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The vestibular system detects head movement in space and maintains visual and postural stability. The semicircular canal system is responsible for registering head rotation. How it responds to head rotation is determined by the rotational axis and the angular acceleration of the head, as well as the sensitivity and orientation of each semicircular canal. The morphological parameters of the semicircular canals are supposed to allow an optimal detection of head rotations induced by some behaviours, especially locomotor. We propose a new method of semicircular canal analysis, based on the computation of central streamlines of virtually reconstructed labyrinths. This method allows us to ascertain the functional structure of the semicircular canal system and to infer its capacity to detect particular head rotations, induced by particular behaviours. In addition, this method is well-suited for datasets provided by any kind of serial sectioning methods, from MRI to μCT scanning and even mechanical serial sectioning, of extant and extinct taxa.

Le système vestibulaire permet de détecter les mouvements de la tête dans l’espace et de maintenir un équilibre visuel et postural. Le système des canaux semi-circulaires est responsable de la détection des rotations de la tête. La façon dont il répond aux rotations est déterminée par l’axe de rotation et l’accélération angulaire de la tête, ainsi que par la sensibilité et l’orientation de chaque canal semi-circulaire. Les paramètres morphologiques des canaux semi-circulaires sont supposés permettre une détection optimale des rotations de la tête induites par certains comportements, en particulier locomoteurs. Nous proposons une nouvelle méthode d’analyse des canaux semi-circulaires, basée sur le calcul de lignes de flux centrales, à partir de reconstructions virtuelles de labyrinthes. Cette méthode permet de déterminer la structure fonctionnelle du système des canaux semi-circulaires et d’en déduire sa capacité à détecter des rotations de tête induites par des comportements particuliers. Elle est applicable à des jeux de données issus de tout type de méthodes de sections sériées, de l’IRM aux scanners μCT en passant par le sectionnement sérié mécanique, et ce, pour des taxons actuels et éteints.

The vestibular system of vertebrates is involved in the coordination of movement, gaze control, and maintaining balance. One of its main tasks is to compensate for uncontrolled head movements to maintain head and gaze stability in space (

The semicircular canal system is commonly described in jawed vertebrates (gnathostomes) as an idealized structure that consists of three functional pairs of roughly circular canals which retain specific angular relationships between them and with the reference planes of the head (

The real structure of the semicircular canal system is much more complex than the idealized structure described above. For the purpose of functional consideration, its complicated structure can be simplified. The semicircular canal system basically works with two sets of three interconnected semicircular ducts (

Our proposed method allows us to easily extract morphological and functional parameters from the semicircular canal system of extant and extinct taxa and to use them, along with behavioural information based on extant taxa, to infer locomotor behaviours of fossil vertebrates without using postcranial data (

For the analysis, we need to compute a three-dimensional triangle mesh model of the semicircular canal system. This kind of reconstruction is easily obtained by using datasets provided by virtual serial sectioning methods, such as MRI (

The analysis of the vestibular capacities is based on the central streamlines of the semicircular canal system. The central streamlines consist of unidimensional representations of the studied object. These streamlines retain the topology of the object and are centred inside it (

Several parameters can be calculated with the twelve streamlines. A streamline is composed of ordered, three-dimensional points and its length can be calculated by adding up the distance between each pair of points. The length of an entire semicircular canal and of all of its parts can then be readily measured. We can also plot the cross-section area variation, relative to the position of the section, in order to characterize the different parts of the semicircular canal. The cross-section area can vary from zero to one order of magnitude along a semicircular duct pathway (

Since we have calculated the central streamlines of each semicircular canal, we can now calculate their sensitivity vectors _{
n
} represents the mean cross-section area of the slender part of the semicircular duct, _{
n
} represents the total area of the cupula surface and

Since many kinds of datasets can be analysed by this method, a common coordinate system is needed for suitable comparison. Considering the various kinds of existing systems (

The SCFS can be used in different kinds of analyses in order to infer the behaviour of extinct taxa and to clarify the functioning of the semicircular canal system.

Scale analyses deal with scaling factors, such as length and mass. They consist of bringing the studied objects to the same scale and analysing their differences. One typical scaling factor used in such analyses is the body mass. Actually, there is a law relating the body mass of an animal and the size of its labyrinth (

Shape analyses are not sensitive to scaling effects and can be used to study interspecific variation of conformation without considering body mass or other scaling factors. Unlike the scale analyses which allow studying the variation of parameters between taxa, shape analyses only rely on the grouping of similar conformation and knowledge of the behaviour of extant taxa in order to infer the behaviour of extinct taxa. The shape analysis is mainly based on the study of the octahedral conformation of the SCFS, since it is supposed to be a relevant structure for the function of the semicircular canal system and its behavioural inferences. The main hypothesis here is that taxa which share similar behaviour also share similar octahedral structures. This suggests that, contrary to the general shape of the semicircular canals, which is supposed to be mainly constrained by phylogeny, the SCFS is much more related to the pattern of head movements which occur in each taxon. In order to test this hypothesis, we can analyse the differences in the composition of groups resulting from shape analysis of the semicircular canal system morphology on the one hand, and functional structure on the other. The morphological analysis can be achieved through a geometrical morphometric analysis of the three-dimensional shape variation of the slender parts of the semicircular canals (

The two previous analyses can be used for interspecific comparisons, while the third type of analysis we propose focuses on the response heterogeneity in a single system. Some previous studies suggested that the semicircular canal system may need an increased sensitivity for particular rotations in order to counteract increased imbalance along these axes induced by particular behaviours (

The second subanalysis is named global response analysis (

The third subanalysis explains the pattern of semicircular canal activation caused by the main rotations (

The previously described methods can be used on both the bony labyrinth and the membranous labyrinth. Even though the analyses of the two structures return fairly similar results, the analysis of the membranous labyrinth is in all more conclusive because it is the functional part of the system. The latter preferred method is not always applicable especially in extinct taxa, in which only the bony labyrinth is available for study. The membranous semicircular ducts and bony semicircular canals have different cross-section areas that can vary by over one order of magnitude (

To perform a valid analysis of the SCFS, the vectorial structure must answer to some criteria. The magnitudes of the vectors of homologous semicircular canals must be the same and their directions must be symmetrical relative to the sagittal plane. Different sources of errors can lead to violate these criteria in fossil specimens. They are: (1) the deformation of the structure of the semicircular canal system; (2) the loss of parts of canals or of entire labyrinths due to a bad preservation of the otic capsule; and (3) the difficulty to accurately reconstruct the labyrinth based on datasets which show a poor density contrast.

In most fossil amniotes the labyrinths are often protected against deformation by the solidly built otic capsule. Dislocation and relative displacement of the bones of the braincase, however, often occur, leading to an asymmetric placement of the two labyrinths to each other. When the two labyrinths are intact, how well they overlay in lateral view will give us an indication of the quality of preservation and of the quality of the reconstruction. This is an easy way to readily see if the semicircular canal system of a fossil specimen has been deformed and if the reconstruction has been correctly done. Moreover, the analysis of the magnitude of the vectors resulting from the reconstruction of fossil semicircular canal systems and the analysis of the angles between these vectors can help us to estimate quantitatively the accuracy of the SCFS. The angular relationship between the three ipsilateral semicircular canals of a labyrinth inform us about the deformation of the labyrinth. If the angular relationship between ipsilateral semicircular canals of the right labyrinth is the same as the angular relationship between ipsilateral semicircular canals of the left labyrinth, this means that the labyrinths are not deformed. However, even if the labyrinths are not deformed, the structure of the semicircular canal system can be deformed if the bones of the braincase experience some displacement and deformation. The angular relationship between the functional pairs of vertical semicircular canals can help us to quantify the deformation of the semicircular canal system. If the angle between the right anterior semicircular canal and the left posterior semicircular canal is the same as the angle between the left anterior semicircular canal and the right posterior semicircular canal, this generally means that there is no deformation of the semicircular canal system. It should be noted that a simple translation of one labyrinth in space does not affect the SCFS, but a rotation of one labyrinth does. Finally, the comparison of the angular relationships between the ipsilateral semicircular canals of the two labyrinths and the comparison of the magnitudes of the sensitivity vectors between the homologous semicircular canals informs us about the quality of the virtual reconstruction. Similar angular relationships of ipsilateral canals and similar magnitudes for the vectors of homologous semicircular canals indicate an accurate reconstruction of the semicircular canal system.

If a complete and non-deformed semicircular canal system cannot be reconstructed, it is, however, possible to estimate the SCFS. The SCFS can be estimated if there is information about: (1) the sensitivity vector of at least one anterior semicircular canal; (2) the sensitivity vector of at least one posterior semicircular canal; (3) the sensitivity vector of at least one lateral semicircular canal; and (4) the position of the bilateral symmetry plane. When the semicircular canal system is deformed, we need a different estimation for the angular relationship between each labyrinth, or part of labyrinth, and the bilateral symmetry plane. If the above criteria are satisfied, it is then possible to reconstruct the SCFS by mirroring each vector of a pair of homologous semicircular canals relative to the respective bilateral symmetry plane in order to reconstruct the missing vector of the pair. This method can then be applied in the case of missing semicircular canals, deformation of labyrinths or semicircular canal systems and heterogeneity in the quality of reconstruction between the right and the left labyrinths. This method, however, implies an increase in the number of hypotheses and is then less precise than the computation of the SCFS based on a complete and non-deformed semicircular canal system.

Semicircular canal functioning can be described by six morphofunctional parameters which consist of: (1) the axes of maximal response; (2) the sensitivity along these axes; (3) the maximal gain (ratio between cupula bending and impulse rotation); (4) the natural frequencies of head rotation where these gains occur; and (5) the upper and (6) lower corner frequencies which constrain the range of head rotation effectively detected by the system (

It is noteworthy to mention here that, for a same body mass, more agile taxa possess longer semicircular canals (

Bilateral symmetry applies some constraints on the system's response. Any change in one side of the head is mirrored in the other side. Thus, for all rotations around axes included in the sagittal plane (i.e. roll, yaw and all intermediate rotations), the response of each canal of one labyrinth will be the exact opposite of the homologous canal of the other labyrinth. This can be seen at the level of the bilateral symmetry plane where each projection of a vector of a pair of homologous semicircular canals possesses the same magnitude and direction than the projection of the other one, but shows an opposite sense (see

Bilateral symmetry also constrains the configuration of the vectors of sensitivity of the semicircular canals in space. In the reference system of the labyrinth, each component of the vectors of the SCFS shows the signs presented in

There are only two possibilities to modify the sensitivity of the semicircular canal system: one can either change the orientation of the vectors and/or change their magnitude. A problem posed is the increasing of the sensitivity along a natural axis without modifying its value along the other two. A solution to this problem is to modify both the orientation and magnitude of the vectors of a pair of homologous semicircular canals. If only one parameter is modified, e.g., orientation or magnitude, already the sensitivity along two main axes of rotation will be slightly modified. The same sensitivity along an axis of rotation can be achieved by different orientation and magnitude of the semicircular canals vectors. However, few deviations of orientation seem to modify the sensitivity much more than the increase in magnitude (

In summary, we have proposed a method based on central streamlines extraction of three-dimensional triangulated surface meshes of the labyrinth in order to extract the semicircular canal system functional structure and to study it through three main analyses. The use of central streamlines is not a novelty, but was previously applied to analyse the dynamic response of the semicircular canal system to particular rotations (

We thank Philippe Taquet, Gaël Clément and Didier Geffard-Kuriyama for the invitation to contribute to this volume. We thank the two anonymous reviewers for their helpful comments which have helped to greatly improve this article. Special thanks to Karin Peyer for comments and suggestions. This work has in part been supported by Legs Prévost of the Muséum national d’histoire naturelle in Paris, and by the European project CLONS. The scan of the specimen 1910.12 of

The 2 appendixes, biomechanical and technical, are supplied as supplementary material to the electronic version of the article.

Lateral view of a schematic labyrinth. The bony labyrinth is in grey.

Vue latérale d’un labyrinthe schématique. Le labyrinthe osseux est en gris.

Dorsal view of the labyrinths of a crocodile showing one synergistic functional pair composed of the left anterior semicircular canal and the right posterior semicircular canal.

Vue dorsale des labyrinthes d’un crocodile montrant une paire fonctionnelle synergiste, composée du canal semi-circulaire antérieur gauche et du canal semi-circulaire postérieur droit.

Sketch of a semicircular duct which shows the main morphological parameters that determine its sensitivity.

Schéma d’un canal semi-circulaire membraneux présentant les principaux paramètres morphologiques influençant sa sensibilité.

Functional structure of the semicircular canals system (SCFS) of an idealized model. The tips of the vectors correspond to the vertices of an octahedron.

Structure fonctionnelle du système des canaux semicirculaires (SCFS) d’un modèle idéal. Les sommets des vecteurs forment les sommets d’un octaèdre.

Lateral view of the left labyrinth of

Vue latérale du labyrinthe gauche d’

Sketch of a semicircular canal duct which shows the main parameters which allow the calculation of the sensitivity vector

Schéma d’un canal semi-circulaire membraneux présentant les principaux paramètres qui permettent le calcul du vecteur de sensibilité

Functional structure of the semicircular canals system of

Structure fonctionnelle du système des canaux semi-circulaires d’

The three natural rotations. (A) Pitch, (B) yaw and (C) roll.

Les trois rotations naturelles. (A) Le tangage, (B) le lacet et (C) le roulis.

Landmarks used in the shape analyses based on (A) the semicircular canals morphology, (B) the functionnal structure of the semicircular canals system and (C) the total response of the system for the three natural rotations.

Points de repère utilisés pour les analyses de forme basées sur (A) la morphologie des canaux semi-circulaires, (B) la structure fonctionnelle du système des canaux semi-circulaires et (C) la réponse totale du système pour les trois rotations naturelles.

A. Total response analysis of the SCFS of

A. Analyse de réponse totale du SCFS d’

Total response analysis of the semicircular canals system of

Analyse de réponse totale du système des canaux semi-circulaires d’

Possible signs of the components of the vectors of the SCFS in the labyrinth reference system.

Signes possibles des composantes des vecteurs du SCFS dans le système de référence du labyrinthe.