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Curvature Sensitive Cells in the Modular Structures of The Visual Cortex

Giovanna Citti, Vasiliki Liontou

Abstract

We propose a model of the functional architecture of curvature-sensitive cells in the primary visual cortex. The model accounts for the modular and hierarchical organization of the cortex, the horizontal connectivity, and the shape of receptive profiles of these cells as Gabor-type filters. We construct a canonical affine subbundle of the cotangent bundle of the manifold of oriented contact elements of the retina as a geometric model for these cells, and show that this subbundle carries an Engel structure related to that of the Cartan prolongation. On an open submanifold of the Cartan prolongation, we identify generators of the Engel distribution whose iterated Lie brackets span the Lie algebra of SIM(2). The identification of sim(2) as the Lie algebra of these generators determines SIM(2) as the natural symmetry group for curvature-sensitive cells. Finally, we characterize the receptive profiles of curvature-sensitive cells as minima of a SIM(2)-adapted uncertainty principle applied to the generators of the Engel structure.

Curvature Sensitive Cells in the Modular Structures of The Visual Cortex

Abstract

We propose a model of the functional architecture of curvature-sensitive cells in the primary visual cortex. The model accounts for the modular and hierarchical organization of the cortex, the horizontal connectivity, and the shape of receptive profiles of these cells as Gabor-type filters. We construct a canonical affine subbundle of the cotangent bundle of the manifold of oriented contact elements of the retina as a geometric model for these cells, and show that this subbundle carries an Engel structure related to that of the Cartan prolongation. On an open submanifold of the Cartan prolongation, we identify generators of the Engel distribution whose iterated Lie brackets span the Lie algebra of SIM(2). The identification of sim(2) as the Lie algebra of these generators determines SIM(2) as the natural symmetry group for curvature-sensitive cells. Finally, we characterize the receptive profiles of curvature-sensitive cells as minima of a SIM(2)-adapted uncertainty principle applied to the generators of the Engel structure.
Paper Structure (19 sections, 13 theorems, 121 equations, 2 figures)

This paper contains 19 sections, 13 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Orientation Selective Odd Simple Cells
  3. Gauge Coordinate System
  4. Contact Structure and Lifting Contours from the retina to the Cortex
  5. Receptive Profiles of Orientation Sensitive Simple Cells and the SE(2) transform
  6. The Ehresmann connection and cells sensitive to Curvature
  7. Ehresmann Connection and Characterization of Retinal Curves lifted to M
  8. Intrinsic description of Propagation of Neural Activity
  9. Singed Curvature in Horizontal Lifts of Retinal Contours
  10. Curvature Selective Cells: Lifting from M to an affine sub-bundle of T*M
  11. Curvature and Curvature Radii Space: Engel Structure and Associated Lie Algebra
  12. Cartan Prolongation and Curvature Radii
  13. Engel Structure
  14. Associated Lie Algebra Structure
  15. Action of Similitude Group on the Engel structure
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\gamma:I\rightarrow B$ be a regular retinal curve, $u:I\rightarrow {\mathbb S} B, ~ t\mapsto \frac{\dot{\gamma}(t)}{\|\dot{\gamma}(t)\|}$ its unit tangent vector field and $\Gamma: I\rightarrow M$ its horizontal lift, then $\Gamma(t)=\flat(u(t)), \text{for every } t$. Moreover, for any $q\in M_

Figures (2)

  • Figure 1: The Frenet approximation of a planar curve $\gamma$, shows that the best quadratic approximation is a parabola which expressed in the $(x,y)$ plane, has the shape $y=\kappa \frac{x^2}{2}$ near $\gamma(0)$ where $\kappa$ is its signed curvature. When the curve is not flat, the parabola is approximated by the osculating circle at $\gamma(0)$Parent. The radius of the osculating circle is the scale parameter $\delta$ of the Lie group $SIM(2)$.
  • Figure 2: The two-layer detection hierarchy. A visual stimulus (contrast) is first processed by orientation-selective cells via the $SE(2)$ -transform, producing a response in $H\subset L^2(SE(2))$. The operator $\mathcal{K}$ then maps this response to $L^2(\text{SIM}(2))$, where curvature-selective cells read out position, orientation, and curvature simultaneously. The composite map from contrast to curvature is equivalent to the $\text{SIM}(2)$ -wavelet transform with effective kernel $f$, when the orientation mother profile satisfies the $SIM(2)$ covariance.

Theorems & Definitions (38)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • ...and 28 more