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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.


Related in: MedlinePlus

Correlation spectra of orientation maps in macaque and tree shrew V1. a and b are from Niebur and Worgotter’s 1994 paper [33]: in a, the solid and dashed lines are spectra obtained by two different methods (direct measurement of correlations and Fourier analysis) from an experimental map obtained by Blasdel in macaque monkey, the power spectrum of which is displayed on b. Images c and d are from Schnabel’s 2008 thesis [30, p. 104]. Methods for obtaining c and d from measurements on Tree Shrews are explained precisely by Schnabel in [30, Sects. 5.3 and 5.4]. The green- and blue-shaded regions code for bootstrap confidence interval and 5 % significance level, respectively. The power spectrum in d has standard deviation around 0.2 in the unit displayed on the horizontal axis and determined by the location of the maximum; the mean and quadratic wavenumbers in this spectrum are in the intervals  and , respectively
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Fig2: Correlation spectra of orientation maps in macaque and tree shrew V1. a and b are from Niebur and Worgotter’s 1994 paper [33]: in a, the solid and dashed lines are spectra obtained by two different methods (direct measurement of correlations and Fourier analysis) from an experimental map obtained by Blasdel in macaque monkey, the power spectrum of which is displayed on b. Images c and d are from Schnabel’s 2008 thesis [30, p. 104]. Methods for obtaining c and d from measurements on Tree Shrews are explained precisely by Schnabel in [30, Sects. 5.3 and 5.4]. The green- and blue-shaded regions code for bootstrap confidence interval and 5 % significance level, respectively. The power spectrum in d has standard deviation around 0.2 in the unit displayed on the horizontal axis and determined by the location of the maximum; the mean and quadratic wavenumbers in this spectrum are in the intervals and , respectively

Mentions: In spite of this, we shall stick to the geometry of maps sampled from Gaussian Random Fields (GRFs) in our short paper. We have several reasons for doing so. A first remark is that a better understanding of maps sampled from them can be helpful in understanding the general principles underlying more realistic models, or helpful in suggesting some such principles. A second remark is that with the naked eye, it is difficult to see any difference between some maps sampled from GRFs and actual visual maps (see Fig. 2), and that there is a striking likeness between some theorems on GRFs and some properties measured in V1. A third is that precise mathematical results on GRFs can be used for testing how close this likeness is, and to make the relationship between GRFs and mature V1 maps clearer. Fig. 2


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

Afgoustidis A - J Math Neurosci (2015)

Correlation spectra of orientation maps in macaque and tree shrew V1. a and b are from Niebur and Worgotter’s 1994 paper [33]: in a, the solid and dashed lines are spectra obtained by two different methods (direct measurement of correlations and Fourier analysis) from an experimental map obtained by Blasdel in macaque monkey, the power spectrum of which is displayed on b. Images c and d are from Schnabel’s 2008 thesis [30, p. 104]. Methods for obtaining c and d from measurements on Tree Shrews are explained precisely by Schnabel in [30, Sects. 5.3 and 5.4]. The green- and blue-shaded regions code for bootstrap confidence interval and 5 % significance level, respectively. The power spectrum in d has standard deviation around 0.2 in the unit displayed on the horizontal axis and determined by the location of the maximum; the mean and quadratic wavenumbers in this spectrum are in the intervals  and , respectively
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4388110&req=5

Fig2: Correlation spectra of orientation maps in macaque and tree shrew V1. a and b are from Niebur and Worgotter’s 1994 paper [33]: in a, the solid and dashed lines are spectra obtained by two different methods (direct measurement of correlations and Fourier analysis) from an experimental map obtained by Blasdel in macaque monkey, the power spectrum of which is displayed on b. Images c and d are from Schnabel’s 2008 thesis [30, p. 104]. Methods for obtaining c and d from measurements on Tree Shrews are explained precisely by Schnabel in [30, Sects. 5.3 and 5.4]. The green- and blue-shaded regions code for bootstrap confidence interval and 5 % significance level, respectively. The power spectrum in d has standard deviation around 0.2 in the unit displayed on the horizontal axis and determined by the location of the maximum; the mean and quadratic wavenumbers in this spectrum are in the intervals and , respectively
Mentions: In spite of this, we shall stick to the geometry of maps sampled from Gaussian Random Fields (GRFs) in our short paper. We have several reasons for doing so. A first remark is that a better understanding of maps sampled from them can be helpful in understanding the general principles underlying more realistic models, or helpful in suggesting some such principles. A second remark is that with the naked eye, it is difficult to see any difference between some maps sampled from GRFs and actual visual maps (see Fig. 2), and that there is a striking likeness between some theorems on GRFs and some properties measured in V1. A third is that precise mathematical results on GRFs can be used for testing how close this likeness is, and to make the relationship between GRFs and mature V1 maps clearer. Fig. 2

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