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Elements of Neurogeometry pp 275–346Cite as

Functional Architectures II: Horizontal Connections and Contact Structure

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Abstract

This chapter studies the second component of the functional architecture of primary visual areas , namely what are called cortico-cortical ‘horizontal’ connections. They join together neurons that detect orientations at different points which are parallel, and even approximately coaxial. This coaxial ‘parallel transport’ has been confirmed by psychophysical experiments on what David Field, Anthony Hayes, , and Robert Hess called the association field. To a first approximation, coaxial parallel transport implements what is known in differential geometry as the contact structure of the fibre bundle \(\mathbb {V}_\mathrm{J}\) of 1 -jets of curves in the plane. It is invariant under the action of the group SE(2) of isometries of the Euclidean plane. The contact structure is associated with a non-commutative group structure on \(\mathbb {V}_\mathrm{J}\), the action of SE(2) then being identified with the left-translations of \(\mathbb {V}_\mathrm{J}\) on itself. This group is isomorphic to the ‘polarized’ Heisenberg group. It is a nilpotent group belonging to the class of what are called Carnot groups. The definition of an SE(2)-invariant metric on the contact planes thus defines what is called a sub-Riemannian geometry. Using this, one obtains a natural explanation for the phenomenon whereby the visual system constructs long-range illusory contours. For illusory contours with curvature, this leads to variational models. At the end of the 1980s, David Mumford already proposed a first model, defined in the plane \(\mathbb {R}^{2}\) of the visual field, which minimized a certain functional of curve length and curvature . We propose a model in the contact bundle \(\mathbb {V}_\mathrm{J}\) that rests on the hypothesis that illusory contours are sub-Riemannian geodesics. The Cartan ‘prolongation’ of this structure, called the Engel structure, corresponds to the 2 -jets of curves in the plane \(\mathbb {R}^{2}\). It must be taken into account if we want to report on the experimental data showing that, in the primary visual system, there exist not only orientation detectors (tangents and 1-jets), but also curvature detectors (osculating circles and 2-jets). We then point out a few properties of the functional architecture of areas V2, V4 (for colour), V5, and MT (for motion). We also discuss Swindale’s model for directions. Finally, we briefly describe the genetic control of the neural morphogenesis and axon guidance of the functional architectures.

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Notes

  1. 1.

    Recall that \(\log \left( 0\right) =-\infty \), so if \(p\in \left[ 0,1\right] \) is small, then \(-\log \left( p\right) \) is a large positive number.

  2. 2.

    \(90\times 90\) is the number of ways to position the little square, 1/2 is the probability of black or white for each pixel, \(\left( 1/2\right) \!{}^{100}\) is the probability that the 100 pixels of the small square are black, and \((1/2){}^{900}\) is the probability that the 900 pixels remaining in the large square are white.

  3. 3.

    Named after George Boole.

  4. 4.

    In the rest of this section, A and B will denote the ends of curves.

  5. 5.

    The reader should not confuse \(\pi \) the projection \(\pi :R\times P\rightarrow R\) and \(\pi \) the component of a tangent vector t.

  6. 6.

    If \(\omega _{1}\) and \(\omega _{2}\) are two 1-forms, \(\omega _{2} \wedge \omega _{1}=-\omega _{1} \wedge \omega _{2}\) (and hence \(\omega \wedge \omega =-\omega \wedge \omega =0\)), and if \(\omega =\mathrm{d}f\) is the differential of a function f(x, y, z), then \(\mathrm{d}\omega =\mathrm{d}^{2}f=0\).

  7. 7.

    The term ‘pacman’ comes from a famous Japanese video game in which a disc with a variable angular sector representing a stylized mouth (the ‘pac-man’) has to find its way through a maze while ‘eating pac-dots’ and at the same time avoiding enemies.

  8. 8.

    Friedrich Engel was one of the main disciples and collaborators of Sophus Lie.

  9. 9.

    There are some technical problems in the definition of the energy, but these are well known in signal theory. To obtain good results, the even/odd filters must be refined, for example, taking the Hilbert transform of \(G_{2}\) instead of \(G_{3} \).

  10. 10.

    For an introduction to structuralism in biology from Waddington to Thom, see [78].

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    Jean Petitot

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Petitot, J. (2017). Functional Architectures II: Horizontal Connections and Contact Structure. In: Elements of Neurogeometry. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-319-65591-8_5

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