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Monochromaticity of orientation maps in v1 implies minimum variance for hypercolumn size.

Afgoustidis A - J Math Neurosci (2015)

Bottom Line: The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout.We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance.The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns.

View Article: PubMed Central - PubMed

Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7 Denis Diderot, 75013 Paris, France.

ABSTRACT
In the primary visual cortex of many mammals, the processing of sensory information involves recognizing stimuli orientations. The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout. This repetitive pattern is understood to be fundamental for basic non-local aspects of vision, like the perception of contours, but important questions remain about its development and function. We focus here on Gaussian Random Fields, which provide a good description of the initial stage of orientation map development and, in spite of shortcomings we will recall, a computable framework for discussing general principles underlying the geometry of mature maps. We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance. Referring to studies by Wolf, Geisel, Kaschube, Schnabel, and coworkers, we also show that spectral thinness is not an essential ingredient to obtain a pinwheel density of π, whereas it appears as a signature of Euclidean symmetry. The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns. A measurement of this property in real maps is in principle possible, and comparison with the results in our paper could help establish the role of our minimum variance hypothesis in the development process.

No MeSH data available.


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Computer-generated map, sampled from a monochromatic field. This figure shows an orientation map which we have drawn from a simulated Invariant Gaussian Random Field with circular power spectrum. We used 100 plane waves with frequency vectors at the vertices of a regular polygon inscribed in a circle, and random Gaussian weights (see the Appendix); with respect to the unit of length displayed on the x- and y-axes, the wavelength of the generating plane waves is
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Fig3: Computer-generated map, sampled from a monochromatic field. This figure shows an orientation map which we have drawn from a simulated Invariant Gaussian Random Field with circular power spectrum. We used 100 plane waves with frequency vectors at the vertices of a regular polygon inscribed in a circle, and random Gaussian weights (see the Appendix); with respect to the unit of length displayed on the x- and y-axes, the wavelength of the generating plane waves is

Mentions: Although this is far from being an infinitely thin spectrum, it is not absurd to look at the extreme situation where we impose the spectral thinness to be zero. Figure 3 shows a map sampled from a monochromatic invariant GRF, in which Γ is one of the maps of the previous paragraph, in other words the inverse Fourier transform of the Dirac distribution on a circle: monochromatic (or almost monochromatic) invariant GRFs yield quite realistic-looking maps, at least to the naked eye. Fig. 3


Monochromaticity of orientation maps in v1 implies minimum variance for hypercolumn size.

Afgoustidis A - J Math Neurosci (2015)

Computer-generated map, sampled from a monochromatic field. This figure shows an orientation map which we have drawn from a simulated Invariant Gaussian Random Field with circular power spectrum. We used 100 plane waves with frequency vectors at the vertices of a regular polygon inscribed in a circle, and random Gaussian weights (see the Appendix); with respect to the unit of length displayed on the x- and y-axes, the wavelength of the generating plane waves is
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4388110&req=5

Fig3: Computer-generated map, sampled from a monochromatic field. This figure shows an orientation map which we have drawn from a simulated Invariant Gaussian Random Field with circular power spectrum. We used 100 plane waves with frequency vectors at the vertices of a regular polygon inscribed in a circle, and random Gaussian weights (see the Appendix); with respect to the unit of length displayed on the x- and y-axes, the wavelength of the generating plane waves is
Mentions: Although this is far from being an infinitely thin spectrum, it is not absurd to look at the extreme situation where we impose the spectral thinness to be zero. Figure 3 shows a map sampled from a monochromatic invariant GRF, in which Γ is one of the maps of the previous paragraph, in other words the inverse Fourier transform of the Dirac distribution on a circle: monochromatic (or almost monochromatic) invariant GRFs yield quite realistic-looking maps, at least to the naked eye. Fig. 3

Bottom Line: The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout.We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance.The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns.

View Article: PubMed Central - PubMed

Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7 Denis Diderot, 75013 Paris, France.

ABSTRACT
In the primary visual cortex of many mammals, the processing of sensory information involves recognizing stimuli orientations. The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout. This repetitive pattern is understood to be fundamental for basic non-local aspects of vision, like the perception of contours, but important questions remain about its development and function. We focus here on Gaussian Random Fields, which provide a good description of the initial stage of orientation map development and, in spite of shortcomings we will recall, a computable framework for discussing general principles underlying the geometry of mature maps. We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance. Referring to studies by Wolf, Geisel, Kaschube, Schnabel, and coworkers, we also show that spectral thinness is not an essential ingredient to obtain a pinwheel density of π, whereas it appears as a signature of Euclidean symmetry. The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns. A measurement of this property in real maps is in principle possible, and comparison with the results in our paper could help establish the role of our minimum variance hypothesis in the development process.

No MeSH data available.


Related in: MedlinePlus