Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux

Petitot, Jean; Tondut, Yannick

Mathématiques informatique et sciences humaines, Tome 145 (1999), pp. 5-101.

Ce travail propose certains modèles variationnels pour les processus corticaux d’intégration des contours subjectifs modaux (de type contours illusoires à la Kanizsa), modèles fondés sur les concepts géométriques de fibration et de structure de contact. La structure rétinotopique des hypercolonnes d’orientation de l’aire $V_1$ (telle qu’elle est décrite depuis les travaux pionniers de Hubel, Wiesel et Mountcastle) est une architecture fonctionnelle qui peut être mathématiquement idéalisée par la fibration $\pi :E\to M$ ayant pour base le plan $M$ de la rétine et pour fibre $F$ la droite projective ${\mathbb{P}}^1$ des directions du plan, l’espace total $E$ de $\pi$ étant isomorphe au produit direct $M\times F$. Au-dessus de chaque position rétinienne se trouve implémenté un exemplaire (discrétisé) de $F$. Les connexions horizontales cortico-corticales implémentent ce que l’on appelle la trivialité locale de cette fibration et sans doute également une connexion (au sens d’Elie Cartan) définissant un transport parallèle. Après avoir rappelé ces données, le papier se focalise sur l’interprétation géométrique des résultats de Field, Hayes et Hess sur le champ d’association. Ces travaux semblent montrer que ce que l’on appelle en géométrie symplectique la structure de contact de la fibration $\pi :E\to M$ se trouve neuralement implémenté. Le champ d’association correspond dans ce cadre à une condition d’intégrabilité des courbes dans $E$ : elles doivent être les relevées de leur projection sur le plan rétinien $M$. Ce modèle d’une fibration munie d’une structure de contact naturelle est ensuite appliqué à l’interprétation des contours subjectifs modaux et conduit à des variantes du modèle dit de l’elastica développé par B.K.P. Horn et D. Mumford. L’idée est que les contours subjectifs modaux ont des relevées qui sont «géodésiques» dans le fibré cortical $E$, c’est-à-dire de longueur minimale (pour une métrique appropriée) dans la classe des courbes satisfaisant la condition d’intégrabilité. Les modèles «géodésiques» sont ensuite reformulés, à la suite de R. Bryant et P. Griffiths, dans un cadre géométrique plus fondamental, celui des groupes de Lie et du repère mobile d’Elie Cartan. Quelques possibilités de test expérimentaux sont enfin considérées.

This work presents some variational models for the cortical algorithms processing Kanizsa modal subjective contours. These models are based on the geometric concepts of fibration and contact structure. The retinoptic structure of the orientation hypercolumns in the visual area $V_1$ is a functional architecture which can be mathematically idealized by the fibration $\pi :E\to M$ having the retinian plane $M$ as base and the projective line ${\mathbb{P}}^1$ as fiber $F$. The total space $E$ of $\pi$ is isomorphic to the direct product $M\times F$. The cortico-cortical horizontal connections implement what is called the local triviality of this fibration, and also a Cartan connection defining a parallel transport between neighboring fibers. Then, the paper focuses on the geometrical interpretation of the results of Field, Hayes and Hess concerning the association field. It shows that the latter implements what is called the contact structure of the fibration $\pi :E\to M$. The association field expresses an integrability condition for the skew curves in $E$ : they have to be a lifting of their projection on the retinian plane $M$. This model of a fibration endowed with a contact structure is then applied to the modal subjective contours and provides a variant of the elastica model developped by B.K.P. Horn and D. Mumford. The key idea is that the lifting of subjective contours satisfy a «geodesic» condition in the cortical fibration $E$ : they have to be of minimal lenght (for an appropriate metrics) among the class of curves satisfying the integrability condition. These «geodesic» models are then reformulated, according to R. Bryant and P. Griffiths, in the more fondamental geometric framework of Lie groups and Cartan’s “repère mobile” (Vielbein). Finally, some experimental possibilities are suggested.

Mot clés : champ d'association, condition d'intégrabilité, contours subjectifs, elastica, équations d'Euler-Lagrange, fibration, géodésique, groupe de Lie, modèles variationnels, repère mobile, structure de contact
Mots clés : association field, contact structure, elastica, Euler-Lagrange equations, fibration, geodesic, integrability condition, Lie groups, subjective contours, variational models, vielbein

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