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Elements of Neurogeometry pp 45–111Cite as

Receptive Fields and Profiles, and Wavelet Analysis

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

Abstract

This chapter discusses receptive fields and receptive profiles of visual neurons, starting with the photoreceptors and the ganglion cells of the retina, and proceeding via the neurons of the lateral geniculate nucleus to those of V1. It explains how they act on the optical signal as filters, and broaches the problem of their linearity or nonlinearity. In the linear case, the effect they have on the signal falls to a large extent under the rule of what is known as ‘wavelet analysis’ in signal processing. The chapter gives an outline of this fundamental notion, then discusses how receptive profiles can be interpreted within the framework of information theory as a means of optimizing the processing of natural images, which have very particular statistical properties. To a certain extent, the geometric formatting of the signal by means of a certain kind of wavelet (essentially, partial derivatives of Gaussians) is a way of optimizing the compression of natural images.

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Notes

  1. 1.

    Note that these are Aristotelian categories. We shall return to this philosophical aspect in the conclusion of the second volume.

  2. 2.

    The paper contains many other examples of retinal cells.

  3. 3.

    \(A-B\) indicates a colour opposition between the centre and the periphery of the GC.

  4. 4.

    Some specialists like Nathans think that the blue/yellow (B/Y) opposition evolved from single-celled organisms in which it controlled the circadian (day/night) response by signalling large spectral changes in the sun’s light.

  5. 5.

    Even though the names are the same, the colour labels of the cones should not be confused with those of the GCs. The mechanisms are not the same. The first refer to spectral absorption properties and the second to circuitry and a receptive field structure.

  6. 6.

    One voxel is typically \(2.5\times 2.5\times 3\) mm\(^3\) in fMRI.

  7. 7.

    Readers wishing to experiment with derivatives of Gaussians are referred to the Gaussian Derivative package of Mathematica, by Bart M. ter Haar Romeny and Markus van Almstick. These techniques in which derivatives of Gaussians are used as filters have applications in image compression (JPEG). See, for example, the reference books by Mallat [53] or Morgan et al. [54].

  8. 8.

    For the definition of \( L^{2}\left( \mathbb {R}\right) \) see below Sect. 3.4. \(H_{n}\left( x\right) \mathrm{e}^{-{x^{2}/2}}\) can be in \(L^{2}\left( \mathbb {R}\right) \) because \(\mathrm{e}^{-{x^{2}/2}}\) decreases more quickly at infinity than any polynomial can increase there.

  9. 9.

    This algorithm can be found today in any image-processing software.

  10. 10.

    The signals correspond to real-valued \(f\left( x\right) \). However, it is convenient also to allow the \(f\left( x\right) \) to be complex-valued. \(\overline{f}\left( x\right) \) is then the complex conjugate of \(f\left( x\right) \).

  11. 11.

    The \(\mathrm{e}^{I\omega x}\) are not actually in this Hilbert space, but we can ignore this well-known technical detail for our present purposes.

  12. 12.

    See also Mallat [53].

  13. 13.

    See, for example, Wickerhauser [68].

  14. 14.

    Named after Ludwig Otto Hesse.

  15. 15.

    \(\left\langle \ \right\rangle \) is the average over signals.

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  1. CAMS, EHESS, Paris, France

    Jean Petitot

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Petitot, J. (2017). Receptive Fields and Profiles, and Wavelet Analysis. In: Elements of Neurogeometry. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-319-65591-8_3

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