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Elements of Neurogeometry pp 113–273Cite as

Functional Architecture I: The Pinwheels of V1

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Part of the book series: Lecture Notes in Morphogenesis ((LECTMORPH))

Abstract

This chapter presents a first set of experimental data on the functional architecture of area V1 and, in particular, on what is referred to as its ‘pinwheel’ structure. Most of the so-called simple neurons in V1 detect positions and orientations in the visual field, with those detecting the various orientations for each position being grouped together into functional micromodules that can be defined anatomically and are called orientation hypercolumns or pinwheels. In this sense, V1 implements a 2D discrete approximation of the 3D fibre bundle \(\pi :\mathbb {V}=R\times \mathbb {P} ^{1}\rightarrow R\), with the retinal plane R as its base space and the projective line \(\mathbb {P}^{1}\) of orientations in the plane as its fibre. V1 then appears to be a field of orientations in a plane (called an ‘orientation map’ by neurophysiologists), a field whose singularities are the centres of the pinwheels . This chapter studies these singularities, gives their normal forms , and specifies the distortions and defects of their networks. One way to model such orientation maps is to treat them as phase fields, analogous to those encountered in optics, whose singularities have been thoroughly analyzed by specialists such as Michael Berry. These fields are superpositions of solutions to the Helmholtz equation, whose wave number depends in a precise manner on the mesh of the pinwheel lattice. They enable the construction of very interesting models, such as those proposed by Fred Wolf and Theo Geisel. These are explained here. However, in these models of phase fields , orientation selectivity must vanish at singularities. Yet many experimental results show that this is not the case. The chapter thus presents another model based on the geometric notion of blow-up. It also explains how the fibration that models the orientation variable interferes with other fibrations (other visual ‘maps’) that model other variables such as direction , ocular dominance , phase, spatial frequency, or colour. For spatial frequency, it presents the dipole model proposed by Daniel Bennequin. V1 therefore implements fibrations of rather high dimension in two-dimensional layers. This leads to the problem of knowing how to express the independence of these different variables. A plausible hypothesis relies on a transversality principle. The chapter ends with data on two other aspects of neurobiology: (i) the relation between the cerebral hemispheres through callosal connections and (ii) the primary processing of colour in the ‘blobs’ of V1.

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Notes

  1. 1.

    When we don’t need to distinguish between R and M, we shall set \(R=M\) and \(\chi =Id\).

  2. 2.

    Van Hooser’s paper also discusses the rat (rodent, nocturnal, laterally placed eyes), the night monkey, also known as the owl monkey or douroucoulis (New World primate, nocturnal, frontally placed eyes), the ferret, etc.

  3. 3.

    In the geometric models of neural functional architectures, there are many problems of terminology. Lexical items such as ‘fibre’, ‘projection’, ‘connection’ are used in different ways by mathematicians and neurophysiologists. In general, the meaning should be clear from the context.

  4. 4.

    Named after Brook Taylor.

  5. 5.

    We could distinguish between the retinal plane R and the cortical layer M (the base space of V) to which it projects. However, to simplify, we shall not do so, considering the retinotopic map \(\chi :R\rightarrow M\) as the identity.

  6. 6.

    The transition from recordings of a few isolated neurons to a visualization of the overall activity of a piece of brain area is analogous to the leap forward in meteorology when recordings made by weather balloons were replaced by satellite imaging. No need for further comment.

  7. 7.

    There are two classes of primates: on the one hand, monkeys and humans, and on the other, the prosimians.

  8. 8.

    For an introduction to simulated annealing, see, for example, the Bourbaki lecture by Robert Azencott [53].

  9. 9.

    For more on such questions of dimensions, the reader could consult my 1982 review [59] and references therein.

  10. 10.

    Given the assumptions of continuity and smoothness, we need consider only connectedness by arcs.

  11. 11.

    Here, the authors use basic theorems of general topology going back to Bolzano, Weierstrass , Heine , Borel, and Lebesgue. For separated topological spaces, the continuous image of a compact (connected) set is compact (connected).

  12. 12.

    Phase singularities are generically points in 2D and lines in 3D because they are specified by two conditions and so have codimension 2 (see below).

  13. 13.

    Named after Carl Gustav Jacob Jacobi.

  14. 14.

    Let \(H_{\varphi }=\left( \begin{array}{cc} a &{} b \\ b &{} c \end{array} \right) \). The eigenvalues are solutions of the quadratic equation \(\text{ Det }\left( H_{\varphi }-\lambda I\right) =0\), which can be written \(\lambda ^{2}-\lambda \left( a+c\right) +\left( ac-b^{2}\right) =0\). The discriminant \(\left( a+c\right) ^{2}-4\left( ac-b^{2}\right) =\left( a-c\right) ^{2}+4b^{2}\) is always non-negative, and the roots are therefore real.

  15. 15.

    \(\left\langle \ {,} \ \right\rangle \) is the natural pairing between vectors and dual covectors. It can also be expressed by a dot.

  16. 16.

    For Hebb’s law, see Sect. 3.6.2.

  17. 17.

    We have already encountered this density \(\pi /\Lambda ^{2} \) in Sect. 4.6.11.

  18. 18.

    We have to use the two variables Z and \(\overline{Z}\) because the functions \(F_{j}\) are not necessarily analytic functions depending only on Z.

  19. 19.

    In the expansion of L, the powers of partial derivatives like \(\left( {\partial }/{\partial x}\right) ^{n}\) mean repeated differentiation \({\partial ^{n}}/{\partial x^{n}}\). The linear PDE \({\partial Z}/{\partial t}=L\left( Z\right) \) for \(L=\mu \) describes exponential damping towards 0 for \(\mu <0\) (stability) and exponential growth for \(\mu >0\) (instability). The PDE \({\partial Z}/{\partial t}=L\left( Z\right) \) for \(L=-\Delta \) is a diffusion equation.

  20. 20.

    For a didactic introduction to non-standard analysis, see, for example, Petitot [93] and the references therein.

  21. 21.

    Rather as in the Kaluza–Klein field theories of physics.

  22. 22.

    I thank Guy Wallet and Michel Berthier for this reference.

  23. 23.

    Lengths are measured in degrees of the visual field.

  24. 24.

    We have already encountered this problem of minimizing the wiring in Sect. 4.4.5.1.

  25. 25.

    For an adequate treatment of this point, one must introduce the rather technical geometric notion of a Lagrangian sub-manifold. We shall say a little more about this in the second volume. Here we only make elementary remarks about the geometry.

  26. 26.

    Recall that the optic nerve contains about 1.5 million axons, so less than a hundredth of the number.

  27. 27.

    On generation and corruption, I, 5, 312b.

  28. 28.

    All the details can be found in René Thom’s two books [118] and [119]. For a didactic introduction, see [130].

  29. 29.

    The pinwheel structure is based on the dichotomy between regular and singular points, but this concerns field lines of the orientation field. Here we are talking about something quite different.

  30. 30.

    Recall (see Sect. 3.2.5 of Chap. 3) that the three kinds of cones in humans are L/M/S, L red, M green, S blue.

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    Jean Petitot

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Petitot, J. (2017). Functional Architecture I: The Pinwheels of V1. In: Elements of Neurogeometry. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-319-65591-8_4

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