ORCID Maxime Mauduy
email: maxime.mauduy@unicaen.fr.

Virginie Bagneux

Cécile Sénémeaud

Fostering Victim-Defending Behaviors among School Bullying Witnesses: A Longitudinal and Experimental Test of Two New Strategies for Changing Behavior

Keywords: school bullying, witness, victim-defending behavior, induced hypocrisy, implementation intention

In school bullying, witnesses play a fundamental role because their defensive reactions can stop bullying situations. However, a great majority of witnesses remain passive despite their prodefense personal beliefs. To address this gap between witnesses' beliefs and behaviors, we developed and tested two social psychology strategies, namely, the induced hypocrisy and implementation intention. In an experimental and longitudinal study (N = 101), we randomly divided 7 th -and 8 th -grade students into three conditions, namely, control, induced hypocrisy, or implementation intention. Linear mixed models showed that the two strategies immediately increased planned defending behaviors as well as self-reported defending behaviors at three months but with a stronger effect at three months for the induced hypocrisy strategy. These new prevention avenues for practitioners are discussed according to current approaches.

Fostering Victim-Defending Behaviors among School Bullying Witnesses:

A Longitudinal and Experimental Test of Two New Strategies for Changing Behavior

Introduction

School bullying is usually defined as intentional and repeated aggressive action among students that involves an observed or perceived power imbalance [START_REF] Salmivalli | Participant Roles in Bullying: How Can Peer Bystanders Be Utilized in Interventions? Theory Into Practice[END_REF]. Despite the development of sensibilization campaigns, school bullying remains widespread around the world (e.g., [START_REF] Demaray | Bullying Participant Behaviors Questionnaire (BPBQ): Establishing a Reliable and Valid Measure[END_REF][START_REF] Kubiszewski | Prévalence des élèves témoins de harcèlement scolaire et difficultés associées : résultats préliminaires. [Prevalence of students witnessing school bullying and associated difficulties: preliminary results[END_REF]. Taking many forms, school bullying is a major societal issue because it is associated with several negative consequences for victims, perpetrators, and witnesses (see [START_REF] Shaw | The forms of bullying scale (FBS): Validity and reliability estimates for a measure of bullying victimization and perpetration in adolescence[END_REF] for a review). For prevention purposes, research initially focused on perpetrators and victims, but a great deal of research has also examined the role of witnesses in preventing bullying (e.g., [START_REF] Kubiszewski | Witnessing school bullying: to react or not to react? An insight into perceived social norms regulating self-predicted defending and passive behaviours[END_REF][START_REF] Polanin | A Meta-Analysis of School-Based Bullying Prevention Programs' Effects on Bystander Intervention Behavior[END_REF] and has shown the important role played by witnesses' reactions [START_REF] Salmivalli | Participant Roles in Bullying: How Can Peer Bystanders Be Utilized in Interventions? Theory Into Practice[END_REF]. Witnesses can adopt three different reactions to bullying situations [START_REF] Demaray | Bullying Participant Behaviors Questionnaire (BPBQ): Establishing a Reliable and Valid Measure[END_REF], namely, helping the ringleader bully or reinforcing bullying (i.e., probullying reactions), watching passively or walking away (i.e., passive reactions) and standing up for the victim (i.e., defending reactions). Nevertheless, while defending reactions can put an end to school bullying [START_REF] Espelage | Willingness to Intervene in Bullying Episodes Among Middle School Students: Individual and Peer-Group Influences[END_REF][START_REF] Salmivalli | Participant Roles in Bullying: How Can Peer Bystanders Be Utilized in Interventions? Theory Into Practice[END_REF], the large majority of witnesses remain passive [START_REF] Demaray | Bullying Participant Behaviors Questionnaire (BPBQ): Establishing a Reliable and Valid Measure[END_REF]. Consequently, fostering victim-defending behaviors among witnesses seems promising to prevent school bullying.

1.1.Need to Move Beyond Witnesses' Personal Beliefs to Foster Victim-Defending

Behaviors

With the aim of preventing school bullying, researchers have worked to identify psychological variables associated with victim-defending behaviors among witnesses. Several variables have been evidenced (see [START_REF] Lambe | Standing up to bullying: A social ecological review of peer defending in offline and online contexts[END_REF] for a review), such as positive antibullying, provictim and victim-defending attitudes [START_REF] Nickerson | Empathetic responsiveness, group norms, and prosocial affiliations in bullying roles[END_REF][START_REF] Rigby | Expressed Readiness of Australian Schoolchildren to Act as Bystanders in Support of Children who are Being Bullied[END_REF] or subjective norms for victim-defending [START_REF] Kubiszewski | Witnessing school bullying: to react or not to react? An insight into perceived social norms regulating self-predicted defending and passive behaviours[END_REF]. As a result, many programs or prevention actions aim at reinforcing these witnesses' positive personal beliefs to make them defend victims [START_REF] Garandeau | Le Programme Anti-Harcèlement KiVa [The KiVa Anti-Bullying Program[END_REF]. However, research suggests that the majority of witnesses already hold prodefense beliefs [START_REF] Kubiszewski | Witnessing school bullying: to react or not to react? An insight into perceived social norms regulating self-predicted defending and passive behaviours[END_REF][START_REF] Salmivalli | Connections between attitudes, group norms, and behaviour in bullying situations[END_REF], which does not necessarily result in witnesses standing up for school bullying victims or in their intention to act [START_REF] Mauduy | Unmasking School Bullying Witnesses : Five Different Psychological Profiles Related to Intention to Defend Victims Unmasking School Bullying Witnesses : Five Different Psychological Profiles[END_REF]. Consequently, new strategies, no longer based on the strengthening of beliefs but rather on the strengthening of the link between beliefs and action, would seem appropriate.

Research in the field of social psychology, beyond pointing out and understanding such gaps between individual personal beliefs (e.g., attitude and intention) and behaviors (see [START_REF] Gollwitzer | Implementation Intentions and Goal Achievement: A Metaanalysis of Effects and Processes[END_REF], has proposed some levers to support witnesses' victim-defending behaviors. The first lever would be to bridge the gap between attitudes and behaviors by making students aware that they do not act according to their antibullying attitudes. The second lever would be to address the gap between intention to defend victims and actual behaviors by helping students plan defense strategies. These two levers correspond to two classic strategies of behavioral change in social psychology, namely, induced hypocrisy (IH, [START_REF] Priolo | Three Decades of Research on Induced Hypocrisy: A Meta-Analysis[END_REF] and implementation intention (II, [START_REF] Gollwitzer | Implementation Intentions and Goal Achievement: A Metaanalysis of Effects and Processes[END_REF], respectively. Thus, we aim to test two developed interventions targeting the attitude-behavior gap and the intention-behavior gap to increase defending behaviors among school bullying witnesses.

1.2.Induced Hypocrisy and Implementation Intention: Two Psychosocial

Strategies of Behavioral Change

First, the IH paradigm was created by [START_REF] Aronson | Overcoming denial and increasing the intention to use condoms through the induction of hypocrisy[END_REF] because everyday life is filled with situations in which our behaviors do not match our beliefs, such as being against school bullying but not defending victims (see [START_REF] Priolo | Three Decades of Research on Induced Hypocrisy: A Meta-Analysis[END_REF] for a meta-analysis). It is a two-step procedure in which individuals advocate a prosocial cause and then recall their own past failures to comply with it. This inconsistency between behaviors and personal beliefs generates the hypocrisy effect, which creates a psychological state that motivates people to adopt behaviors in accordance with their prosocial beliefs. A meta-analysis indicated a medium to large hypocrisy effect (d =.70, [START_REF] Priolo | Three Decades of Research on Induced Hypocrisy: A Meta-Analysis[END_REF], making it a powerful strategy for encouraging prosocial behaviors in many fields (e.g., environmental protection, road safety, and health-related issues). It has also been shown that IH is effective in reducing discriminatory behaviors [START_REF] Mauduy | When Combining Injunctive and Descriptive Norms Strengthens the Hypocrisy Effect: A test in the Field of Discrimination[END_REF][START_REF] Hing | Inducing hypocrisy to reduce prejudicial responses among aversive racists[END_REF], but to the best of our knowledge, it has never been tested for school bullying prevention.

Second, to resolve the intention-behavior gap, which is one of the most well-known weaknesses of predictive behavioral models, [START_REF] Gollwitzer | Implementation intentions: Strong effects of simple plans[END_REF] developed the II strategy. It does not affect individuals' intentions but ensures that the intended behavior is triggered at the appropriate time in relevant situations by asking individuals to engage in action planning. II procedures involve making "if-then" plans that encourage people to identify critical situations (e.g., a bullying situation) and associate them with appropriate responses (e.g., alerting an adult). Due to the projected plans, the accessibility of the critical situation is enhanced when people encounter it, leading to automatically related responses. Forming such plans has a medium to large effect size (d =.65, see [START_REF] Gollwitzer | Implementation Intentions and Goal Achievement: A Metaanalysis of Effects and Processes[END_REF] meta-analysis) to translate intentions into specific actions in a range of contexts (e.g., reducing binge drinking, cigarette smoking or discrimination). Nevertheless, this strategy has not yet been tested in school bullying prevention.

1.3.The present study

This study aims to test the effectiveness of IH and II, two behavioral change strategies, in the context of school bullying. More specifically, we expect that, on the one hand, IH by targeting the witnesses' gap between their prodefense attitudes and passive behaviors, and on the other hand, II addressing witnesses' gap between their intention to defend victims and their actual behaviors, will increase victim-defending behaviors. Since developing and testing these two strategies in the school bullying prevention context is new, we were interested in whether they could stand alone for fostering victim-defending behaviors among witnesses. We therefore opted to test their effectiveness independently of each other.

Method

Sample and Study Design

A total of 118 students at the 7 th and 8 th grade levels from one middle school in Normandie (France) participated on a voluntary basis in the study on school bullying.

Ultimately, 101 students (Mage = 12.7, SD = 0.77; 47.5% of girls; 49.5% in 7 th grade and 50.5% in 8 th grade) were included in the analyses. Six participants were excluded for not fully completing the experimental conditions, ten because they did not witness school bullying, and one participant because he or she did not indicate his or her past reactions to bullying. This sample size allowed us to achieve a sufficient statistical power of .82 (see supplemental materials for more details). The students came from a priority area school ("Réseau d'Education

Prioritaire" in French), with most of them from socioeconomic lower-class families.

An experimental longitudinal trial was conducted over a 3-month period. Students were approached collectively in classrooms and were balanced across three conditions: control (n = 26; two classes), II intervention (n = 36; three classes) and IH intervention (n = 39; three classes). Students in the same class were assigned to the same experimental condition for two reasons. First, in a theoretical view, school bullying is a group phenomenon that often takes place within the class group and the strategies used can be more effective when carried out in a group (vs. individually). Second, for ecological reasons, it was easier to mobilize the students collectively within their class rather than individually, especially for practitioners and educational staff who are likely to adopt these strategies in a group class and not individually due to organizational constraints. Grade level and gender are equally balanced across the three conditions (see the Supplemental Table 1 in supplemental materials for more details). The first measure (i.e., intentions of defending behaviors) was administered at the end of the intervention, and the second measure (i.e., self-reported defending behaviors) was administered three months later. At the end of the study, the students were debriefed to describe the purpose of this study and the scope of the results and to address any questions. The school nurse and psychologist were present at the school to address any requests from students following the study; however, no such requests were made.

2.2.Ethics

The survey was conducted in full agreement with the ethical standards set by the Psychology Department that follow the American Psychological Association's Ethical Principles of Psychologists and the Code of Conduct (APA, 2019) for the ethical treatment of human participants. The protocol was submitted to the school departments of the Normandy Region as well as to the school principal and teachers of each class involved in the study. Prior to data collection, individual consent for participation and active parental consent were requested and obtained. The participants were assured of the data confidentially and informed that their participation in the study was voluntary and that they could withdraw from the study at any time. Finally, none of the 118 participating students left the study over the course of the survey.

2.3.Materials

Induced-Hypocrisy Intervention

First, the students performed the normative step of the IH strategy (i.e., advocacy). They were collectively invited to develop normative arguments against school bullying (see the supplemental materials). Advocating a prosocial cause is the classical procedure used to activate the hypocrisy effect. It is most effective when it is carried out collectively (vs. individually, [START_REF] Fointiat | Being together in a situation of induced hypocrisy[END_REF] and made public (vs. private, [START_REF] Stone | To Practice What We Preach: The Use of Hypocrisy and Cognitive Dissonance to Motivate Behavior Change[END_REF]. Hence, in subgroups of four or five, the students were asked to list arguments demonstrating why school bullying is disapproved of and were told that these arguments would be reused in antibullying programs for schoolchildren. Second, the students performed the "transgressions recall" step of the IH strategy. They were invited to individually and anonymously (as recommended by [START_REF] Stone | To Practice What We Preach: The Use of Hypocrisy and Cognitive Dissonance to Motivate Behavior Change[END_REF]) complete a questionnaire concerning their past passive behaviors in witness situations. In this questionnaire, we presented them six passive witness situations (i.e., hurtful teasing, unpleasant name calling, deliberate exclusion, spreading of lies about an individual, threats of harm, physical injury, and prevention of someone from talking) 1 . In order to facilitate students' recall of their past passive behaviors, they were asked to specify for each situation whether it had happened to them as well as when they occurred (see supplemental materials).

The Implementation Intention Intervention

The II procedure was based on the technique of the Volitional Help Strategy [START_REF] Armitage | A volitional help sheet to encourage smoking cessation: a randomized exploratory trial[END_REF]) combined with coping planning (i.e., dealing with situational cues that usually lead to unintended responses, [START_REF] Schwarzer | Coping planning as an intervention component: A commentary[END_REF]. First, a collective discussion with the students was held to identify school bullying situations (e.g., "a student is deliberately excluded"), the barriers witnesses encounter in defending victims of school bullying (e.g., "I don't think it's my concern"), and the appropriate responses to defend victims, including the "where" and "when" (e.g., "I will alert an adult during recess"). Each element identified by the students was noted on the school board by the experimenter under the headings "situations", "barriers" and "appropriate responses + where and when". Asking the students to identify these three elements 1 A pilot study conducted on witnesses of school bullying (N = 321) allowed the identification of the situations when the witnesses had most frequently reacted passively. Hurtful teasing was observed by 50.4% of witnesses (68.1% of passive reactions), deliberate exclusion was observed by 43.9% (58.9% of passive reactions), prevention from talking was observed by 30.5% (63.3% of passive reactions), unpleasant name calling was observed by 30.2% (70.1% of passive reactions), threats of warm harm were observed by 26.8% (68.3% of passive reactions), and insults were observed by 22.4% (75.0% of passive reactions). themselves allowed us to specifically apply the II strategy to the students' experiences. Second, the students each received an action planning sheet (see supplemental materials) and were asked to choose, write and draw lines connecting three situations, barriers and appropriate responses by using those previously listed. The instructions stated that the more precise, concrete and personal their formulated plans were, the greater their plans could help them.

Third, they were asked to write their three plans under the following format: "IF [situation] BUT [barrier], THEN [appropriate response]". Finally, the students were told to memorize their plans carefully.

Dependent Variable Measures

Behavioral Intention to Defend School Bullying Victims

Adapted from [START_REF] Rigby | Expressed Readiness of Australian Schoolchildren to Act as Bystanders in Support of Children who are Being Bullied[END_REF], a questionnaire was created to measure behavioral intention to defend victims of school bullying. First, participants were instructed to imagine that they encounter three fictive situations of witnessing school bullying (i.e., "Students are hurtfully teasing another student for his or her appearance", "Secrets were told about a student to others to hurt him/her", "A student was not able to eat his/her meal because his/her food was damaged"). For each situation, the participants were asked to indicate what their reaction would be by selecting one of the following four reactions: assisting the bully, reinforcing bullying, remaining passive, and defending the victim [START_REF] Mauduy | Unmasking School Bullying Witnesses : Five Different Psychological Profiles Related to Intention to Defend Victims Unmasking School Bullying Witnesses : Five Different Psychological Profiles[END_REF]. The calculation of the ratio of defending reactions to the three situations allowed us to establish a score of defending behaviors' intention. Concretely, we divided the number of students' planned defending behaviors by the number of bullying situations (i.e., three). Ranging from 0 to 1, a score of 0 for the defending reactions indicates that students not report behavioral intention to defend school bullying victims, and a score of 1 indicates he/ she indicates a strong behavioral intention.

Self-Reported of Victim-Defending Behaviors (at three months)

We assessed both students' witnessing of school bullying and their reactions to it [START_REF] Mauduy | Unmasking School Bullying Witnesses : Five Different Psychological Profiles Related to Intention to Defend Victims Unmasking School Bullying Witnesses : Five Different Psychological Profiles[END_REF]. First, after the adolescents were provided with a comprehensive definition of school bullying (see supplemental materials), 12 items were presented with 5point scales (1-never, 2-only once or twice, 3-two or three times a month, 4-about once a week, 5-several times a week). These items have traditionally been used to assess school bullying situations, describing negative behaviors such as hurtful teasing, unpleasant name calling, deliberate exclusion, the spreading of lies about someone, threats and harm, and physical injury.

The adolescents were asked to indicate how often they had observed each of these 12 situations in the last three months. Second, for each of the 12 observed school bullying situations specified above, students were asked to report their potential resort to one of the four reactions: (1) assisting the aggressor, (2) reinforcing the bullying situation, (3) remaining passive, and (4) defending the victim. When the students reported that they had encountered a situation, they were asked to indicate their reactions by checking one of the four reactions. Finally, a score of defending reactions was calculated from the ratio of the number of past defending reactions to the total number of bullying situations witnessed.

2.4.Data Analysis

We used the Helmert contrast method [START_REF] Brauer | L'utilisation des contrastes dans l'analyse des données : Comment tester les hypothèses spécifiques dans la recherche en psychologie ? [The Use of Contrasts in Data Analysis: How to Test Specific Hypotheses in Psychological Research ?[END_REF] to process our data.

According to it, with the first contrast, control = -2, II = 1, IH = 1, we compared the control condition to the two interventions considered together in order to test their effectiveness, and with the second contrast, control = 0, II = -1, IH = 1, we tested whether a difference was observed between the two interventions.

Two linear mixed models with the bootstrapping method in Jamovi [START_REF] Gallucci | GAMLj: General analyses for linear models[END_REF] were constructed for behavioral intention to defend victims and self-reported defending behaviors at three months, with the two contrasts and gender included as fixed effects and students' grade level as a random effect. These mixed models allow us to test the effects of our interventions by controlling for variance in our outcomes explained by students' grade level.

Results

Linear Mixed Model for Defending Behaviors' Intention

The results of the mixed model for defending behaviors' intention (see Table 1 for all statistical details) indicated that only the Helmert 1 contrast was significant after controlling for the random effect of grade level. More specifically, when facing fictitious situations of school bullying, the students who participated in the II (M = 61.6%, SD = 38.4) and IH (M = 60.7%, SD = 33.6) interventions reported greater intentions to defend victims of school bullying than the students in the control condition (M = 37.2%, SD = 37.0). No significant difference was observed between the II and IH interventions (Helmert 2).

Linear Mixed Model for Self-Reported Defending Behaviors at Three

Months

The results of the mixed model for self-reported victim-defending behaviors at three months (see also Table 1) indicated that both Helmert 1 and Helmert 2 contrasts were significant. More specifically, the students in both experimental conditions reported more victim-defending behaviors three months after the interventions compared to the control condition (M = 14.4%, SD = 22.2). Additionally, the students in the IH intervention reported more defending behaviors (M = 50.3%, SD = 33.6) than those in the II intervention (M = 33.3%, SD = 27.9) (see Figure 1).

Discussion

The present study aimed to test the effectiveness of two psychosocial strategies of behavioral change for the first time in promoting school bullying witnesses' victim-defending behaviors. The results of the experimental longitudinal study indicated that both the IH and II strategies seem to be effective in a short time, although the effect of the IH strategy seems to last longer (three months after the intervention). Two important issues need to be discussed.

First, we provide insights into the greater effectiveness of IH on II. Second, we place the possible contribution of these new prevention strategies for practitioners in the context of current prevention approaches.

First, the lower effectiveness of II compared to IH may be related to the strength of students' preexisting intention to defend victims. We know that the II strategy is more effective when individuals have a strong intention to perform the behavior [START_REF] Gollwitzer | Implementation Intentions and Goal Achievement: A Metaanalysis of Effects and Processes[END_REF]. However, Mauduy et al.' study (2021) shows that only one witness out of two (i.e., 52.55%) declares intent to defend victims of bullying and, furthermore, that the strength of this intention varies across witness profiles. Thus, a possible explanation for the lower effectiveness of II compared to IH could be that II is a suitable strategy for fewer bullying witnesses, given that it requires a prior (strong) intention to defend the victims. The IH, however, would be even more suitable for the bullying situation since [START_REF] Mauduy | Unmasking School Bullying Witnesses : Five Different Psychological Profiles Related to Intention to Defend Victims Unmasking School Bullying Witnesses : Five Different Psychological Profiles[END_REF] showed that nearly three-quarters of witnesses hold a positive attitude toward victim defense (i.e., 73.2%). Moreover, the IH has the particularity of (1) reinforcing the intention to perform a behavior and (2) reinforcing the realization of that same behavior (see [START_REF] Priolo | Three Decades of Research on Induced Hypocrisy: A Meta-Analysis[END_REF]. Thus, an interesting avenue for future research would be to combine these two strategies to (1) reinforce the intention to support victims via the IH strategy and then (2) translate this intention into action via the II strategy. To our knowledge, combining these two strategies has never been studied in the field of bullying or in any other field.

Second, we believe that the strategies proposed and tested in this study can improve bullying prevention and be directly useful to practitioners by departing from current prevention approaches. Current prevention programs, such as the Kiva program [START_REF] Kärnä | Effectiveness of the KiVa Antibullying Program: Grades 1-3 and 7-9[END_REF], the most widespread program in Europe, aim to strengthen students' personal beliefs, such as prodefense attitudes, normative beliefs, sense of self-efficacy, and intention to defend school bullying victims [START_REF] Garandeau | Le Programme Anti-Harcèlement KiVa [The KiVa Anti-Bullying Program[END_REF]. In other words, current prevention programs do not offer any solutions to bridge gaps that exist in some witnesses between their prodefense attitude and actual behaviors or their intention to defend victims and actual behaviors.

Therefore, adding two modules based on IH and II would be effective at a minimum time cost (one-hour session), directly available and easily integrated by school educational staff, and highly complementary to current approaches. Testing whether these new strategies can supplement current approaches would be an interesting avenue for future research to improve our effectiveness in addressing school bullying.

Limitations and Future Research

This study has three main limitations that we should note. First, to adapt and preliminarily test two new psychosocial strategies for bullying prevention, this study was conducted in a single school. The generalizability of the findings, both throughout France and internationally, is a first limitation, and the effectiveness of these strategies needs to be confirmed. Second, students' initial behavioral intention and defending behaviors were not measured. Although the comparability of our three conditions with respect to the dependent measures is assumed because of the study experimental design (i.e., randomization), controlling students' initial values of dependent measures would strengthen the study findings. Third, we based our study on the literature showing that the school bullying rate (victim and perpetrator) is reduced through witnesses' reactions (e.g., [START_REF] Polanin | A Meta-Analysis of School-Based Bullying Prevention Programs' Effects on Bystander Intervention Behavior[END_REF]. However, we did not test whether our interventions allowed us to globally reduce the school bullying rate. Including such measures in future studies would be necessary to test the overall effectiveness of these strategies in preventing school bullying.

Figure 1

 1 Figure 1

Acknowledgements

We are grateful to the parents, teachers, and school administrators who made the research possible, and the students who contributed to this study.

Declaration of interest statement

The authors have no interest conflict.

Author Biographical Statements

Dr. Maxime Mauduy is currently a postdoctoral researcher in social psychology at the University of Caen Normandie (LPCN, Caen, France) and a lecturer at University of Lausanne (FSSP, Lausanne, Suisse). His research interests are cognitive dissonance, school bullying, and students' substance use (alcohol and tobacco), where he studies their underlying psychological processes to develop prevention strategies.

Dr. Virginie Bagneux, is an Assistant Professor at the University of Caen Normandie (LPCN, Caen, France). Her research interests are emotional and cognitive biases in information processing and critical thinking.

Dr. Cécile Sénémeaud, is a Professor at the University of Caen Normandie (LPCN, Caen, France). She is an expert in psychological processes at play in behavioral change, especially in the public health domain (blood donation, addictions, bullying). She more specifically studied the effectiveness of various communication strategies on behavioral change as a function of individual differences.

Tables and Figures


T Fouchet

Lesia

B B Ézard

P Drossart

R Mel- Chiorri

T Encrenaz

M Combes

F Poulet

Y Langevin

B Gondet

IAS J.-P Bibring

F Altieri

G Bellucci

M Äättänen

OMEGA LIMB OBSERVATIONS OF THE MARTIAN DUST AND ATMO-SPHERIC COMPOSITION

Introduction

The martian dust is one of the most difficult atmospheric components to model and observe. Indeed, a remote observer, to infer the dust optical thickness and properties, must discriminate photons scattered by aerosols among photons reflected by the martian surface. Various methods have been use to solve this problem. Here we present results obtained by looking at the Mars limb. Any light detected have been scattered by aerosols, although a fraction of the scattered light may have been reflected on the surface.

But limb observations do not only allow to more acrrurately disentangle surface and dust contributions to the martian reflectance. They also allow the observer to retrieve the vertical profile of the dust and aerosols (particle density as a function of altitude), while nadir observations are usually only sensitive to column integrated density. Moreover, when the dust slant optical thickness reaches saturation, the reflectance only depends on the single scattering albedo and the phase function of the particles. In these conditions, limb observations become powerful tool to constrain the particle optical properties. Finally, limb observations increase the sensitivity to minor gaseous species, and allows again to measure vertical distribution of some atmospheric constituents like water or oxygen.

The OMEGA spectro-imaging system aboard the Mars Express mission has observed during several limb sessions. OMEGA operates between 0.35 and 5.1 m with a spectral resolution ranging from 7 to 20 nm. The vertical resolution of the limb observations varies from session to session between 2 and 9 kilometers. Various seasons, latitudes, and local time have been sampled.

Model

We use a two-stream radiative transfer code. The source function are calculated in plane parallel geometry (the solar zenith angle is never higher than 60 degrees). Then, the radiative transfer equation is integrated for spherical geometry. Gaseous opacity is calculated from a line-by-line radiative transfer code. For the particle optical properties we first assume a single scattering albedo and an asymmetry parameter extracted from the Ockert-Bell model (Ockert-Bell et al. (1999)). The dust vertical profile is inverted according the method proposed by [START_REF] Jaquin | The vertical structure of the limb hazes in the martian atmosphere[END_REF]. Our code does not allow yet to include the thermal emission which starts to dominate the spectrum at wavelengths larger than ¡ £¢ m.

To interpret the optical depth measured at a given altitude as a function of wavelength, we have designed a deconvolution tool. We take advantage of the optical thickness being the sum of the cross-section due to particles of all sizes, hence a convolution of the particle size distribution by the cross-section function. We have assumed a cross-section calculated from the Mie theory for the real and imaginary refractive index from Ockert-Bell et al. (1999). The deconvolution was stabilized and regularized using algorithm described in [START_REF] Craig | Inverse problems in astronomy[END_REF].

This allows us to recalculate the single scattering albedo and the asymmetry parameter as a function of wavelength and altitude and to iterate the inversion.

We will present the inverted dust vertical profiles. As shown in Figure 1, a wide variety of profiles have been retrieved some rather homogeneous, some exhibiting one or several detached haze layers. In Figure 2 we present one example of optical thickness as a function of altitude and wavelength. The dependence of the optical thickness with wavelength increases with altitude, showing the decrease of particle size with altitude.

Figure 1 :

 1 Figure 1: Figure 1 : Optical thickness as a function of altitude for different seasons and local time. The OMEGA data present a huge variability in terms of total optical thickness, the presence of detached layers some of which associated with water ice.

Figure 2 :

 2 Figure 2: Figure 2: Optical thickness as a function of altitude and wavelengths for ¤ ¦¥ ¨ § © © . The detached layer at 60 km presents the spectral signature of water ice at 1.5 m.


Victoriya Kashtanova

Mihaela Pop

Ibrahim Ayed

Patrick Gallinari

Maxime Sermesant
email: maxime.sermesant@inria.fr

APHYN-EP: Physics-based deep learning framework to learn and forecast cardiac electrophysiology dynamics

Keywords: Physics-based learning, Deep Learning, Electrophysiology, Simulations

Biophysically detailed mathematical modeling of cardiac electrophysiology is often computationally demanding, for example, when solving problems for various patient pathological conditions. Furthermore, it is still difficult to reduce the discrepancy between the output of idealized mathematical models and clinical measurements, which are usually noisy. In this paper, we propose a fast physics-based deep learning framework to learn cardiac electrophysiology dynamics from data. This novel framework has two components, decomposing the dynamics into a physical term and a data-driven term, respectively. This construction allows the framework to learn from data of different complexity. Using 0D in silico data, we demonstrate that this framework can reproduce the complex dynamics of transmembrane potential even in presence of noise in the data. Additionally, using ex vivo 0D optical mapping data of action potential, we show the ability of our framework to identify the relevant physical parameters for different heart regions.

Introduction

Multi-physics phenomena involved in cardiac function can be studied using mathematical models of different complexity. Among them, several electrophysiology (EP) models can accurately reproduce the electrical behaviour of the heart at cellular, tissue or organ level. For instance, in order to describe the dynamics of transmembrane voltage, current and different ionic concentrations in the cardiac cell, detailed biophysical models such as the Ten Tusscher-Panfilov model [START_REF] Ten Tusscher | A model for human ventricular tissue[END_REF][START_REF] Ten Tusscher | Alternans and spiral breakup in a human ventricular tissue model[END_REF] have been proposed. However, these models are intricate and computationally expensive, and have numerous hidden variables which are nearly impossible to measure all, making the model parameters difficult to personalise.

Alternatively, one can use phenomenological models that involve simplified descriptions derived from biophysical models. Examples of such models are the FitzHugh-Nagumo, Aliev-Panfilov, and Mitchell-Schaeffer models [START_REF] Fitzhugh | Impulses and physiological states in theoretical models of nerve membrane[END_REF][START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF][START_REF] Aliev | A simple two-variable model of cardiac excitation[END_REF][START_REF] Nash | Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias[END_REF][START_REF] Mitchell | A two-current model for the dynamics of cardiac membrane[END_REF], which all employ fewer variables and parameters, and are therefore especially useful for rapid computational modelling of wave propagation at tissue and organ level. However, these are less realistic and, consequently, need a complementary mechanism to enable their fitting to measured data. Machine learning and in particular deep learning (DL) approaches could help provide such correction mechanisms. Thus, the combination of rapid phenomenological models and machine learning components may allow the development of rapid and accurate models of transmembrane dynamics.

For this reason, researchers have started to use coupled physico-statistical approaches for cardiac electrophysiology simulations with a high precision and at low cost. For example, one group designed a neural network that approximates the FitzHugh-Nagumo model [START_REF] Court | Design of the monodomain model by artificial neural networks[END_REF], others used a physics-informed neural network for cardiac activation mapping accounting for underlying wave propagation dynamics [START_REF] Sahli Costabal | Physicsinformed neural networks for cardiac activation mapping[END_REF], while another group proposed an approach to create a nonlinear reduced order model with the help of deep learning algorithms (DL-ROM) designed for cardiac EP simulations [START_REF] Fresca | POD-enhanced deep learningbased reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium[END_REF]. Lastly, other researchers presented a physics-informed neural network for accurate simulation of action potential and correct estimation of model parameters [START_REF] Herrero Martin | EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks[END_REF]. However, the majority of these coupled approaches is based on high-fidelity physical models. Fitting those to the data could be not only computationally expensive but also difficult to properly manage large discrepancies between simulated and real data.

To alleviate this limitation, here we propose to use the APHYN-EP framework, a DL framework based on a fast low-fidelity (or incomplete) physical model. This framework has two components, decomposing the dynamics into a physical term and a data-driven term, respectively. Notably, the data-driven deep learning component is specifically designed to capture only the information that cannot be modeled by the incomplete physical model. In our previous paper [START_REF] Kashtanova | Deep learning for model correction in cardiac electrophysiological imaging[END_REF] we showed that this framework is able to reproduce with good precision the 2D dynamics simulated by the Ten Tusscher -Noble -Noble -Panfilov ionic model. But we applied the framework only on simulated data and on the depolarisation part of a cardiac action potential cycle, neglecting the repolarisation part.

In this paper, we specifically explore the capability of APHYN-EP framework to learn the full cardiac action potential cycle (i.e., depolarisation and repolarisation phases). Using 0D in silico data, we demonstrate that our framework can reproduce the complex dynamics of transmembrane potential even in presence of noise in the data. Additionally, using ex vivo 0D epicardial optical fluorescence mapping data, we show the capability of this framework to identify relevant physical parameters for different anatomical zones of the heart.

Learning Framework

In order to learn the cardiac electrophysiology dynamics (X t ), here we solve an optimization problem via our physics-based data-driven APHYN-EP framework. This particular framework combines a physical model (F p ) representing an incomplete description of the underlying phenomenon with a neural network (F dl ), where the latter complements the physical model by capturing the information that cannot be modeled by the physics-described component:

min Fp∈Fp,F dl ∈F dl F dl subject to ∀X ∈ D, ∀t, dX t dt = F (X t ) = (F p +F dl )(X t ). (1)
Our incomplete physical model is the two-variable (v, h) model by [START_REF] Mitchell | A two-current model for the dynamics of cardiac membrane[END_REF] for cardiac EP simulations [START_REF] Camara | Inter-model consistency and complementarity: Learning from ex-vivo imaging and electrophysiological data towards an integrated understanding of cardiac physiology[END_REF]. The variable v represents a normalised (v ∈ [0, 1]) dimensionless transmembrane potential while the "gating" variable h controls the repolarisation phase (i.e, the return to initial resting state):

dv dt = hv 2 (1 -v) τ in - v τ out + J stim (t) dh dt = 1-h τopen if v < v gate -h τ close if v > v gate (2)
where J stim is a transmembrane potential activation function, which is equal to 1 during stimulation time (t stim ). This physical model has been successfully used in patient-specific modelling [19], covering general EP dynamics. Furthermore, in contrast to very detailed ionic/cellular models, this model is flexible in terms of spatial and temporal steps. Thus, assuming the initial conditions for this system (2) v(t = 0) = 0 and h(t = 0) = 1 we can calculate an approximation of h for any time point t with the help of a simple integration scheme.

The framework's deep learning component (F dl ) is a Residual Network (Res-Net) [START_REF] He | Deep residual learning for image recognition[END_REF], because it can accurately reproduce a complex dynamics due to residual connections in its architecture. The choice of data-driven component is not limited by ResNet architecture. We performed preliminary experiments with other types of networks (out of scope for this paper), but overall the ResNet model was the most stable along the different simulations.

In APHYN-EP framework the physical and the data-driven components are trained simultaneously using automatic differentiation tools provided by the Pytorch library [START_REF] Paszke | Pytorch: An imperative style, highperformance deep learning library[END_REF]. Here, the Loss function (L) for training consists of 2 parts: trajectory-based loss and loss on norm of F dl , being represented as following:

L = λ * L traj + F dl = λ * N i=1 T /∆t h=1 ||X (i) h∆t - X(i) h∆t || + F dl where each state X(i) h∆t = X (i) 0 +h∆t X (i) 0 (F θp p + F θa a )(X s ) dX s is calculated from the initial state X (i)
0 via a differentiable ODE solver [START_REF] Chen | Neural ordinary differential equations[END_REF]3]. The role of the λ coefficient is to balance the two parts of the loss. During training, we use λ in the dynamic mode as λ j+1 = λ j + γL traj (θ j+1 ) (j is an epoch number) to artificially increase the importance of L traj at the beginning of training and gradually decrease it, changing the focus of optimization on norm of F dl . This training algorithm is described in detail in [START_REF] Yin | Augmenting physical models with deep networks for complex dynamics forecasting[END_REF][START_REF] Kashtanova | Deep learning for model correction in cardiac electrophysiological imaging[END_REF].

Experimental settings

To test the performance of APHYN-EP framework, we chose two types of experiments. First, using in silico data, we tested the ability of the framework to reproduce the complex dynamics of transmembrane potential even in the presence of noise in the data. Second, using ex vivo optical mapping data, we showed that our framework can identify the relevant physical parameters for different anatomical zones of the unhealthy heart. The details of data collection and training settings used for the experiments are included below.

In silico data

Data collection To evaluate our method, we used a dataset of transmembrane potential activation simulated with a monodomain reaction-diffusion equation and the Ten Tusscher -Panfilov ionic model [START_REF] Ten Tusscher | Alternans and spiral breakup in a human ventricular tissue model[END_REF], which represents 12 different transmembrane ionic currents. The simulations were performed with finetwave software 7 , described in detail in the second chapter of [START_REF] Nezlobinsky | Software for cardiac modelling and patch clamp experiments, and its application to study conduction in the presence of fibrosis[END_REF]. The computational domain represents a 2D slab of cardiac tissue (isotropic), with 24 × 24 elements in size. One stimulation was applied for 1 ms in the selected area (left top corner for training and near the center for validation datasets, see Figure 1) for transmembrane potential activation. Simulations were conducted for 400 ms (of a heart beat) to achieve full depolarization-repolarization cycle (videos showing simulated dynamics across time can be seen in the Supplementary Material). Next, we saved a time sequence for each pixel of the cardiac slab in separate files, creating two databases: for training and validation, respectively. To simplify the workflow for our framework, we removed the section of time sequences where the potential is equal to zero (Figure 1(c)). To test the ability of the framework to operate with noisy data we add to each time sequence a 5 percent random noise with normal Gaussian distribution.

The data simulated via the Ten Tusscher -Panfilov model with added noise were considered here as the ground truth. The objective was then to learn the complex dynamics generated via this model with the APHYN-EP framework, combining a simplified physics description with a deep learning component. This should result in a low computational cost surrogate model of the computationally intensive biophysically detailed Ten Tusscher -Panfilov model. Training settings The physical model (F p ) of Eq. ( 2) was implemented with a standard finite-difference scheme. We estimated only τ in , τ out and τ close as unknown parameters in (2), since they control the major part of model dynamics and thus the main difference between the Mitchell-Schaeffer and the Ten Tusscher -Panfilov models in our simulations. The initial Mitchell-Schaeffer model parameters were taken as in the original paper [START_REF] Mitchell | A two-current model for the dynamics of cardiac membrane[END_REF]: τ in = 0.3, τ out = 6, τ open = 120, τ close = 150, v gate = 0.13 and t stim = 1. For the deep learning component (F dl ) of the framework, we used a ResNet with 8 filters at the initial stage and 3 intermediary blocks, and started with a re-weighted orthogonal initialisation for its parameters. We used a time resolution of 0.1 ms to compute the forecast given by APHYN-EP framework. The training was performed using a horizon of 350 ms. We trained the framework until full model convergence (about 100 epochs) using an ADAM optimiser [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] with initial learning rate of 10 -3 . The hyper-parameters λ 0 and γ of the algorithm were set to 1 and 10, respectively.

It is important to notice that, while the framework takes a few hours to train (about 2h on Nvidia Quadro M2200 GPU), once this is done the inference is quick (less than 10 sec to compute 350 ms of forecasting) and does not require any recalibration.

Ex vivo data

Data collection We tested APHYN-EP framework performance on ex vivo datasets from optical fluorescence imaging of action potential. Briefly, the optical signals were recorded ex vivo on an heart explanted from a juvenile swine ( 25kg in weight). The heart was attached to a Langendorff perfusion system and a voltage sensitive dye (i.e., di-4-ANEPPS) was injected into the perfusate. In order to avoid cardiac motion artifacts during the electrophysiological recordings, the heart contraction was suppressed by a bolus of saline and Cytochalasin D (an electro-mechanical uncoupler). All optical images were acquired epicardially using a high-speed CCD camera (MICAM02, BrainVision Inc. Japan), with hightemporal resolution (3.7ms) as well as a high-spatial resolution (i.e., pixel size 0.7mm). The action potential was then derived at each pixel from the relative change in the intensity of fluorescence signal. The experiments are described in more detail in [START_REF] Pop | Fusion of optical imaging and mri for the evaluation and adjustment of macroscopic models of cardiac electrophysiology: A feasibility study[END_REF]. The recorded signals were exported from the BV-Ana acquisition software and further signal analysis was performed using in-house scripts written in Matlab and Python.

For this experiment, we used a heart with an ischaemic region. We manually selected two rectangular regions of interest (ROIs) with different action potential dynamics across time (see Figure 2). Next, we normalised the optical signal, to obtain a [0, 1] min/max interval for transmembrane potential (while keeping the noise in the data). We took a first full cardiac cycle and removed the parts with zero potential keeping only time-sequences of 300 ms per experiment. Then, we saved a time sequence for each pixel from each ROI in separate files, creating two databases (ROI A and ROI B) containing each about 10 and 5 time-sequences for training and validation respectively. The optical data were considered here as the ground truth. Our specific objective was to learn the complex dynamics of measured action potential, and then to identify the relevant physical parameters for different parts of the heart.

Training settings

The training settings for this experiment were similar to the ones described in section 3.1, except for training horizon (300 ms) and parameter γ, which was set to 1 for better equilibrium.

Results

In silico data

Results demonstrated that APHYN-EP framework was able to accurately reproduce several key features, the wave morphology and the electrical conduction properties of the trasnmembrane potential generated by the Ten Tusscher -Panfilov model (see Figure 3), even in presence of noise in data. Figure 3(a) shows that in absence of parameter τ close (controlling the repolarisation), the data-driven component (ResNet model) completed the dynamics generated by physical component. In the presence of τ close the error of dynamics learned by data-driven component is minimal (Figure 3(b)). The predicted dynamic was generated via an Euler integration scheme, by assimilating only one first measurement of the transmembrane potential dynamics. The framework showed robust resistance to noise in the data, and, as a result, it neglected rapid changes in transmembrane potential activation (see Figure 3, first 40 ms).

Ex vivo data

Using optical imaging mapping data, APHYN-EP framework was able to reproduce the observed action potential dynamics for different ROIs within the heart, identifying the 3 major physical dynamics parameters (τ in , τ out and τ close ). Figure 4 demonstrates that the framework correctly estimated the difference in value of parameter τ close which increases APD or shortens it, respectively. Table 1 shows the mean squared error (MSE) results for our framework forecasting on train and validation data samples. To calculate this error, for each data sample, we fed the framework with only one initial measurement, then let it predict 300 ms forward without any additional information. We also added for comparison a baseline model corresponding to the "incomplete" physical model trained alone on the same dataset as APHYN-EP framework, described in 3.2. As we can observe, our framework outperform the physical model for every dataset, while the contribution of F dl component is still minimal. It is important to notice that we cannot use the Ten Tusscher -Panfilov ionic model [START_REF] Ten Tusscher | Alternans and spiral breakup in a human ventricular tissue model[END_REF] applied on ex vivo data as a baseline model due its parameter personalisation difficulties [START_REF] Camara | Inter-model consistency and complementarity: Learning from ex-vivo imaging and electrophysiological data towards an integrated understanding of cardiac physiology[END_REF]. Despite the expectations and suggestions indicated in [START_REF] Camara | Inter-model consistency and complementarity: Learning from ex-vivo imaging and electrophysiological data towards an integrated understanding of cardiac physiology[END_REF][START_REF] Niederer | Simulating human cardiac electrophysiology on clinical time-scales[END_REF], these difficulties have not been overcome. Complex biophysical models have too many parameters and instabilities to enable robust parameter estimation from such data.

Discussion and Conclusion

In this work, we presented a learning framework that is able to learn the considered dynamics. Overall, our results suggest that automated learning of cardiac electrophysiology dynamics is feasible and has great potential. We showed that APHYN-EP framework can learn and forecast the EP data of different origin (i.e, data simulated via the Ten Tusscher -Panfilov ionic model, as well as real optical mapping data), even in presence of noise in the data. This may be useful for applications concerning fast parameterization of computational heart models.

The main advantage of our proposed framework is its coupled architecture, which allows us to use a simplified low-fidelity electrophysiological model as a physical component of the framework. Such framework opens up possibilities in order to introduce prior knowledge in deep learning approaches through explicit equations, as well as to correct physical model errors from data.

However, we acknowledge some limitations of our model. For instance due to data-driven architecture of the framework, its training is not regular. This could lead to the local minimum for the parameters of the physical component and additional involvement of the deep learning component. A solution for this problem may be given by a more advanced training protocol, rigid boundaries on physical model parameters and performing ablation studies including deep learning component architectural changes, which will be addressed in our future work.

Fig. 1 :

 1 Fig. 1: (a,b) 2D cardiac slab with transmembrane potential activation (yellow) and resting phase (dark blue), for train and validation dataset, respectively. (c) Typical temporal sequence for the experiment (without noise), normalised amplitude of the transmembrane potential on the Y-axis and time (ms) on the X-axis.

Fig. 2 :

 2 Fig. 2: Example of optical mapping data (tracings of action potential waves) recorded ex vivo in a porcine heart. ROI B represents an ischaemic region characterized by a shorten action potential duration (APD) compared to the normal APD recorded in ROI A.

Fig. 3 :

 3 Fig. 3: Validation results of trained framework with learning of (a) 2 (τ in and τ out ) and (b) 3 (τ in , τ out and τ close ) physical parameters. Ground truth (GT), prediction of the framework (Prediction FW), decomposition of prediction on physical (F p ) and DL (F dl ) parts.

Fig. 4 :

 4 Fig. 4: Validation results of the framework trained on: (a) ROI A data and (b) ROI B data. Ground truth (GT) data, prediction of the framework (Prediction FW), decomposition of prediction on physical (F p ) and DL (F dl ) parts.

Table 1 :

 1 Mean-squared error (MSE) of normalised transmembrane potential forecasting (forecasting horizon of 300 ms).

		ROI A	ROI B
		Train	Validation Train	Validation
	APHYN-EP	9.12 * 10 -3	5.72 * 10 -3	1 * 10 -2	8 * 10 -3
	framework (||F dl || 2 )	(1.6 * 10 -4 ) (8 * 10 -5 )	(8 * 10 -5 )	(7 * 10 -5 )
	Physical model only	1.4 * 10 -2	1 * 10 -2	1.45 * 10 -2	9.3 * 10 -3

https://github.com/TiNezlobinsky/Finitewave

Acknowledgements This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, through the "Research and Teaching chairs in artificial intelligence (AI Chairs)" funding for DL4Clim project. The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support and to Dr. A. Panfilov and his PhD student T. Nezlobinsky for showing us their software FiniteWave to easily simulate the EP data.