of every glyph are presented. For each, the x and y coordinates represent the mean D statistic for the dysgraphic and non-dysgraphic children, respectively. Each glyph close to the blue line (where dysgraphic children and "average" children obtain the same score) is approximately of the same difficulty for dysgraphic and nondysgraphic children. Those above the yellow line are harder to draw for the group of children with writing difficulties.

Figure 3: Discriminative glyphs (based on RNN model). The comparison of the difficulty in writing particular glyphs for dysgraphic and nondysgraphic children. Glyphs above the yellow line are harder to draw for the group of children with writing difficulties. There are no glyphs below the blue line since each letter is easier for non-dysgraphic children. However, the digits '0' and '4' are nearly of the same difficulty for both groups (it may be initially surprising that the digit '0' is harder than the digit '4', but this is the case because children are expected to write down cursive glyphs; similarly, the letter 'o' and the digit '0' are found to be hard).

Therefore, the glyphs 'e', 's', and 'k', for example, are discriminative glyphs according to our RNN model. This knowledge may be used to prepare even a better prototype, for example, a weighting factor may be assigned proportionally to the discriminative power of the glyphs. Another possibility would be to ask children to write down only the most discriminative glyphs. This technique was applied to create Figure 4. The 15 most discriminative glyphs were used to compute the D Statistic. This technique reduces the diagnostic test duration while providing an accuracy similar to when all glyphs are used for the diagnosis. As mentioned in Subsection 5.1, we used a cross-validation technique by training the model for five folds. For each fold, the most discriminative glyphs found can be slightly different since they are within this particular model and validation set. This means that using the same discriminative glyphs to assess the model performance on the same validation set would bias the model. Hence, to alleviate this issue, we selected 15 of the most discriminative glyphs found for another fold, where each fold was used to obtain the most discriminative glyph only once.

It is interesting to notice that some of the most discriminative letters found are not used in the original BHK ('y', 'k', 'q', 'w', and 'x'). We believe the utilisation of this subset in our prototype is one of the reasons why our model requires fewer data compared to the BHK for classification.

Adding the handwriting dynamic reduces confusion between visually similar glyphs

Certain glyphs might look very similar when we only have access to the final trace. This is, for example, the case between the letters 'e' and 'l' or between 'g' and the digit '9'. Thus, we hypothesize that the final trace is sometimes not enough The estimated threshold is also included. Each point represents D statistic for one child (averaged on each fold, the error bar represents the standard deviation). The results are similar to the case where all glyphs are used (see Figure 1). The key difference is a higher variance that reflects a smaller number of the instances for each child.

to discriminate between these classes, so, in these cases, the dynamics of handwriting must be fundamental. For these glyphs whose final traces look similar, the CNN will make inaccurate classifications while the RNN will have better results due to its access to the movement dynamics. We believe that this is one of the reasons explaining the superiority of the RNN model over CNN. giving similar values in our case, we decided to use their average to obtain the final misclassification values plotted in Figure 5. For example, a confusion of 16% for the CNN model is seen between the letters 'e' and 'l' meaning that the misclassification of 'e' in 'l' or the misclassification of 'l' in 'e' occurs in 16% of the cases. This graph appears to confirm our hypothesis as we can see that the RNN generates fewer confusions between the pairs of glyphs that looks visually similar, likely due to its access to the handwriting dynamics. Concerning the other pairs of glyphs that do not look visually similar, the confusion between the two models is essentially the same and close to 0% of misclassification.

Figure 5: Confusion between the six most similar pair of glyphs (those that bring the highest confusion) for the RNN and CNN models.

Additional class

We noticed that it is beneficial to add a class to the training set, which we call the star or, simply, the '*' class. This class consists of random hybrids of real glyphs obtained by combining two or three drawings to create a non-existing glyph, such that the beginning part of the first glyph trajectory is associated to the middle part of the second one, for example.

This data augmentation makes our model suspicious about glyphs that look strange and labels them as '*'. In other words, our model cannot assume that the analysed object is a glyph. We believe this strategy helps in situations when a dysgraphic child would perform poorly on a glyph but still makes a good guess since the mistake is easy to figure out. Examples include triple 'w' or a letter 'i' with additional dots.

Discussion

This tablet-based test might pave the way to the systematic diagnosis of dysgraphia in schools. Our aim would be to minimize the impact of dysgraphia on childrens school experiences by providing a rapid, simple and cheap diagnostic tool that will allow for earlier remediation. For example, this tool could be employed in conjunction with other screening tests that are already in use in schools, such as auditory and ophthalmologic tests, facilitating the early detection and remediation of handwriting issues.

Contrary to all the techniques used for the diagnosis of dysgraphia (particularly the BHK test) that only include the final trace as input and, therefore, exclude key parts of the information, our test exploits the dynamic of the handwriting. We showed that adding the dynamics of the movement improves the accuracy and decreases the necessary amount of information needed to deliver the identification of dysgraphic children. The RNN using the time-frame in addition to the trace allows improved results (90% of dysgraphic children successfully labelled dysgraphic) over the CNN using only the static image (less than 30% of dysgraphic children successfully labelled dysgraphic). This is the primary reason that suggests developing a digital approach is valuable for this field as the dynamics of the movement is difficult to interpret with the final trace alone, even if an expert is observing the child while writing.

A digital approach to the diagnosis offers additional potential interests. First, it enables to remove human subjectivity as the model cannot be biased by any external parameters that can influence a human. The diagnosis of dysgraphia is important as it has significant implications for the lives of children and their parents. Hence, providing a non-subjective method is certainly not negligible. Moreover, digitally diagnosing dysgraphia will enable to save time as trained therapists spend around 10 minutes per BHK test to analyse the handwriting of the children.

Other information, such as the pressure or inclination of the pen during handwriting extractable with devices like the Wacom(C) graphics tablet could be integrated with the model to improve efficiency further. Currently, the children are asked to write all the letters of the alphabet. As seen in Section 5.3, even if we could remove certain nondiscriminatory letters, some vital information may remain in the way letters or words are linked together. Our model does not consider this potential connection, and it would be interesting to incorporate this aspect in future work.

During this project, the data acquisition utilised a Wacom(C) tablet with an A5 paper sheet affixed to the screen. This device was selected because, in addition to the x, y, and time coordinates, it can accurately extract pressure and pen inclination. However, our current model only incorporates the first three properties. Data acquisition with a high-end tablet, such as the one we use, may soon become obsolete as an available Android device that is less expensive and more accessible may offer a practical solution for anyone wishing to apply the model. However, it is important to note that the friction between the pen and a tactile surface or paper is not the same between the two classes of device. So, data extracted from the two solutions can differ in ways that would mislead the model. In future work, it may be required to compare the handwriting output difference between the two approaches and, if necessary, train a new model with data extracted from a tactile tablet.

Future Work

In a general context, this approach could have great potential for educational purposes, such as the CoWriter project [START_REF] Hood | When children teach a robot to write: An autonomous teachable humanoid which uses simulated handwriting[END_REF][START_REF] Jacq | Building successful long child-robot interactions in a learning context[END_REF][START_REF] Johal | Child-robot spatial arrangement in a learning by teaching activity[END_REF][START_REF] Lemaignan | Learning[END_REF]. This project aims to teach handwriting by using an original approach: the learner becomes the professor of a robot needing help to improve writing techniques. This paradigm, known as learning by teaching, takes advantage of the Protégé effect by making the child feel responsible for the robot, thus encouraging the child to be more engaged in the task. Words the child needs to teach to the robot, using a digital tablet to write handwriting examples, are randomly chosen and might not be appropriate for the child's skill level or needs. We imagine applying this work to create a child's handwriting profile using the D-statistic of each letter to represent the child's proficiency for the letter. With this type of handwriting profile, we can then choose more relevant words that would make the child focus on the targeted letters the model detects with deficiencies. and to all children and their parents for participating in the study.

Statements of ethical approval

The present study was conducted in accordance with the Declaration of Helsinki, was approved by the University ethics committee (agreement n 2016-01-05-79). It was conducted with the understanding and written consent of each child's parents and the oral consent of each child and in accordance with the ethics convention between the academic organisation (LPNC-CNRS) and educational organisations by teaching a robot: The case of handwriting. IEEE Robotics & Automation Magazine, 23(2):56-66, 2016.

  .

4. 3

 3 Analysis of handwriting and identification of dysgraphic children 4.3.1 Scarcity of annotated data

Figure 1 :

 1 Figure 1: With RNN, quantile function of averaged D statistic on validation and dysgraphic sets. The blue horizontal line represents the estimated threshold. Each point represents the D statistic for a given child (averaged on each fold, the error bar represents the standard deviation). By definition, 8.6% of the validation results are below threshold which separates the children into dysgraphic and non-dysgraphic groups, such that in the perfect case, all dysgraphic children should score below this threshold.

Figure 2 :

 2 Figure 2: With CNN model, quantile function of D statistic on validation and dysgraphic sets.The estimated threshold is included, and each point represents D statistic for one child (averaged on each fold, the error bar represents the standard deviation). By definition, 8.6% of the validation results are below the threshold, which divides the children into dysgraphic and nondysgraphic groups such that, in the perfect case, all dysgraphic children should score below the threshold. In our case, the model is not very efficient as less than 30% of dysgraphic children are correctly diagnosed.

Figure 4 :

 4 Figure 4: With RNN, quantile function of D statistic on the validation and dysgraphic sets calculated for the 15 most discriminative glyphs.The estimated threshold is also included. Each point represents D statistic for one child (averaged on each fold, the error bar represents the standard deviation). The results are similar to the case where all glyphs are used (see Figure1). The key difference is a higher variance that reflects a smaller number of the instances for each child.

Figure 5

 5 plots the confusion between the six most similar pairs of glyphs (representing the pairs of glyphs with the greatest confusion) of the CNN and RNN models. The confusion for a pair of glyph A-B is the number of misclassification (B instead of A or A instead of B) over the entire set of As and Bs. The misclassification of A in B over all As and the misclassification of B in A over all Bs are different. However, as it was

Figure 6 :

 6 Figure 6: An example of glyph 'w' (top) with consecutive RNN label predictions (bottom).

Figure 6

 6 Figure6presents the evolution of consecutive predictions of our RNN model for the letter 'w'. In the beginning, the most probable answers are star class since the usual existing glyphs are always longer. Afterward, the model switches to 'r' because the first curve of the glyph 'w' looks similar to the beginning of the glyph 'r'. The model further evolves to 'v', 'u', and finally the correct answer 'w'. Employing the dynamics in the model not only improves the predictive power but also makes the model interpretable.

Acknowledgments

We would like to thank the Swiss National Science Foundation for supporting this project through the Swiss National Center of Competence in Research Robotics, the French National Center for Scientific Research (CNRS), and the University Grenoble-Alpes (France). Konrad Żo lna would also like to thank the National Science Centre, Poland (2017/27/N/ST6/00828, 2018/28/T/ST6/00211). We are all grateful to the directors and teachers of the participating schools, to the Reference Center for Learning and Speech Disorders at the Grenoble hospital,

Appendix

Recurrent neural network details

Architecture

The recurrent neural network is a 2-layer LSTM architecture with hidden sizes of 100 (for both layers). The output of the second LSTM layer is followed by a feed-forward neural network with one hidden layer (of size 40), a dropout (p=0.5), and ReLU (rectified linear unit) nonlinearity to obtain a final output of size 37. The final output is then softmaxed to achieve a probability distribution over all possible glyphs (26 letters, 10 digits, and the special character '*').

Learning procedure

All parameters were initialised randomly using a uniform distribution on the interval (-0.08, 0.08). The Adam optimizer [START_REF] Kingma | A method for stochastic optimization[END_REF] was used with default hyper-parameters except for the learning rate where 0.005 was used instead. Gradient clipping was also used (threshold equals 10 for L ∞ norm), and the batch size was constant at 20. The learning procedure was stopped after 15 epochs without progress on the validation set (considered to be early stopping). The hyper-parameters were found using a random grid search, and the span of the checked values was not very wide due to limited computation power so that these initialization results may be improved.

Cross-validation

A k-fold (k = 5) cross-validation was performed for the training. For each fold, the ratio between the training and validation data was fixed to 80% -20% (see Subsection 4.3.3 for more de-tails). Since k was set to 5 and 20% of the data were used for the validation in each run, every child record in our database was used in the validation set exactly once.


C Jolly

C Piscicelli

L Mathevon

C Bérenger

D Pérennou

Interpreting spatial dysgraphia after stroke: straight-ahead of straight above?

Introduction

Results

Materials and Methods

Discussion and Conclusion

Case presentation JW was a 75 year-old male, right-handed, who underwent a hemorragic right hemisphere stroke causing a total left hemiplegia with pronounced hemianesthesia, left hemianopsia and signs of unilateral spatial neglect (UNS). At entry, balance disorders were particularly severe, with a pusher syndrome. Stroke also induced a spatial dysgraphia characterized by a counterclockwise tilt of the writing lines (Fig. 1). At 3 months, JW's handwriting was slow and presented an upward tilt and an increased left-hand margin due to a global tilt of his orthogonal scheme. The spatial features of his handwriting were maintained 9 months after stroke, while his writing speed was normalized. JW's handwriting inclination was neither related to spatial neglect nor to a rotated straight-ahead, but was clearly improved by a transient modulation of JW's verticality perception. After stroke, a tilted handwriting may thus be due to a tilted representation of the vertical. Spatial dysgraphia, frequently observed after a right hemisphere stroke (1), associates signs of spatial compression in relation to spatial neglect, and a tilted writing which remains to be explained. Here we present a case study showing that tilted writing is due to a tilted representation of the vertical. IQR (°) 1.7 3.9 1.9

Handwriting evaluation

Lines up 12°m

edian (°)

  -Tasks: Copy of the 5 first lines of the BHK test, at M3 and M9, and after modulation of verticality representation (M3). Several types of cueing were tested: blank paper (reference condition) and spatially indexed paper (12°or 24 upwardly or downwardly sloping lines). -Parameters analysed: inclination of each writing line in respect to Earth vertical (°) or to the cueing lintes, inclination of the left-hand margin in respect to the Earth vertical, and the mean time to write a letter (sec). -Statistics: comparison of JW's results with a peered healthy participant (JB). Representation of the verticality -Postural vertical (VP) [2] -Visual vertical (VV) [3] -Modulation: tilt for 10 minutes at 30°in the dark UNS Body and non-body UNS evaluation using a battery of well-known tests.

[ 1 ]

 1 Pisella L, Mattingley JB. The contribution of spatial remapping impairments to unilateral visual neglect [2] Pérennou DA, Mazibrada G, Chauvineau V, et al. Lateropulsion, pushing and verticality perception in hemisphere stroke: a causal relationship? Brain J Neurol. 2008;131. [3] Piscicelli C, Barra J, Sibille B, Bourdillon C, Guerraz M, Pérennou DA. Maintaining Trunk and Head Upright Optimizes Visual Vertical Measurement After Stroke. Neurorehabil Neural Repair. 2016 Jan; 30: 9-18 -Inclination of handwriting lines on blank paper and downward sloping lines, both at M3 and M9 (Tab.1) -No significant line inclination on upwardly sloping lines (Tab.1) -Inclination of the left-hand margin on blank paper at M3 and M9 and conservation of the orthogonality (Tab.2) -Writing speed slower than the healthly subject at M3 but no lonter at M9 (Tab.3) -Signs of UNS at M3 but not M9 (not shown) -Verticality representation: altered at M3 and M9 (not shown) -Lines inclination, margin tilt and writing speed clearly improved after PV normalization (Tab.4)

Tilt of the left-hand margin p

									JB	JW-M3	JB vs. JW-M3	JW-M9	JB vs. JW-M9	JW-M3 vs. JW-M9	C	Before modulation	After modulation
								Blank paper			p		p	p	Postural vertical	p
				0.008*		0.008*	0.08	median (sec/lett) IQR (sec/lett)	0.9 0.3	1.7 0.7	0.008*	0.9 0.4	0.548	0.008*	mean (°) SD (°)	-9.8 3.6	0.6 3.7
								Lines down 24°m						Writing lines inclination	p
				0.008*		0.008*	0.548	edian (sec/lett)	0.7	1.5	0.008*	0.8	0.151	0.008*	median (°)	-9.4	-5.2
								IQR (sec/lett)	0.2	0.2		0.3			IQR (°)	4.6	2.7
								Lines down 12°m
				0.008*		0.008*	0.89	edian (sec/lett)	0.7	1.4	0.008*	0.8	0.222	0.008*	median (°)	-9.4	-3.2
								IQR (sec/lett)	0.3	0.8		0.2			IQR (°)	4.6	4.4
								Lines up 12°m
		0	-1.8	0.016	-0.6	0.016	0.08	edian (sec/lett)	0.7	1.4	0.016	0.9	0.095	0.008*
	IQR (°)	0.7	2.1		1			IQR (sec/lett)	0.3	0.7		0.2
	Lines up 24°m							Lines up 24°m
	edian (°)	-1.1	-0.3	0.548	-1	0.222	0.04	edian (sec/lett)	0.7	1.1	0.032	0.8	0.222	0.222
	IQR (°)	1.4	1.2		1.1			IQR (sec/lett)	0.3	0.5		0.4
		JB	JW-M3	JB vs. JW-M3	JW-M9	JB vs. JW-M9	JW-M3 vs. JW-M9
	Tilt of the left-hand margin		p		p	p
	median (°) IQR (°)	-0.1 2.2	-9.4 4.6	0.008*	-8.5 1	0.008*	0.416
	Angle between the margin and the writing lines
	median (°) IQR (°)	92 1	90 2	0.102	91 1	0.129	1

Angle b/w the margin and the writing lines p

				0.001
				0.043*
				0.008*
	median (°) IQR (°)	90 2	92 1	0.279
	Writing speed			p
	median (sec/lett) IQR (sec/lett)	1.7 0.7	0.9 0.1	0.008*

* Figure 1. Sample of JW's handwriting at M3 on blank paper.

Table 1 .

 1 Inclination of JW's writing lines at M3 and M9 in the different conditions.

Table 3 .

 3 JW's handwriting speed at M3 and M9 in the different conditions.

Table 2 .

 2 Tilt of JW's left-hand margin and writing orthogonality at M3 and M9 in the 'blank paper' condition.

Table 4 .

 4 Spatial and temporal features of JW's handwriting after verticality normalization.


Caroline Jolly

Céline Piscicelli

Rémi Gimat

Monica Baciu

Dominic Perennou
email: dperennou@chu-grenoble.fr

Remi Gimat

Spatial Dysgraphia

Laure Mathevon

Eric Guinet

Claire Berenger

Anne Chrispin

Tilted writing after stroke, a possible sign of biased verticality representation

Keywords: stroke, right hemisphere, spatial neglect, vertical representation, spatial dysgraphia, spatial cognition, tilted writing

Handwriting is a complex task involving motor, linguistic, perceptual, and attentional skills predominantly controlled by the left hemisphere but requiring a spatial organization that depends on the right hemisphere [START_REF] Rapp | The neurotopography of written word production: an fMRI investigation of the distribution of sensitivity to length and frequency[END_REF] . Handwriting is often affected after right-hemispheric lesions, with a spectrum of signs related to the spatial layout of the written language, [START_REF] Ardila | Spatial agraphia[END_REF][START_REF] Cubelli | Afferent dysgraphia after right cerebral stroke: an autonomous syndrome?[END_REF][START_REF] Yoon | Language-specific dysgraphia in Korean stroke patients[END_REF][START_REF] Jang | Language-specific dysgraphia in Korean patients with right brain stroke: influence of unilateral spatial neglect[END_REF] constituting spatial dysgraphia/agraphia. [START_REF] Ardila | Spatial agraphia[END_REF][START_REF] Hécaen | Disorders of written language following right hemisphere lesions[END_REF][START_REF] Rode | Prism adaptation improves spatial dysgraphia following right brain damage[END_REF] These signs are multi-faceted. [START_REF] Ardila | Spatial agraphia[END_REF][START_REF] Cubelli | Afferent dysgraphia after right cerebral stroke: an autonomous syndrome?[END_REF][START_REF] Jang | Language-specific dysgraphia in Korean patients with right brain stroke: influence of unilateral spatial neglect[END_REF][START_REF] Rode | Prism adaptation improves spatial dysgraphia following right brain damage[END_REF] Most deal with spatial compression or mental rotation of the space and are related to spatial neglect: omitting the left half of the paper, overwriting or compressing some words, and omitting and substituting letters or graphemes. Others such as tilted writing and a progressive increase of the left margin cannot be interpreted as signs of spatial neglect and remain to be understood.

Here we present a case suggesting that these handwriting signs after stroke might be due to a counterclockwise tilt in representing the vertical, transposed on the sheet of paper referring to top and bottom.

JW was a 75-year-old right-handed male who had a right fronto-parietal hematoma (Fig. 1A and 1B) causing left hemiplegia with hemianesthesia, left hemianopsia and spatial neglect (US National Institutes of Health Stroke Scale score 13). At entry in rehabilitation (day 13 after the stroke), JW showed spatial neglect and pusher syndrome interpreted as a result of an extreme bias in the internal model of verticality [START_REF] Pérennou | Lateropulsion, pushing and verticality perception in hemisphere stroke: a causal relationship?[END_REF] . He behaved as if he implicitly aligned his body posture onto a representation of the vertical, tilted counterclockwise. JW also presented a consistent and global counterclockwise tilt of his written production (Fig. 1E-G): drawing (-12°) and writing (left margin -9° with respect to the vertical, lines -11° with respect to the horizontal). He wrote without any space compression, deletions or omissions.

JW agreed to a further clinical investigation of these troubles to guide their rehabilitation.

Several domains of spatial cognition were assessed at 2, 3 and 9 months post-stroke (M2, M3 and M9): spatial neglect by means of a battery of tests including the representation of the subjective straight ahead (SSA), [START_REF] Jeannerod | The directional coding of reaching movements. A visuomotor conception of spatial neglect[END_REF] representation of the vertical (visual vertical [VV] and postural vertical [PV]) [START_REF] Pérennou | Lateropulsion, pushing and verticality perception in hemisphere stroke: a causal relationship?[END_REF] , and drawing and writing on blank and cued (lined) paper as well as after a modulation of the verticality bias. Assessments involved using validated tests, devices, and protocols, all routinely used (details in additional material), except for a novel procedure to test handwriting. The SSA (10 trials) was compared to the actual straight ahead (0°), whereas deviations affecting the VV and PV (10 trials each) were compared to published normal values. [START_REF] Pérennou | Lateropulsion, pushing and verticality perception in hemisphere stroke: a causal relationship?[END_REF] To interpret JW's drawing and writing, we tested 12 right-handed healthy individuals (5 males), matched in age (mean [SD] 75. JW's spatial neglect was severe at M2, affecting body and non-body spaces (Table ), without any sign of spatial alexia. Only a few neglect signs persisted at M3 but no longer at M9, when the representation of the SSA, not tested earlier, was also normal (-0.7° [2.8], p=0.739).

JW's representation of verticality was tested under baseline conditions at M2 and M9 and also after a modulation procedure (M3). Thresholds of significance were corrected at 0.05/4=0.012. At M2, JW showed a severe transmodal counterclockwise tilt affecting the VV (-13.1°; 11.9*T-scores, p<10 -6 ) and PV (-11.2°; 12.4*T-scores, p<10 -6 ). This bias remained at M9 but was attenuated (VV= -5.3°; PV= -5.7°). As expected [START_REF] Barra | Humans use internal models to construct and update a sense of verticality[END_REF] , the PV was modulated after 10 min of ipsilesional (right) whole body tilt in the dark at 30° (-9.8° before, 0.5° immediately after). JW's writing (9 lines and margins) was better oriented 20 min after the modulation procedure (before -10.2° (2.2) vs after -6° (1.5); p=0.008).

In controls and JW, drawing orientation was quantified by the tilt with respect to the horizontal for the 5 horizontal segments of the landscape to be copied (Fig. 1E) [START_REF] Gainotti | Patterns of drawing disability in right and left hemispheric patients[END_REF] . The corrected P-value was 0.05/4=0.012. The drawing was tilted counterclockwise: slightly for controls (-1.5° [0.57]; t=8.5, p<10 -6 ) and markedly for JW at M3 (-4.5°, t=5 and p=0.008; 5.3 * T-scores, p<10 -6 ) and M9 (-8.4°, t=3.8 and p=0.02; 12.1 * T-scores, p<10 -6 ). Because M3 and M9 data did not differ (p=0.08), they were pooled to show a substantial and remaining counterclockwise tilt (-6.7°) greatly deviated from the horizontal (t=4.9, p<10 -3 ), much greater than the control deviation (9.1 * T-scores, p<10 -6 ).

To analyze writing orientation, controls and JW were asked to copy the first 5 lines of the French version of the Brave Handwriting Kinder test [START_REF] Hamstra-Bletz | Concise Evaluation Scale for children's handwriting[END_REF] (Fig. 1G), a text standardized to diagnose dysgraphia in children. They were seated comfortably with the head and trunk aligned and wrote at a comfortable speed on a paper sheet that was blank or cued (lined) and was affixed to a graphic tablet (Wacom© Intuos 4 A5 USB) carefully centered in front of them on a slanted table (30°). The written samples were digitized, and 3 criteria were measured by an operator (CJ) who was blinded to the writer: the orientation of each of the 5 lines with respect to the horizontal, the orientation of the left margin with respect to the vertical (4 segments defined between the beginning of lines averaged to give a margin orientation per individual), and the mean time to write a letter calculated by using Ductus software [START_REF] Guinet | Ductus: a software package for the study of handwriting production[END_REF] .

For writing on blank paper, thresholds of significance were 0.05/6=0.008 for orientation and p=0.05/3=0.016 for time. The lines were tilted counterclockwise for all individuals: slightly for controls (-1.8° [1]; t=6.4, p<10 -4 ) and markedly for JW at M3 (-11.1° [2.6], 9.3 * T-scores, p<10 -6 ) and M9 (-8.2° [0.7], 6.4 * T-scores, p<10 -6 ). Because M3 and M9 data were comparable (p=0.080), they were pooled. JW showed a substantial deviation, far away from the horizontal (-9.9°; t=13.8, p<10 -6 ), and much greater than that for controls (7.4 * T-scores, p<10 -6 ). The left margin was vertical for controls (-0.4 [2.9]; t=0.49, p=0.64) but tilted counterclockwise for JW (8 values pooled for M3-M9: -8.1°, t=20.4, p<10 -6 ). After a transformation (90°), margins and lines did not differ for controls (p=0.182) or JW (p=0.02), which indicates that they remained orthogonal. Writing a letter took 0.5 (0.16) sec for controls but more time for JW at M3 (1.3 sec; 5 * T-scores, p=10 -6 ) and M9 (0.9 sec; 2.5 * T-scores, p=0.012).

For writing on a paper spatially cued, individuals were asked to copy the same text following lines tilted at 12° and 24° (Fig. 1H-K), counterclockwise (upward) or clockwise (downward).

The corrected p-value was 0.05/10=0.005. Friedman ANOVA revealed no effect in controls (p=0.118); they followed the lines with negligible deviations at -24° (-0.1° [0.3]; t=1.3, p=0.2), -12° (-0.1° [0.3]; t=1.3, p=0.2), 12° (0° [0.6]; t=0.23, p=0.82), and 24° (-0.6° [0.7°]; t=3.03, p=0.01). In contrast, cueing had a strong effect on JW (p<10 -3 for pooled data for M3-M9). He wrote above lines, and the deviation increased with the downward cueing (Fig. 1J,K). He followed lines at 24° (-0.6°; t=-2.9, p=0.017) but not lines at -12° (-1.3° [1.1]; t=-3.9, p=0.004), 12° (-4.1° [2]; t=-6.5, p<0.001), or 24° (-8.8° [2.5°]; t=11.2, p<0.001).

JW never wrote with the smallest sign of spatial dysgraphia but with a consistent vertically oriented bias persistent for at least 9 months (although attenuated), even though the spatial neglect disappeared and representation of the SSA was normal. This observation excluded a causal association between spatial neglect and tilted writing and helps in understanding why prism adaptation improves neglect-related signs of spatial dysgraphia but not line inclinations [START_REF] Rode | Prism adaptation improves spatial dysgraphia following right brain damage[END_REF] . In contrast, JW showed disorders related to verticality bias, with a counterclockwise tilt consistent in magnitude for 2 modalities of verticality perception: drawing and both components of handwriting (i.e., left margin and lines), the writing keeping isotropic dimensions and orthogonality between margin and line deviations. Two ways to modulate spatial representation brought consistent additional information. When submitted to a procedure that should transiently attenuate his verticality bias, JW wrote with an attenuated tilt. When instructed to follow tilted lines, he failed to write on lines tilted to the side opposite his own verticality bias and only succeeded with the lines that greatly tilted in the same direction as his own verticality bias. This case study strongly demonstrates that tilted handwriting may result from a biased representation of the vertical. It argues for a multidetermined impaired spatial organization of writing after right hemisphere lesions [START_REF] Croisile | Spatial or afferent agraphia without left-sided neglect[END_REF] , responding to several mechanisms: spatial neglect, feed-back related errors, or motor direction bias and also bias in the representation of the vertical. JW correctly constructed letters that excluded feed-back-related errors. His drawing was also homogeneously tilted, which excluded a motor bias error. This tilt had detrimental consequences on executing the task, slower than for controls.

This study extends to handwriting the close link between perception and action with respect to the vertical, until now documented only for postural control. The detection of tilted handwriting could signal a biased representation of the vertical, which should be explored [START_REF] Piscicelli | Visual verticality perception after stroke: A systematic review of methodological approaches and suggestions for standardization[END_REF] .

In conclusion, our case study brings new insights into tilted writing, which must be differentiated from other signs of spatial dysgraphia and may be caused

  2 [2.8] years) and sociocultural level. All participants were tested in accordance with the Helsinki Declaration after giving their written consent. Statistical analysis involved using SPSS v23 (IBM Corp., Armonk, NY, USA). The existence of a systematic deviation (tilt) by comparison to the reference was tested by one-sample t tests. Negative values indicated a counterclockwise tilt (leftward, upward). Amplitudes and the significance of JW's tilts were analyzed with Z-scores (VV, PV) or T-scores (drawing, writing) calculated with control data. Conditions were compared by Wilcoxon test for paired samples or Friedman ANOVA. Significance was set according to Bonferroni corrections (bilateral tests). Data are presented as mean (SD).

  by a bias in the representation of the vertical. Our observations open a new avenue for both clinical practice and research.

Figure 1 .

 1 Figure 1. Some puzzling behaviors presented by JW after a right hemisphere stroke. A) Axial Flair MRI slice and B) coronal FSE T2-weighted MRI slice 2 months after stroke, showing a right parietal hematoma in resorption. C) Leftward lateropulsion and pusher behavior in standing. D) Line bisection 1 month after stroke. The 21-mm ipsilesional deviation indicated severe spatial neglect (cut-off 6.5 mm). E) Copying a landscape (Gainotti and Tiacci, 1970) 1 month after stroke, showing signs of spatial neglect (left parts of the first tree and the house) associated with a -12° inclination of the main axis of the drawing. F) Short text written by JW before the stroke. Main line axes were approximately horizontal (0.5°). G) Short text written by JW after stroke showing a global counterclockwise tilt (plain blue lines). The dashed blue line represents the horizontal reference. Plain blue lines indicate the inclination of the orthogonal coordinate system. The mean tilt of lines was -9.4° and mean tilt of the left margin -9°. Finally, counterclockwise tilts observed after stroke were all congruent in direction and amplitude: drawing, margin and lines. H-K) JW's handwriting on cued (lined) paper, with lines traced 24° upward (H), 12° upward (I), 24° downward (J) and 12° downward (K).

TABLES AND FIGURES

Asterisks indicate pathological scores. Details on these methodological references are in the references section of additional material.

M2 M3 M9

Body neglect scores

Bisiach test (0 to 3, cut-off > 0)

Comb test (% neglect index, cut-off > 11%)

Razor test (% neglect index, cut-off > 11%)

Fluff-Test (omissions 0 to 15, cut-off > 2)

Non-body neglect scores

Bells cancellation test (total omissions, cut-off ≥ 6)

Line bisection (ipsilesional deviation in mm, cut-off ≥ 7 mm)

Landscape copying (omissions 0 to 5, cut-off > 0)

Text reading (omissions, cut-off > 0)

Overlapping figures test (omissions, cut-off > 0)

Neglect in daily life score