Invariants for primary visual cortex
type: Item
instance of: scholarly article
title: Invariants for primary visual cortex
zbMATH Open document ID: 1405.53038
author: Nadiia Konovenko
author: Valentin Lychagin
published in: Differential Geometry and its Applications
publication date: 2018-08-07T00:00:00Z
review text: The authors consider a 2-dimensional manifold \(M\) as a model for the retina. Simple neurons detect any oriented line in the tangent or cotangent plane in an arbitrary point \(a\) of the retina. In the paper under review the authors work with oriented lines in cotangent planes to \(M\), which form a 3-dimensional contact manifold \(S (M) = (T^*M\setminus 0) /R^+\), called the spherization of the cotangent bundle, replaced by the projectivization of the cotangent bundle \(P(M)=(T^*M\setminus 0) /\mathbb{R}^*\), when the lines are nonoriented. According to [\textit{D. Alekseevsky}, ``Light, eye, brain. Geometric problems in vision'', in: Proceedings of the 8th seminar on geometry and topology, GTS. Teheran: Amirkabir University of Technology. 10--32 (2015); \textit{J. Petitot}, Chaos Solitons Fractals 50, 75--92 (2013; Zbl 1310.92012)], simple neurons operate as filters and the optic signal can be considered as convolution with the Gaussian or Gabor filters. Thus, the manifold \(M\) has not only a contact structure, but also a metric or, at least, a conformal structure. These structures naturally related to primary visual cortex, as well as rational differential invariants of distributions under the action of the structure group, are treated in the paper under review. For example, it is shown that Euclidean differential invariants of (non)oriented distributions on the plane are generated by basic differential invariants \(J_1 = u_x \cos u + u_y \sin u , J_2 = -u_x \sin u + u_y \cos u\) and invariant derivations \(\nabla_1 = \cos u\frac{d}{dx} + \sin u\frac{d}{dy}, \nabla_2 = -\sin u\frac{d}{dx} + \cos u\frac{d}{dy}\), satisfying the relation \(J_{12} - J_{21} + J^2_1 + J^2_2 = 0,\) where \(J_{kl}=\nabla_k(J_l), k, l=1, 2\). In other words, any Euclidean differential invariant is a rational function of invariants \(\nabla^\mu(J_1)\) and \(\nabla^\mu(J_2)\), where \(\mu=(m_1,m_2)\) is a multi-index and \(\nabla^\mu=\nabla_1^{m_1}\nabla_2^{m_2}\). Moreover, the authors …
Mathematics Subject Classification ID: 53B50
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 58A17
zbMATH DE Number: 6914488
zbMATH Keywords: cotangent bundle
zbMATH Keywords: differential invariants
zbMATH Keywords: orbit
zbMATH Keywords: retina
reviewed by: S. L. Druţă-Romaniuc
MaRDI profile type: Publication
Wikidata QID: Q115355123
full work available at URL: j.difgeo.2018.04.009
OpenAlex ID: W2809528129
cites work: The visual cortex is a contact bundle
cites work: Neurogeometry of neural functional architectures
cites work: Invariants of projective actions and their application to recognition of fingerprints
cites work: Möbius invariants in image recognition
cites work: Geometry of differential equations
cites work: Global Lie-Tresse theorem
cites work: Q5735629
DOI: 10.1016/J.DIFGEO.2018.04.009
description: scientific article; zbMATH DE number 6914488