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Conformal models for hypercolumns in the primary visual cortex V1

D. V. Alekseevsky, A. Spiro

Abstract

We propose a differential geometric model of hypercolumns in the primary visual cortex V1 that combines features of the symplectic model of the primary visual cortex by A. Sarti, G. Citti and J. Petitot and of the spherical model of hypercolumns by P. Bressloff and J. Cowan. The model is based on classical results in Conformal Geometry.

Conformal models for hypercolumns in the primary visual cortex V1

Abstract

We propose a differential geometric model of hypercolumns in the primary visual cortex V1 that combines features of the symplectic model of the primary visual cortex by A. Sarti, G. Citti and J. Petitot and of the spherical model of hypercolumns by P. Bressloff and J. Cowan. The model is based on classical results in Conformal Geometry.

Paper Structure

This paper contains 15 sections, 1 theorem, 51 equations.

Table of Contents

  1. Hoffman's pioneering model of the V1 cortex
  2. Petitot's contact model of the V1 cortex
  3. Sarti, Citti and Petitot's symplectic model
  4. Bressloff and Cowan's spherical model for the hypercolumns of the V1 cortex
  5. A short introduction to the conformal geometry of the sphere
  6. The Riemannian spinor model of the conformal sphere
  7. The stereographic projections of the sphere
  8. Basics of conformal geometry of the sphere
  9. The conformal model and the reduced model of a hypercolumn
  10. The conformal model as an extension of Bressloff-Cowan's model
  11. The reduced model
  12. The "globalised" reduced model
  13. Relations with Sarti, Citti and Petitot's model and possible developments
  14. A comparison between the reduced model and Sarti, Citti and Petitot's symplectic model
  15. Reduced conformal models for the V1 cortex and a modification of Sarti, Citti and Petitot's symplectic model

Key Result

Theorem 6.1

Let $\rho>0$ be a constant such that all elements $A = \left(\smallmatrix 1 0\\ c 1 \endsmallmatrix\right)$ of the compact subset $K^- \subset G^-$ satisfy $|c | < \rho$. For any given $\varepsilon >0$, the disc $\Delta_\frac{\varepsilon}{1 + \rho \varepsilon}\subset \mathbb{C} \simeq \widehat{

Theorems & Definitions (2)

  • Theorem 6.1
  • proof