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Orientation Maps in V1 and Non-Euclidean Geometry.

Afgoustidis A - J Math Neurosci (2015)

Bottom Line: In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others.In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing.Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry.

View Article: PubMed Central - PubMed

Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Universite Paris 7 Denis Diderot, 75013, Paris, France, alexandre.afgoustidis@imj-prg.fr.

ABSTRACT
In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non-Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.

No MeSH data available.


The “composite distance” to a point of the boundary. Definition of the quantity  if x is a point of  and b a point of its boundary:  is the horocycle through x which is tangent to the boundary at b, and  is the segment joining the origin O to the point on  which is diametrically opposite b; the number  is, up to a sign, the hyperbolic length of this segment
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Fig3: The “composite distance” to a point of the boundary. Definition of the quantity if x is a point of and b a point of its boundary: is the horocycle through x which is tangent to the boundary at b, and is the segment joining the origin O to the point on which is diametrically opposite b; the number is, up to a sign, the hyperbolic length of this segment

Mentions: The indications we gave for the proof of the Iwasawa decomposition led Helgason to call a (signed) composite distance; the definition and its relationship with the hyperbolic distance are illustrated on Fig. 3. Fig. 3


Orientation Maps in V1 and Non-Euclidean Geometry.

Afgoustidis A - J Math Neurosci (2015)

The “composite distance” to a point of the boundary. Definition of the quantity  if x is a point of  and b a point of its boundary:  is the horocycle through x which is tangent to the boundary at b, and  is the segment joining the origin O to the point on  which is diametrically opposite b; the number  is, up to a sign, the hyperbolic length of this segment
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4469697&req=5

Fig3: The “composite distance” to a point of the boundary. Definition of the quantity if x is a point of and b a point of its boundary: is the horocycle through x which is tangent to the boundary at b, and is the segment joining the origin O to the point on which is diametrically opposite b; the number is, up to a sign, the hyperbolic length of this segment
Mentions: The indications we gave for the proof of the Iwasawa decomposition led Helgason to call a (signed) composite distance; the definition and its relationship with the hyperbolic distance are illustrated on Fig. 3. Fig. 3

Bottom Line: In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others.In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing.Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry.

View Article: PubMed Central - PubMed

Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Universite Paris 7 Denis Diderot, 75013, Paris, France, alexandre.afgoustidis@imj-prg.fr.

ABSTRACT
In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non-Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.

No MeSH data available.