Lucas Raynal
email: lucasraynal@gmail.com

Evelyne Clément
email: evelyne.clement@u-cergy.fr

Emmanuel Sander
email: emmanuel.sander@unige.ch

Structural similarity superiority in a free-recall reminding paradigm

Keywords: analogical retrieval, structural similarity, abstract encoding, free-recall reminding paradigm

The high proportion of retrievals of situations sharing surface similarity in previous experiments gave rise to the view that surface similarity predominantly drives access. In contrast, we claim that the retrievals of those situations are due to the structural similarity they still preserved. We tested our alternative structural superiority hypothesis while isolating the influence of structural and surface similarity by assessing whether participants predominantly retrieve situations sharing only structural similarity (superficially dissimilar analogs) or situations sharing only surface similarity (superficially similar disanalogs). Contrary to previous experiments instructing participants to produce analogies, we used a free-recall reminding paradigm in which participants had to recall any situations a target cue reminded them of. Results demonstrated that a greater proportion of participants predominantly retrieved situations sharing only structural similarity than situations sharing only surface similarity. Results are discussed relatively to the current debate regarding the retrieval of analogs preserving or not surface similarity.

Introduction

Imagine that someone tells you that, after saying to herself every day that tomorrow she will go to the Do It Yourself store to buy the lightbulb she needs, she finally bought it at least two weeks later. Would you be likely to be reminded of that moment when you had to answer an e-mail, what you finally did many days later, after saying to yourself several times a day that you would send it later during the day? Or would you be more incline to retrieve this situation where you broke a lightbulb in a friend's house and bought him one back the same day in the Do It Yourself store? A great deal of research led in the field of analogy has aimed to assess whether analogies can be spontaneously processed between a present (a target cue) and a stored situations (a source candidate situation, such as between the buying a lightbulb and the answering an e-mail previous situations) sharing an abstract pattern of relations (structural features, e.g. an activity that should be done now is postponed), even though they imply different objects and object attributes (surface features, e.g. lightbulb and Do It Yourself Store as opposed to e-mail and computer). Most of it has converged on the implausibility to retrieve superficially dissimilar analogs demonstrating that retrieval, being predominantly oriented towards surface similarity, often leads to access mereappearance matches sharing surface but no structural similarity (Gentner, Ratterman & Forbus, 1993, e.g. the breaking a lightbulb at a friend's house situation). Following [START_REF] Forbus | Extending SME to handle large-scale cognitive modeling[END_REF], "memory retrieval is strongly influenced by content, and only weakly influenced by relational structure" (p 1164). Determining the type of similarity that is used to retrieve is fundamental to understand human cognition since it is tightly linked to the question of the nature of the encoding [START_REF] Gentner | Reviving inert knowledge: Analogical abstraction supports relational retrieval of past events[END_REF][START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF], the usefulness of reminding for transfer [START_REF] Gick | Schema induction and analogical transfer[END_REF] and the possibility of an experience-based conceptual development [START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF][START_REF] Loewenstein | Structure mapping and vocabularies for thinking[END_REF].

Source-target paradigms

The question of whether structural or surface similarity is dominant in retrieval has initially been addressed through source-target paradigms where source candidate situations are presented before target cue situations sharing structural, surface or both similarities.

In classical story-recall tasks, some source candidate situations share only structural similarity while others are supposed to share only surface similarity with the target cue. Results showed that superficially similar disanalog source candidate situations were mainly retrieved whereas superficially dissimilar analog source candidate situations were only marginally accessed, leading to the conclusion that surface similarity is more influential than structural similarity in retrieval [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF]. The main effect of surface similarity on retrieval has been replicated in studies using similar methodologies [START_REF] Catrambone | The effects of surface and structural feature matches on the access of story analogs[END_REF], whereas the secondary role attributed to structural similarity varied across studies. Namely, it has been shown that a structurally similar source candidate situation, sharing surface similarity or not, is preferentially retrieved over a structurally dissimilar source candidate situation sharing the same level of surface similarity (high level in [START_REF] Wharton | Below the surface: Analogical similarity and retrieval competition in reminding[END_REF]low level in Wharton, Holyoak & Lange, 1996).

In problem-solving, the influence of structural and surface similarity has been mainly addressed through the comparison of spontaneous transfer between two analog problems sharing surface similarity or not. A source problem is provided to the participant with its solution before the analog target problem has to be solved. A recurrent finding is that without hint to use the source problem solution, transfer rarely occurs between two superficially dissimilar analog problems [START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Gentner | Learning and transfer A general role for analogical encoding[END_REF][START_REF] Gentner | Reviving inert knowledge: Analogical abstraction supports relational retrieval of past events[END_REF]. Adding surface similarity between the two analog problems strongly increases the amount of transfer [START_REF] Keane | On retrieving analogues when solving problems[END_REF].

In sum, data from story-recall tasks indicate that mereappearance matches are preferentially retrieved over superficially dissimilar analog source candidate situations, and results from analogical problem-solving suggest that recalling a source problem with its solution is highly dependent on the presence of surface similarity.

Real-world analogies

As opposed to studies using classical experimental paradigms, observational studies led out of the experimental context have put emphasis on the importance of structural similarity when retrieving analog source situations. Their methodology generally consists in extracting analogies that are generated during expert discussions and analyzing their structural and superficial components. [START_REF] Dunbar | How scientists think: Online creativity and conceptual change in science[END_REF] study has shown that although the great majority of the analogies produced by molecular biologists in their lab are intradomain analogies, they contain a deep structural similarity. Similar findings were obtained when analyzing analogies generated by both experts and novices in management who had to solve business management problematic situations [START_REF] Bearman | The structure and function of spontaneous analogising in domain-based problem solving[END_REF]. [START_REF] Christensen | The relationship of analogical distance to analogical function and preinventive structure: The case of engineering design[END_REF] found an important proportion of interdomain analogies in engineers' discussions working on the design of medical plastics, almost as frequent as intradomain analogies. Analogies extracted from discussions between experts in economy were also equally distributed among intradomain and interdomain (Kretz & Krawczyk, 2014). Further analysis demonstrated that superficially similar features were rarely mentioned between the source and the target cue situations. [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF] showed that a high majority of the analogies used in newspapers dealing with the referendum on the Quebec independence were taken from a distant semantic domain from the one of politics. Hence, observational studies introduced two major considerations concerning the role of surface and structural similarities in analogical retrieval. First, superficially similar analogs represented an important proportion among the source situations that were used [START_REF] Dunbar | How scientists think: Online creativity and conceptual change in science[END_REF][START_REF] Bearman | The structure and function of spontaneous analogising in domain-based problem solving[END_REF]. Second, analogies without surface similarity were more frequent than in laboratory studies, suggesting that structural similarity may play a greater role in access than previously shown [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Christensen | The relationship of analogical distance to analogical function and preinventive structure: The case of engineering design[END_REF]Kretz & Krawczyk, 2014).

Production paradigms

The contrasted findings stemming from experimental and observational studies motivated researchers to identify the critical factors allowing subjects to drive structurally-based retrievals in ecological conditions, while constraining participants to use superficial cues to retrieve source candidate situations in experimental settings. In this spirit, [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF] created a production paradigm to bring more ecological validity in an experimental context. In this production paradigm, participants had to produce analogies using familiar source situations from their experience instead of unfamiliar source candidate situations which are presented during the experiment. In that study, participants were asked to act as political consultants to produce analogies with familiar situations that could help convincing the population that a target political strategy (prozero or anti-zero deficit strategy) is well-founded. In sharp contrast with previous findings from experimental studies, they found that 67% of the analogies generated by the participants did not share surface similarity with the target situation, belonging to a distant semantic domain from the one of politics. From this result, the authors suggested that the dominance of surface similarity that had been observed until then was a consequence of the unfamiliarity of the experimental stimuli, the impossibility to rely on one's own source of analogy and the absence of a goal inciting the analogizer to use deeper cues to retrieve.

However, their findings were recently contested by [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] who pointed out that the source situations proposed by the participants could have been the consequence of analogs' creations rather than analogical retrieval of real memories. Replicating the production paradigm's design, the researchers provided either a superficially similar or dissimilar analog target cue situation to elicit the retrieval of a source candidate situation from the participant's experience and which presence in memory is controlled. For instance, the retrieval of having consumed so much of a new food with the consequence of becoming disgusted of it or of having played a video game so much with the consequence of getting fed up with it was tested by instructing participants to find an analogy to prevent someone who enjoys passion fruit so much that he is interested in incorporating it into cheesecakes, toppings and daiquiri from doing so. Results showed that retrievals of the critical analog source candidate situation were more frequent when it was superficially similar than when it was not. The authors concluded that the superiority of structural similarity in retrieval reported in Blanchette and Dunbar's studies was due to a lack of experimental control, and that their results reconcile with the ones obtained with traditional experimental paradigms in terms of the dominance of surface similarity in retrieval.

The implausibility of surface similarity-based access

Findings revealing the difficulty of retrieval based on structural similarity are generally explained in terms of an encoding that would be focused on surface features. This position can be summed up with Gentner et al.'s (2003, p 393) words: "In short, our ability to take advantage of our prior experiences is highly limited. One explanation for the low degree of appropriate recall is that people often encode cases in a situation-specific manner, focusing mainly on their surface features" (see also [START_REF] Forbus | Extending SME to handle large-scale cognitive modeling[END_REF][START_REF] Loewenstein | Structure mapping and vocabularies for thinking[END_REF]. Such a surface level encoding would seem very disabling since it implies that our comprehension of the situations rarely goes beyond the consideration of the situation's objects. Also, a retrieval based on these objects would lead to access many mere-appearance matches which structure is sterile to help understanding a present situation.

However, structural features of the situations are encoded when they involve relations which are usually encountered in daily-life. For instance, it has been shown that solution transfer between isomorphic problems sharing an abstract "random distribution principle" is mediated by the presence of objects assigned to humans in both problems, since it enables to grasp a common "get" familiar structure [START_REF] Bassok | Judging a book by its cover: Interpretative effects of content on problem-solving transfer[END_REF]. These data suggest that the inability to drive structurally-based access could be the consequence of the unfamiliarity of the structures which are implemented between the superficially similar analogs in the experiments [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF]. In familiar situations, such as the ones we encounter in dailylife, our encoding might go beyond the objects we perceive to grasp the relations they display in light of the usual relations we have learned to associate to them [START_REF] Popov | The Relational Luring Effect: Retrieval of Relational Information during Associative Recognition[END_REF]. Referring to the answering an email or the buying a lightbulb situations we presented above, it seems likely that the activation of familiar concepts such as activities that must be done, lazyness and postponing allows one to encode the structure "an activity that should be done now is postponed" and thus permits structurally-based retrievals.

Superficially similar analogs' retrievals

In [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF] and [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] studies, the question of the dominance of structural versus surface similarity has been addressed by directly asking participants to generate analogies (after explaining them what an analogy is), and then comparing the proportion of superficially similar versus dissimilar analogs. However, using this methodology to assess whether surface or structural similarity predominantly drives spontaneous access may present the bias of inciting participants to use structural similarity in their retrievals. In addition, the preponderance of superficially similar over superficially dissimilar analogs' retrievals, as shown in some observational studies [START_REF] Dunbar | How scientists think: Online creativity and conceptual change in science[END_REF][START_REF] Bearman | The structure and function of spontaneous analogising in domain-based problem solving[END_REF] as in [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] production study, cannot lead to clear conclusions regarding the type of similarity that elicited them. Notably, while [START_REF] Dunbar | The in vivo/in vitro approach to cognition: The case of analogy[END_REF] insist on the role of structural similarity in superficially similar analogs' retrievals ("It is important to note that even though the scientists were basing their analogies on sources and targets within the domain of biology, the analogies were based upon underlying sets of structural relationships rather than any superficial similarity between clams and plasmodium", p. 336), [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] claim that the preponderance of superficially similar analogs' retrievals is the hallmark of the surface similarity superiority ("our results run counter to the claim that the dominance of superficial similarities in retrieval is rooted in the artificiality of the tasks and materials used in traditional experiments", p. 21; see also [START_REF] Bearman | The structure and function of spontaneous analogising in domain-based problem solving[END_REF][START_REF] Loewenstein | Structure mapping and vocabularies for thinking[END_REF] for similar conclusions).

As suggested by [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF], a way to compare the influence of structural versus surface similarities in retrieval is to compare the retrievals' frequency of mereappearance matches and superficially dissimilar analog source candidate situations. As mere-appearance matches are only superficially similar, their retrieval can only be the consequence of a surface-similarity based retrieval, and as superficially dissimilar analog source candidates are only structurally similar, their retrievals can only be provoked by a structural similarity-based retrieval. In traditional experimental contexts, it has been shown that mereappearance matches' retrievals were preponderant [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF]. However, a closer look at the stimuli used in these experiments reveals that structural similarity is still present in mere-appearance matches. For instance, Karla the hawk target cue situation relates the story of a hawk which gives feather to a hunter with a bow who is grateful and promises never to attack eagle. The mere-appearance match deals with an eagle which gives tail feathers to a sportsman with a crossbow who is grateful and promises never to attack eagle, but who finally shoots at the eagle. Hence, the mereappearance match also shares a structure with the target cue (e.g. make a deal to avoid a bad situation) although an additional twist is presented in the eagle story (the sportsman finally breaks the deal). [START_REF] Raynal | Challenging the superficial similarities superiority account for analogical retrieval[END_REF] demonstrated that when structural and surface similarities are isolated in different source candidate situations, superficially dissimilar analog are better retrieved than superficially similar disanalog source candidate situations.

The present experiment aims at testing the structural superiority hypothesis in a task preserving the production paradigm's ecological benefit regarding the retrieval of source situations from the participants' own experience (in line with Blanchette &[START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF][START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] while isolating the influence of surface and structural similarities by comparing the retrieval of superficially dissimilar analogs and superficially similar disanalogs (in line with [START_REF] Raynal | Challenging the superficial similarities superiority account for analogical retrieval[END_REF]. As superficially similar analogs' retrievals can be both explained by a surface similarity superiority or a structural similarity superiority account for retrieval, they will not be considered to answer our main question of interest. In accordance with the structural superiority hypothesis, we predicted that situations showing structural but no surface similarity will be predominantly retrieved compared to situations sharing surface but no structural similarity.

Experiment Method

Participants 97 participants (78 women and 19 men, M= 24.3 years, SD=7.3 months) took part in the experiment during a university class.

Material

A booklet was presented with, in its first page, the instructions and a target cue presenting a short description of a situation similar to the ones that can be encountered in daily-life. Two sets of the material were built. The first one presented the participants with the following target cue situation: "I had to go to the Do It Yourself store to buy a lightbulb, but every day I was saying to myself that I would rather go there tomorrow, and it is only some two weeks later that I finally bought it". The second one was composed of the following target cue situation: "I had the idea to answer that I forgot my glasses when the photographer offered me to go to his exhibition. In reality, I had my glasses with me but I had no desire to go there". The first and the second pages presented altogether eight plots to be filled with the retrieved memories. Half of the participants received the first set of material and the other half received the second one.

Procedure and experimental design

The task was presented as a memory-recall task. The instructions stated that the participants will have to report a maximum of memories that the target cue situation reminds them of and to report all memories that would come to mind. Also, the instructions stated that the reported situations had to be real memories and not situations invented during the task. Some experiments using an analogy production paradigm have been criticized for inciting participants to create analogs rather than to retrieve real memories [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF]. Indeed, the instruction to generate analogs could become the participant's priority relatively to the instruction of using real memories. In contrast, the participants were only instructed to recall memories in the free-recall reminding paradigm we used here. Hence, the instruction to retrieve memories could not be overshadowed by the instruction to generate analogs. Participants had 10 minutes to fulfill the task.

Results and discussion

Retrieved situations preserving the relational structure of the target cue were coded as superficially similar analogs when they contained at least one object that was semantically close to one of the target cue's objects (involving a light, a store or handiwork for the buying a lightbulb situation, involving photography, glasses or cultural activity for the refusing to go to an exhibition situation), and as superficially dissimilar analogs when they did not show any semantic overlap at the objects' level. Retrieved situations which did not preserve the relational structure (an activity that should be done now is postponed for the buying a lightbulb situation, providing an excuse to avoid a situation for the refusing to go to an exhibition situation) were coded as superficially similar disanalogs if at least one object was semantically close to an object of the target cue, and as superficially dissimilar disanalogs when there was no semantic overlap with any objects.

A total of 304 situations were reported by the participants. The mean number of memories reported for each participant was 3.3. Among all retrieved situations, 50.3% were superficially dissimilar analogs, 26.6% were superficially similar disanalogs, 13.2% were superficially similar analogs and 9.9% were superficially dissimilar disanalogs (Figure 1). It could be criticized that the higher number of superficially dissimilar analogs over superficially similar disanalogs is the consequence of participants who retrieved superficially dissimilar analogs reporting more memories and participants who retrieved superficially similar disanalogs reporting less memories. The comparison of the number of participants predominantly retrieving superficially dissimilar analogs over superficially similar disanalogs and vice versa may be a more appropriate measure to assess the structural similarity superiority hypothesis. In this line, analysis revealed that 58.8% (61.2% in the buying a lightbulb version and 56.3% in the refusing to go to an exhibition version, as can be seen in Figure 2) of the participants retrieved more superficially dissimilar analog than superficially similar disanalog source candidate situations. 24.7% of the participants retrieved more superficially similar disanalog than superficially dissimilar analog source candidate situations (24.5% in the buying a lightbulb version and 25% in the refusing to go to an exhibition version). A chi square performed on the number of participants who retrieved more superficially dissimilar analog than superficially similar disanalog source candidate Results are in accordance with the structural superiority hypothesis in that participants more frequently retrieved preferentially superficially dissimilar analog rather than superficially similar disanalog source candidate situations. It appears that retrievals were more influenced by structural similarity alone than by surface similarity alone. Retrieving Figure 2 : proportion of participants retrieving more superficially dissimilar analog than superficially similar disanalogs and the inverse only one superficially dissimilar analog source candidate situation can be considered as an evidence of the encoding of the situations' structure. In this line, analyses were drawn and revealed that 76.3% of the participants retrieved at least one superficially dissimilar source candidate situation, showing an ability to encode situations in an abstract way. It could be Figure 3: proportion of participants retrieving a superficially dissimilar analog or a superficially similar disanalog for the first reported memory opposed that a global analysis of the distribution of each type of retrievals could hide a higher proportion of surface similarity-based access for the first memory that is accessed. Indeed, access could be primarily based on surface features and then be progressively oriented toward structural features of the target cue situation. However, as shown in Figure 3, further analysis demonstrated that the source candidate situations which were recalled in the first place were significantly more frequently superficially dissimilar analog (48.5%) than superficially similar disanalog source candidate situations (20.6%, X 2 (1, N=66)=10.88, p < .001). These data suggest an early focus on the structural features of the target cue.

Conclusions

Using a free-recall reminding paradigm, we investigated the type of retrieval elicited by familiar situations without instructing participants to draw analogies. Given that both the structural and the surface similarity hypotheses can account for superficially similar analogs' retrievals, we focused on the comparison of the recall of superficially dissimilar analog source candidate situations that can only be driven by structural similarity and superficially similar analog source candidate situations that can only be elicited by surface similarity. As predicted by the structural superiority hypothesis, retrievals of superficially dissimilar source candidate situations were preponderant, while retrievals of superficially similar disanalog source candidate situations were only marginal, failing to support the surface similarity superiority hypothesis.

The difference is striking when comparing the number of participants retrieving the superficially dissimilar analog source candidate situation in [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] second experiment (16.5%) and the number of participants retrieving a (at least one) superficially dissimilar analog source candidate situation in our study (76.3%). A possible explanation deals with the fact that only the retrieval of one superficially dissimilar source candidate situation was considered in their study (e.g. having consumed so much of a new food with the consequence of becoming disgusted of it or of having played a video game so much with the consequence of getting fed up with it), whereas we coded all superficially dissimilar source candidate situations that were retrieved by the participants. Indeed, superficially dissimilar analog source candidate retrievals would have seemed much less frequent in our study if we had only coded one type of superficially dissimilar analog source candidate situation (e.g. an-mail that should be sent now is postponed for the buying a lightbulb target cue situation).

Although the present study's major aim was to determine whether structural or surface similarity is preponderant in retrieval, our results can also shed light on the question of the superiority of superficially similar versus dissimilar analog source candidate situations' retrievals [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF]. At first sight, it appears that our results replicate the preponderance of superficially dissimilar over superficially similar analog source candidate situations' retrievals observed in [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF], in contrast with [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] results. Indeed, it appears that the encoding is abstract in a way that structural similarity on its own can be used to retrieve a situation. If the data demonstrate that structural similarity may be the main factor influencing access, it does not preclude that surface similarity plays an important additional role. Notably, research has suggested that superficially similar analogs are preponderant when driving predictions or identifying problems whereas superficially dissimilar analogs are mainly used with explanatory goals [START_REF] Dunbar | How scientists think: Online creativity and conceptual change in science[END_REF][START_REF] Christensen | The relationship of analogical distance to analogical function and preinventive structure: The case of engineering design[END_REF]Kretz & Krawczyk, 2014). The preference for a superficially similar or dissimilar analog is affected by the goal one is pursuing. Hence, more studies are needed to inform the type of contexts influencing the preference for a superficially similar or dissimilar analog instead of drawing an absolute supremacy between the two types of analogies.

The failure to rely on structural similarity in retrieval, which is referred as the "retrieval gap" [START_REF] Holyoak | Analogy and relational reasoning[END_REF], would confine the possibility to draw analogies to contexts in which two analog situations are presented jointly, allowing the mapping process to occur. On the contrary, the demonstration of a structural similarity-based retrieval suggests that analogs can be accessed from long-term memory, permitting analogies to occur more frequently than it was supposed to under the surface similarity account for retrieval.

Figure 1 :

 1 Figure 1: proportion of retrievals for each type of source candidate situations


Lucas Raynal
email: lucasraynal@gmail.com

Evelyne Clément
email: evelyne.clement@u-cergy.fr

Emmanuel Sander
email: emmanuel.sander@paris8-univ.fr

Challenging the superficial similarities superiority account for analogical retrieval

Keywords: Analogy, analogical retrieval, structural similarity, abstract encoding, story-recall task

The predominant view concerning determinants of analogical retrieval is that it is preferentially guided by superficial cues. In order to test the cognitive plausibility of a structural similarities-based retrieval, we constructed a story-recall task in which life-like scenarios shared structural correspondences. In Experiment 1, we showed that such structural similarities induce retrievals when the participant had several source candidate situations sharing superficial similarities with the target cue. Experiment 2 was designed to test whether the encoding was sufficiently oriented on structural similarities to drive retrievals, even if the participants possess only one source candidate situation with superficial matches in memory. The results of the two present experiments lead us to conclude that in some contexts, abstract encoding induces a superiority of structural similarities over superficial ones in retrieval. Further implications for analogical retrieval approaches are discussed.

Introduction

Analogy has been identified as a key process to perceive the conceptual structure of a new situation by importing it from a familiar analog representation [START_REF] Gentner | Structure-Mapping: A Theoretical Framework for Analogy[END_REF][START_REF] Gick | Schema induction and analogical transfer[END_REF]. Most studies are consensual that the mapping process permitting this transfer is preferentially based on structure rather than surface correspondence. In other words, when comparing two analogs, subjects do not rely on similarities in terms of objects or object attributes, but instead tend to focus on common abstract relations. Figure 1 provides an illustration of a target cue story sharing a structural correspondence with a superficially dissimilar analog source candidate situation ("making a deal to avoid a bad situation"), and a surface correspondence with a superficially similar disanalog source candidate situation belonging to the same semantic domain (raptors, tailfeathers, etc).

Target cue story

Karla, an old hawk, lived at the top of a tall oak tree. One afternoon, she saw a hunter on the ground with a bow and some crude arrows that had no feathers. The hunter took aim and shot at the hawk but missed. Karla knew the hunter wanted her feathers so she glided down to the hunter and offered to give him a few. The hunter was so grateful that he pledged never to shoot at a hawk again. He went off and shot deer instead.

Superficially dissimilar analog source candidate situation

Once there was a small country called Zerdia that learned to make the world's smartest computer. One day Zerdia was attacked by its warlike neighbor, Gagrach. But the missiles were badly aimed and the attack failed. The Zerdian government realized that Gagrach wanted Zerdian computers so it offered to sell some of its computers to the country. The government of Gagrach was very pleased. It promised never to attack Zerdia again.

Superficially similar disanalog source candidate situation

Once there was an eagle named Zerdia who donated a few of her tailfeathers to a sportsman so he would promise never to attack eagles. One day Zerdia was nesting high on a rocky cliff when she saw the sportsman coming with a crowsbow. Zerdia flew down to meet the man, but he attacked and felled her with a single bolt. As she fluttered to the ground Zerdia realized that the bolt had her own tailfeathers on it. Types of similarity implicated in the retrieval of a source representation in memory are the main issues debated in the literature. The predominant view is that the retrieval of a source candidate situation critically depends on superficial similarities, whereas the influence of structural similarities seems more peripheral [START_REF] Gentner | Structure-Mapping: A Theoretical Framework for Analogy[END_REF]Gentner, Ratterman & Forbus, 1993;[START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF]. In [START_REF] Gentner | Analogical processes in human thinking and learning[END_REF] words: "Relational retrieval can be said to be the Achilles' heel of our relational capacity. There is considerable evidence that similarity-based retrieval, unlike the mapping process, is more influenced by surface similarity than structural similarity." However, opposite assertions have arisen in the literature, attributing a major influence to structural similarities, even overcoming that of surface similarities [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Dunbar | The in vivo/in vitro approach to cognition: The case of analogy[END_REF][START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF]Kretz & Krawczyk, 2014). The aim of this paper is to demonstrate that structural matches have a greater influence than superficial ones in the retrievals of life-like situations. This will be shown by creating a competition between source candidate situations sharing either exclusively superficial or structural similarities with the target cue (Figure 2 illustrates this intended feature composition). Before considering the current experiments, we report the major findings stemming from several experimental paradigms.

Analogical Problem Solving

In the problem solving domain, a common experimental design to study analogical retrieval via analogical transfer is the source-target paradigm [START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Holyoak | Surface and structural similarity in analogical transfer[END_REF]. A source problem situation is first proposed with its solution, then an analog target problem is given to be solved. To measure the role of surface correspondence in access, the surface of the source problem is manipulated to either match or not with the target one. The retrieval is considered to have occurred when the participant detects similarities between the target problem and the source problem without further hints from the experimenter, leading to the transfer of the resolution procedure from the source to the target. Results have shown that retrieval is high when the source is both superficially and structurally similar to the target. More precisely, the similarities in terms of the problems' story theme have appeared to be a crucial determinant in access [START_REF] Ross | This is like that: The use of earlier problems and the separation of similarity effects[END_REF]. Inversely, superficially dissimilar analog source problems are seldom retrieved by the participants [START_REF] Gick | Schema induction and analogical transfer[END_REF]. Studies on problem solving have also demonstrated that structural similarities can, in certain circumstances, have a role in retrieval: when two source analogs are presented jointly to be compared, their critical solution principle can be retrieved when faced with the target [START_REF] Catrambone | Overcoming contextual limitations on problem-solving transfer[END_REF]. Also, retrievals are reported to be more frequent when the two superficially dissimilar analog problems share structural similarities at a less abstract level [START_REF] Holyoak | Surface and structural similarity in analogical transfer[END_REF].

Story-Recall Paradigm

Another frequently used paradigm is the story-recall task [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF][START_REF] Wharton | Remote analogical reminding[END_REF][START_REF] Catrambone | The effects of surface and structural feature matches on the access of story analogs[END_REF]. A set of short text stories are presented as source candidates for the retrieval before the introduction of target cue situations which share various similarities with them (Figure 1). Within a problem solving paradigm, the problems generally share the same structure since it is the transfer of an abstract solution being investigated. This is not systematically the case in the story-recall paradigm since the situations are chosen to create a competition between source candidate situations possessing exclusively the surface or exclusively the structure in common with the target cue [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF]. This paradigm showed superficially similar disanalogs were retrieved significantly more often than superficially dissimilar analogs, leading to the conclusion that superficial similarity is the main factor implicated in access. [START_REF] Wharton | Below the surface: Analogical similarity and retrieval competition in reminding[END_REF] argued that the minor role attributed to structural matches in story-recall tasks could be due to the fact that only one source candidate, the superficially similar disanalog, shares some semantic features with the target cue. One can note that in real-life conditions, several source candidate situations, corresponding in superficial features with the perceived situation, are generally available in memory. The authors observed that structural similarities may play a certain role in access because when the analog source candidate also shares some superficial similarities, it is better retrieved than the concurrent source candidate possessing only superficial similarities with the target cue. Structural similarities also seem to play a certain role in access when they are implemented without superficial matches: the retrieval of a superficially dissimilar source candidate is higher when it shares structural features with the target cue [START_REF] Wharton | Remote analogical reminding[END_REF]. However, the role of structural similarities in access has only been shown when two source candidates shared the same amount of surface correspondence. Hence, the reviewed works did not demonstrate the superiority of structural similarities in access, since it would require showing that superficially dissimilar analogs are better retrieved than superficially similar disanalogs.

Production Paradigm

Challenging the ecological nature of traditional experimental conditions (unfamiliar source and target situations, short familiarization time, restrained pool of source candidates), further research focused on the retrieval of situations encoded prior to the experiment in real-life conditions [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF][START_REF] Dunbar | The in vivo/in vitro approach to cognition: The case of analogy[END_REF]Kretz & Krawczyk, 2014). With this configuration, a high structural overlap was generally observed between the source and the target cue situation. For instance, expert discourse in scientific domains (politics, biology, economics) exhibited predominance for structural analogies, though sometimes also sharing a superficial correspondence. Those findings were replicated in experimental conditions with a production paradigm, where participants who were allowed to select their own sources of analogy retrieved significantly more semantically distant analogs than superficially similar analogs [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF]. In other words, those findings not only advocate for the major role of structural similarity but also for the weak impact of superficial similarity when accessing an analog. However, [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF] pointed out that the sources provided by the participants could be invented rather than real memories, and that superficially dissimilar analogs could be more common in memory than superficially similar ones. They tested the potential that while controlling the availability and number of the two types of concurrent source candidates, findings obtained from usual source-target and story-recall paradigms would be replicated when the participant generates analogical retrievals from her or his own experiences. They proposed target situations with either superficial and structural, or solely structural similarities with their memories. In accordance with prior findings, participants more often proposed superficially similar analogs than superficially dissimilar In another study using this paradigm, situations retrieved by management consultants provided with target situations embodying an original negotiation principle manifested only superficial matches [START_REF] Gentner | Reviving inert knowledge: Analogical abstraction supports relational retrieval of past events[END_REF]. Contrary to previous studies using the production paradigm, this experiment reflects a marginal tendency to access structurally similar source situations. These results, in sharp contrast with the ones obtained by [START_REF] Blanchette | How analogies are generated: The roles of structural and superficial similarity[END_REF], indicate that the natural settings of the encoding condition of one of the analog is not in itself the critical parameter influencing the type of retrieval. Namely, whereas it probably promotes the abstract encoding of this analog situation, the access to its structural matches with the other analog which is still provided by the experimenters is not guaranteed.

Encoding through abstract concepts

When faced with a target cue situation, the fail to retrieve an analog situation is generally interpreted as a defect of abstract encoding of the two situations [START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Gentner | Reviving inert knowledge: Analogical abstraction supports relational retrieval of past events[END_REF]. If the participant is unable to grasp the structure of the situations, the abstract similarities could not be used as a cue to retrieve. Indeed, only if the participant is incited to compare either two source analogs or two target analogs, he might extract a schema sufficiently abstract, and subsequently perceive this schema in a superficially dissimilar analog, in the context of the retrieval.

However, even if the schema extracted from the situation is not abstract enough so that the similarities introduced between the analogs by the experimenter will be detected, it cannot be claimed that the encoding of a situation is purely concrete and literal [START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF]. Whereas a wealth of stimulations is permanently available in our environment, one has to select the properties relevant to make sense of the situations. This cannot be done by processing every perceptively available superficial piece of information. Instead, the situation's understanding depends on the properties (whether perceptual or abstract) that are compatible with the conceptual structure in construction and on the neglect of those which are not.

We suggest that familiar concepts' evocation during the encoding is a critical point to account for the abstract information raised by perceived situations. This idea is congruent with findings revealing that source analog problems that participants usually fail to retrieve in experiments are better accessed by expert participants [START_REF] Novick | Analogical transfer, problem similarity, and expertise[END_REF]. When experts have a familiar concept to highlight the abstract properties of these situations, the novice does not have this conceptual door toward the structure. This reveals that the novice participant has not elaborated the specific concept that allows for the encoding of the structure of complex problem situations necessary for the structurallybased retrieval of this type of situations. However, it is very likely that he/she has acquired an expertise in daily-life situations, where one systematically has to deepen the encoding until elementary abstract concepts are established, such as "making a deal", "bad faith", "lie", "authenticity", "prosocial behavior" and so on, to produce adapted behaviors. The activation of these concepts should highlight the structure underlying some daily-life situations in a way that elicit structurally-based retrieval. Whereas most studies have focused on abstract inter-domain analogies that a novice could rarely access, we aim at investigating analogies between situations inspired from social scenarios that can be experienced in different domains of daily-life.

Experiment 1

In order to demonstrate the superiority of structural similarities over superficial similarities in retrieval, we conducted a first experiment where those two types of similarities were in competition in the retrieval of a source situation candidate. In that way, a story-recall task was used so as to control for the highlighting of the structure underlying both the analog source candidate situation and the target cue situation.

Method Material

Social scenarios inspired from [START_REF] Wharton | Remote analogical reminding[END_REF] involving life-like contexts were used. Although the objects of the analog situations were clearly divergent, they shared very similar role at the required level of abstraction for making sense of the stories. In the analog pair reported in Figure 3 for instance, both stories relate the setting of a social competition between two characters (rival cookers or classmates) having the same goal (turnover or seducing someone) and an unusual way to put an end to it by helping the competitor to enhance her/his critical ability (improve Lorenzo's pizza dough or looking after Diane's appearance). If those situations are not directly taken from the participant's experience, s/he still can use her/his general knowledge about social relations to encode such scenarios.

In order to make sure that the two types (superficial or structural) of similarities with the target cue were never confounded in a same source situation, superficially similar source stories structure clearly diverge from the target ones. In this way, contrasting with previous works (e.g. [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF], disanalogs are not modified versions of a same structure story, but describe structurally different scenarios. [START_REF] Wharton | Below the surface: Analogical similarity and retrieval competition in reminding[END_REF] noticed that when semantic correspondence was not only present between the source disanalog and the target cue, the artificial saliency of the superficial matches decreased and resulted in a weaker influence on access. In their experiment, the concurrent superficially similar source candidates were the analog and the disanalog. As previously indicated, we aim at isolating the influence of structure and surface similarities by implementing them in different source candidates. Hence, we multiplied the number of source candidates sharing surface features with the target cue by introducing three superficially similar disanalogs. To respect a symmetry between the number of semantically close and distant source candidates, we also introduced three superficially dissimilar source candidate situations (the analog sharing structural correspondence, and two unrelated distractors).

Target cue story

Luigi holds a pizza truck in a very popular place. Lorenzo, another ambulant pizzaiolo, has placed his truck just beside Luigi's and is detrimental to his turnover. Luigi realizes the dough of Lorenzo's pizzas is bland. Luigi spontaneously gives his personal recipe to Lorenzo so as he can enhance the quality of his product. Since then, his pizza dough is amazingly tasty. The same evening, Lorenzo declares to Luigi that in order to show him how much he found his intention was nice, he will move his truck in another sector, far from this one.

Superficially dissimilar analog source candidate situation

Julie is in love with Victor, her classmate, and she is getting closer to him in order to seduce him. But Diane joins the class in the middle of the year and also has a crush on Victor. Julie remarks that Diane was not very aware of her style and proposes her some relooking advices, showing her fashion photos and taking her for shopping. Diane now looks very cute and chic. Diane is so grateful that she tells Julie that she would stop flirting with Joe.

Superficially similar disanalog source candidate situation

In a market place, the truck called « At Alessandro & Fabio's » has various choices of homemade pizzas. The important clientele going there is fond of the authentic atmosphere steaming from this stand held by the two happy 1 4 participants chose one of the two distractors or did not report any retrieval and were not included in the analyses looking men in Italian traditional suits. However, once they will have left this selling space, the two men will go to another market place after taking care of wearing German traditional clothes to sell sausage specialties. The sign will display « At Hans and Hendrich's ».

Procedure and experimental design

The first two pages of a booklet presented the 6 source stories, then a blank page separated them from the last page comporting the target cue situation. Under each source story was a 5 points scale inviting participants to assess the ease they had to imagine the scene while reading it. As recommended by [START_REF] Wharton | Remote analogical reminding[END_REF], this was done to promote a deep treatment of the situations. The dependent variable was the source retrieved during the reading of the target cue situation.

It was indicated that the task took around 10-15 minutes to fulfill but no time limitations was imposed. After they agreed to participate, participants were given the booklet. They were invited to read the instructions available in the first page. The target cue situation was presented on the last page, followed by the solicitation to indicate if they were reminded of one of the previous situations. If it was the case, they had to restitute any element they could remember about it.

Participants

34 participants (25 women and 9 men, mean age 23.8 years) accepted to take part in the experiment in University libraries (Paris 5 and Paris 8). They were all fluent French speakers.

Results and discussion

Access credit was attributed to the source candidate for which the participant recalled word content. If content word from more than one source was reported, the source containing the more content words in common was credited. If the participant explicitly reported more than one source despite the instruction, his response was excluded. 3 responses were not analyzed for this reason.

Analyses were drawn on a comparison between the superficially and structurally similar source candidates that were retrieved. Structurally similar source candidate situations were much more retrieved (84.61 %) than superficially similar disanalog source candidate situations (15.39 %, see Figure 4). This difference reaches high significance (X 2 (1, N=29)=12.46; P<0.001) 1 .

In real-life condition, one generally has in memory multiple source candidates sharing similarities in terms of superficial objects with the target cue situation. The results reveal that when a pool of semantically similar source candidate situations is available in memory, but those situations do not preserve the structure of the target cue, the retrieval is preferentially guided by structural matches with sources of a distant semantic domain. However, our results cannot help us identifying whether the structural matches of daily-life scenarios are sufficiently blatant to drive the retrieval when only one concurrent source candidate belongs to the same semantic domain as the target cue. Experiment 2 was designed in order to answer this question.

Method

Material

Six source candidate situations were proposed before the target cue situation (taken from experiment 1): 4 unrelated stories (distractors), one superficially similar disanalog story (taken from Experiment 1) and a superficially dissimilar analog story (taken from Experiment 1, c.f. Figure 3). Hence, the design was more similar to the one used in traditional recall tasks [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF], though it differs in the isolation of structural or superficial similarities in different source candidate stories that are in competition.

Procedure and experimental design

The procedure and experimental design were the same as in Experiment 1.

Participants

67 students (52 women and 15 men, mean age 20.8 years) accepted to take part in the experiment during a class (University Paris 8).

Results and discussion

3 responses could not be interpreted since several source candidate situations were reported despite the instruction asking for only one.

2 2 participants chose one of the two distractors or did not report any retrieval and were not included in the analyses Again, analyses were focused on the comparison between surface and structure similarities-based retrievals. As illustrated in Figure 5, the superficially dissimilar analog source candidate was significantly more retrieved than the superficially similar disanalog source candidate (respectively 81.25% and 18.75%, X 2 (1, N=62)=22.41, p<0.001) 2 .

These results share a similar pattern with the ones obtained from Experiment 1. This comparison induces that the presence of multiple situations belonging to the same semantic domain as the target cue was not a determinant factor promoting the superiority of structural similarities in retrieval. Instead, the fact that our stimuli depicted daily-life situations might have been a critical parameter so that the participants may have rely on the abstract structures of the scenarios as retrieval cues.

Conclusion

In Experiment 1, the superiority of structural similarities in retrieval was observed while the source analog was in competition with several source candidates sharing exclusively surface features with the target cue. As noted by [START_REF] Wharton | Below the surface: Analogical similarity and retrieval competition in reminding[END_REF][START_REF] Wharton | Remote analogical reminding[END_REF], providing participants with only one source candidate sharing objects with the target cue may provoke its retrieval. However, in Experiment 2, providing participants with only one surface matching source candidate in competition with the superficially dissimilar analog did not reduce the proportion of structural similarity-based retrievals. Thus, structural similarity-based retrievals are predominant when the situations experimentally provided are close to the ones that are encountered in daily-life.

Experimental studies have widely converged on the conclusion that retrieval is driven by superficial similarities. However, as the analog situations that have been mainly studied are unfamiliar for the participants, the latter conclusion cannot be applied to the more ecological retrievals that are processed in daily-life. Indeed, in analogical problem solving, analog problems share a highly abstract resolution principle [START_REF] Gick | Schema induction and analogical transfer[END_REF][START_REF] Ross | This is like that: The use of earlier problems and the separation of similarity effects[END_REF]. Alternatively, the nature of retrieval can be better informed when meaningful structural similarities are set between situations potentially encountered by the participants in their familiar environment. Under these conditions, the participant's knowledge allows to encode familiar relations that constitute cues for retrieving former episodes, while surface features may be neglected [START_REF] Novick | Analogical transfer, problem similarity, and expertise[END_REF][START_REF] Hofstadter | Surfaces and essences: Analogy as the fuel and fire of thinking[END_REF]. Indeed, a filter has to operate in order to identify the relevant properties constitutive of a concept that allows to make sense of the situation. Some authors have claimed that the commonly observed surface similarity-based retrieval was not so detrimental to our cognition since situations sharing surface generally share structure (the kind world hypothesis; [START_REF] Gentner | Similarity and the development of rules[END_REF][START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF]. It is noticeable indeed that objects in our environment usually interact in regular ways and have typical relations (Bassok, Wu & Olseth, 1995). For instance, situations involving two pizzaioli in the same place potentially induce very closed relations, such as a competition between them (Figure 3). Two situations sharing both surface and structure can only be more structurally similar than two surface dissimilar situations sharing only structure at a certain level of abstraction. Yet, it has been taken as granted for advocating the superficially driven retrieval view that retrievals of structurally and superficially similar situations were more frequent than only structurally similar ones [START_REF] Trench | The role of surface similarity in analogical retrieval: Bridging the gap between the naturalistic and the experimental traditions[END_REF]. An attempt to introduce a source candidate sharing only surface and no structure with the target cue has been made in storyrecall paradigms [START_REF] Gentner | The roles of similarity in transfer: Separating retrievability from inferential soundness[END_REF]). Yet, a closer look at the stimuli (Figure 1) makes apparent that the superficially similar disanalog source candidate situations systematically still shared some relational features with the target cue [START_REF] Wharton | Remote analogical reminding[END_REF]. Their structures are highly similar (making a deal to avoid a bad situation) until opposite conclusions at the end of the stories (betrayal or respect of the deal). However, a set of objects still can induce a heterogeneous panel of relations (e.g. two pizzaioli: rivalry, friendship, etc), while different types of objects can induce very similar relations (e.g. a loving couple can also induce the relation rivalry for instance, c.f. Figure 3). In our experiments, dissociating object similarities and similarities in terms of familiar relations into different source candidate situations demonstrated that it is not the objects in themselves that drive access, but rather the familiar structural relations that link them.

Figure 1 :

 1 Figure 1: Situations sharing different types of similarity inGentner, Ratterman & Forbus (1993)

Figure 3 :

 3 Figure 3: Examples of stories used in the Experiment1

Figure 4 :

 4 Figure 4: % of retrievals of structurally versus superficially similar source candidates in Experiment 1 Experiment 2

Figure 5 :

 5 Figure 5: % of retrievals of structurally versus superficially similar source candidates in Experiment 2


François Boulier
email: francois.boulier@lifl.fr

Marc Lefranc
email: marc.lefranc@univ-lille1.fr

François Lemaire
email: francois.lemaire@lifl.fr

Pierre-Emmanuel Morant
email: morant@phlam.univ-lille1.fr

Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits ⋆

In this paper, a rigorous quasi-steady state approximation method is applied in order to study the absence of oscillations in a family of models describing a gene regulated by a polymer of its own protein. A result presented in a former paper at Algebraic Biology 2007 is thereby generalized. The rigorous method is illustrated over the basic enzymatic reaction.

Introduction

In a former paper [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF], we studied a simple family of models depending on an integer parameter n and featuring a negative feedback loop, one of the core ingredients for generating oscillations [START_REF] Fall | [END_REF]. These abstract models are closely related to models studied by Goodwin and Griffith in the 60's [START_REF] Goodwin | Temporal Organization in Cells[END_REF][START_REF] Goodwin | Advances in Enzyme Regulation[END_REF][START_REF] Griffith | Mathematics of Cellular Control Processes. I. Negative Feedback to One Gene[END_REF]. Griffith considered a model of a gene regulated by a polymer formed of n copies of its own protein. We studied the same problem, but in a slightly more general case, where gene activation is not assumed to be fast. We eventually concluded with the absence of Poincaré-Andronov-Hopf bifurcation in our family of models for n ≤ 8 and their existence for n ≥ 9. The absence/presence of Poincaré-Andronov-Hopf bifurcation for n ≤ 8 is a strong indicator for the absence/presence of oscillating trajectories. Extensive numerical experiments [START_REF] Doedel | AUTO software for continuation and bifurcation problems in ODEs[END_REF][START_REF] Ermentrout | Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students[END_REF] confirmed the absence of oscillations for n ≤ 8 and their existence for n ≥ 9.

In this paper, the models are designed by means of systems of parametric nonlinear ordinary differential equations (ODE) [START_REF] Conrad | Modeling Molecular Interaction Networks with Nonlinear Differential Equations[END_REF]. The approach applied in [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] consisted in two steps: first simplifying the initial system of n + 2 parametric ordinary differential equations as a reduced system of three ODE by means of a quasi-steady state approximation; second, studying the reduced model.

The idea of quasi-steady state approximation is simple: study the dynamics of the slow reactions, assuming that the fast ones are at quasi-equilibrium, thereby removing from the ODE system, the differential equations which describe the evolution of the variables at quasi-equilibrium. Many authors [START_REF] Van Breusegem | Reduced order dynamical modelling of reaction systems: a singular perturbation approach[END_REF][START_REF] Okino | Simplification of Mathematical Models of Chemical Reaction Systems[END_REF][START_REF] Vora | Nonlinear model reduction of chemical reaction systems[END_REF][START_REF] Bennet | Transient Dynamics of Genetic Regulatory Networks[END_REF] state that carrying out rigorously this approximation is far from straightforward. We would rather say that there are different ways to perform this approximation and that it is the problem of ascertaining the domain of validity of each kind of approximation which is not straightforward.

In another paper [START_REF] Boulier | Model Reduction of Chemical Reaction Systems using Elimination[END_REF], the authors reformulated the methods of [START_REF] Van Breusegem | Reduced order dynamical modelling of reaction systems: a singular perturbation approach[END_REF][START_REF] Okino | Simplification of Mathematical Models of Chemical Reaction Systems[END_REF][START_REF] Vora | Nonlinear model reduction of chemical reaction systems[END_REF][START_REF] Bennet | Transient Dynamics of Genetic Regulatory Networks[END_REF], which are equivalent, and made them fully algorithmic, by means of differential elimination methods [START_REF] Ritt | Differential Algebra[END_REF][START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF][START_REF] Boulier | Representation for the radical of a finitely generated differential ideal[END_REF][START_REF] Wang | Elimination Practice: Software Tools and Applications[END_REF][START_REF] Boulier | Differential Elimination and Biological Modelling[END_REF]. An efficient implementation, based on [START_REF] Lemaire | The RegularChains library in MAPLE 10[END_REF], was developed by the third author.

In this paper, we show that the reduction method of [START_REF] Boulier | Model Reduction of Chemical Reaction Systems using Elimination[END_REF] can be applied to our family of models. It yields a reduced model which contains that of [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] as a particular case: our new approximation is more precise. By a very concise proof, we show that the results obtained in [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] also hold for the new model. This paper gives us also the opportunity to widen the audience of our [START_REF] Boulier | Model Reduction of Chemical Reaction Systems using Elimination[END_REF]diffReduce] algorithm by recalling its principle.

2 Our family of models

The initial model

A schematic model describing a single gene regulated by an order n polymer of its own protein is provided in Figure 1. It is borrowed from [1, page 68]. The variables G and H represent the state of the gene. The mRNA concentration and the concentration of the protein translated from the mRNA are represented by M and P . The n types of polymers of P are denoted by P = P 1 , P 2 , . . . , P n . Greek letters represent parameters. First, one translates the diagram as a system of generalized [START_REF] Horn | General mass action kinetics[END_REF] chemical reactions (transcription and translation are not balanced reactions), introducing 2 (n -1) extra parameters k - i , k + i (1 ≤ i ≤ n -1):

G + P n α ⇀ ↽ θ H, G ρ f -→ G + M, H ρ b -→ H + M, M β -→ M + P, M δM -→ ∅, P δP -→ ∅, P i + P k + i ⇀ ↽ k - i P i+1 (1 ≤ i ≤ n -1).
The initial model is obtained by translating the above chemical reactions system as the following system of parametric ordinary differential equations, denoting

A i = (k - i P i+1 -k + i P i P ).
Variables G, H, M, P = P 1 , . . . , P n are time varying functions. The dot appearing over the variables, in the equations left handsides, denotes the derivative w.r.t. the time.

Ġ = θ H -α G P n , Ḣ = -θ H + α G P n , Ṁ = ρ f G + ρ b H -δ M M, Ṗ = β M -δ P P + 2 A 1 + A 2 + • • • + A n-1 , Ṗi = -A i-1 + A i (2 ≤ i ≤ n -1), Ṗn = -A n-1 + θ H -α G P n (1)
The variables G and H should be viewed as "random variables" instead of concentrations. They are bound by the relation G + H = γ 0 where γ 0 is a constant equal to the total quantity of the gene. In [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF], the variable H was replaced by γ 0 -G.

The new quasi-steady state approximation

The quasi-steady state approximation that is performed in this paper relies on the following hypotheses: the n -1 chemical reactions describing the polymerization of the protein are fast while the other ones are slow. This happens when the parameters k + i , k - i are much larger than the other parameters. These hypotheses permit us to compute a reduced model by the following method. First one replaces, in the initial model, the contributions of the fast reactions by new variables F i (1 ≤ i ≤ n -1). This just amounts to rewriting A i = F i in system [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF]. Then one adds the following algebraic equations to the system, in order to express the pre-equilibrium conditions:

0 = k + i P P i -k - i P i+1 , ( 1
≤ i ≤ n -1)
Then one eliminates the new variables F i from the above differential-algebraic system, thanks3 to [START_REF] Boulier | Model Reduction of Chemical Reaction Systems using Elimination[END_REF]diffReduce]. One is led to the raw reduced model:

Ġ = θ H -α K n-1 P n G, Ḣ = -θ H + α K n-1 P n G, Ṁ = ρ b H + ρ f G -δ M M, Ṗ = n θ H -n α K n-1 P n G -δ P P + β M n-1 i=0 (i + 1) 2 K i P i (2)
where

K i = k + 1 • • • k + i k - 1 • • • k - i with the convention K 0 = 1.

Parameters reduction

The raw reduced model ( 2) can now be simplified by rescaling all parameters and variables. The following equations express the old variables as functions of the new ones, which are overlined:

θ = θ δ M , β = β δ M , δ P = δ δ M , ρ b = µ δ θ δ M α β , ρ f = δ θ (µ + λ) δ M α β , δ M = δ M , α = α δ M K n-1 , K i = K i α i θ i (1 ≤ i ≤ n -1), G = G, H = γ 0 -G, M = M δ θ α β , P = θ P α , t = t δ M
Performing these substitutions in the raw reduced model (2), withdrawing the redundant ODE which expresses the evolution of H and removing the bars for legibility, one gets the reduced model (3):

Ġ = θ (γ 0 -G -G P n ), Ṁ = λ G + γ 0 µ -M, Ṗ = n α (γ 0 -G -G P n ) + δ (M -P ) n-1 i=0 (i + 1) 2 K i P i • (3)
Remark 1. In order to recover [1, system (1)], it is sufficient to replace the right handside of the last equation of system (3) by its numerator.

Last observe that one more parameter could be removed from the above system, the software [START_REF] Sedoglavic | Expanded Lie Point Symmetry (MAPLE package)[END_REF] shows. The above reduction is however more convenient in this paper for it permits us to directly apply the results of [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF].

On the existence of Poincaré-Andronov-Hopf bifurcations

One proves in this section that no Poincaré-Andronov-Hopf bifurcation [START_REF] Hale | Dynamics and Bifurcations[END_REF] arises in system (3) for meaningful 4 values of the parameters and the variables if and only if n ≤ 8. Our proof essentially amounts to reducing the study of system (3) to that of [1, system (1)] and then applying the main result of this paper.

According to remark 1, the steady point equations of system (3) are exactly those of [1, section 4]. They write:

γ 0 = G + G P n , M = P, λ = P -µ G -µ G P n G •
The Jacobian matrix of system (3), evaluated at the steady points5 writes:

J =       -θ (1 + P n ) -n θ G P n-1 0 P -µ G -µ G P n G 0 -1 - n α (1 + P n ) B - n 2 α G P n-1 + δ B δ B       where B = n-1 i=0 (i + 1) 2 K i P i .
If one clears the denominators of the last row of the Jacobian matrix J, i.e. if one lets B = 1, then one exactly gets the Jacobian matrix of [1, page 73].

Remark 2. The matrix J is invariant under the following transformation, where ℓ denotes a nonzero arbitrary constant:

B → ℓ B, δ → ℓ δ, α → ℓ α.
Remark 3. The parameters K i only occur in the denominator B of the Jacobian matrix J.

Assume that a Poincaré-Andronov-Hopf bifurcation occurs for system (3), for some meaningful6 values of the system parameters and variables (1 ≤ i ≤ n -1):

(G, P, M, λ, α, θ, δ, γ 0 , µ, K i ) = (G 0 , P 0 , M 0 , λ 0 , α 0 , θ 0 , δ 0 , γ 0 0 , µ 0 , K 0 i ).

Then, by introducing B 0 = n-1 i=0 (i + 1) 2 K 0 i (P 0 ) i , and according to the two remarks above, a bifurcation of [1, system (1)] occurs for the meaningful values (G, P, M, λ, α, θ, δ, γ 0 , µ, ) = (G 0 , P 0 , M 0 , λ 0 , α 0 /B 0 , θ 0 , δ 0 /B 0 , γ 0 0 , µ 0 ).

Conversely, suppose that a birfurcation of [1, system (1)] arises for some meaningful values (G, P, M, λ, α, θ, δ, γ 0 , µ) = (G 0 , P 0 , M 0 , λ 0 , α 0 , θ 0 , δ 0 , γ 0 0 , µ 0 ).

Using the two remarks above and taking ℓ = 2, one can easily find some positive values for K 0 1 , . . . , K 0 n-1 such that B 0 = n-1 i=0 (i + 1) 2 K 0 i (P 0 ) i = 2. Thus a bifurcation for the system (3) occurs when (G, P, M, λ, α, θ, δ, γ 0 , µ, K i ) = (G 0 , P 0 , M 0 , λ 0 , 2α 0 , θ, 2δ 0 , γ 0 0 , µ 0 , K 0 i )

which are meaningful values. Combining the two implications, one concludes that a Poincaré-Andronov-Hopf bifurcation arises for [1, system (1)] if and only if such a bifurcation arises for the system (3), for meaningful values of the systems variables and parameters. The next proposition follows from the results of [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF]:

Proposition 1.
For meaningful values of the parameters and the variables, no Poincaré-Andronov-Hopf bifurcation arises in system (3) if and only if n ≤ 8.

4 On the new quasi-steady state approximation

Principle of the method

Our method is based on simular ideas as in [START_REF] Vora | Nonlinear model reduction of chemical reaction systems[END_REF][START_REF] Van Breusegem | Reduced order dynamical modelling of reaction systems: a singular perturbation approach[END_REF][START_REF] Bennet | Transient Dynamics of Genetic Regulatory Networks[END_REF] (see [1, section 4.2] for more details). The main interest of our method is that it is fully algorithmic. One first builds a system of differential equations involving some one extra variable F i for each fast reaction. Those extra variables are then eliminated by performing elimination. Our original implementation is based on the diffalg [START_REF] Boulier | Computing representations for radicals of finitely generated differential ideals[END_REF] package available in the standard library of MAPLE. A more recent implementation is based on the RegularChains [START_REF] Lemaire | The RegularChains library in MAPLE 10[END_REF] package. This section aims at summarizing [1, section 2]. Consider the classical system of chemical reactions (4) and ( 5) describing the transformation of a substrate S into a product P under the action of the enzyme E (an intermediate complex C is produced):

E + S k1 ⇀ ↽ k-1 C (4)
C k2 -→ E + P (5)
Assume that the reaction (4) is fast. Our method consists in introducing the following system:

" Ė = -F1 + k2 C, Ṡ = -F1, Ċ = F1 -k2 C, Ṗ = k2 C, k1 E S = k-1 C # .
Performing the elimination process (and introducing K = k -1 /k 1 ), one gets:

" F1 = k2 E S (S + K) K (S + E + K) , Ė = k2 E 2 S K (S + E + K) , Ṗ = k2 E S K , Ṡ = - k2 E S (S + K) K (S + E + K) , C = E S K # .
Using the conservation laws E+C = E 0 +C 0 and S+C +P = S 0 +C 0 +P 0 (where the subscript 0 indicates the initial concentration), assuming that C 0 = P 0 = 0 and introducing V m = k 2 E 0 , further computations yield:

Ṡ = - V m S (K + S) K E 0 + (K + S) 2 (6)
which differs from the Henri-Michaëlis-Menten and Briggs-Haldane formulae 7 :

Ṡ(t) = - V m S(t) K + S(t) • (7)
Over this easy example, the benefits are clear since the reduction is automatic and yields the formula (6) which seems more accurate than the classical reductions, especially when the condition S ≫ E 0 is not fulfilled. Observe that formula [START_REF] Ermentrout | Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students[END_REF] is recovered from ( 6) by assuming S ≫ E 0 .

A better new reduced model

The reduced system (2) appears to be more precise than that of [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF], numerical simulations show. Figure 2 shows two different numerical simulations of the variable G(t) for the same parameters and initial conditions values except for the k - i 's (hence for the K i 's). In both cases, three curves are displayed: one for the initial model ( 1) in red, one for the raw reduced model ( 2) in blue and one for [1, last system page 69] in green. In both cases, the blue and the red curves almost coincide. The reduction performed in [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] is clearly less precise than the new one. As the K i 's parameters tend towards zero, the reduction performed in [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] becomes more accurate (the rightmost picture is obtained with values of K i 's smaller than that of the leftmost one). Thus the domain of validity of the reduction performed in [1, last system page 69] is more narrow than that of our new reduction: the K i 's need to be small.

Conclusion

In this paper, the result presented in [START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF] is generalized by applying an algorithmic, accurate, quasi-steady state approximation method.

It is well-known that quasi-steady state approximation is useful for it permits to reduce the size of the differential system to study and the number of its parameters. But quasi-steady state approximation could also be viewed as a way to study the dynamical properties of gene regulatory networks which are invariant for a range of reactions mechanisms since the mechanisms involved in the fast reactions may not need to be known in detail. This issue is important [START_REF] De Jong | Qualitative Approaches to the Analysis of Genetic Regulatory Networks[END_REF]. This shows that the development of algorithmic and accurate quasi-steady state approximation methods is an important research domain in the field of algebraic biology.

2 Fig. 1 .

 21 Fig. 1. A single gene regulated by a polymer of its protein

k 1 1 inFig. 2 .

 112 Fig. 2. Numerical simulations of the variable G(t). The horizontal axis give the time t. The vertical one gives the value of G. For both simulations, n = 3, G(0) = 0.5, M (0) = 1, P (0) = 1, γ0 = 1, α = 1, θ = 10, ρ f = 10, ρ b = 5, β = 50, δM = 5, δP = 10 and k + i = 100 for each i. On the leftmost picture, k - i = 100 thus Ki = (100/100) i = 1 for each i. On the rightmost one, k - i = 2000 thus Ki = (100/2000) i = 0.05 i for each i. The red (system 1) and the blue (system 2) curves almost coincide. The green [1, last system page 69] curve is the lower one on both pictures.

Our algorithm does not handle the generic system with a symbolic n. We computed the reduced system for many different values of n, inferred the general formula and, checked afterwards that the inferred formula is correct.

Following[START_REF] Boulier | On proving the absence of oscillations in models of genetic circuits[END_REF], all variables and parameters are required to be positive apart λ, which may be negative but must anyway be greater than -µ,

Note that the derivative of B w.r.t. to P which appears in the Jacobian of the system (3) disappears after the evaluation at the steady points since it is multiplied by a term which cancels.

In the sense precised above.