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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

Variance is a decreasing function of spectral thinness. This is a plot of the variance of the random variable recording the number of times a given orientation is present on a straight line segment of fixed length. We considered here Invariant Gaussian Random Fields with uniform power spectra, and plotted the variance as a function of spectral width. For each percentage of the mean wavenumber, we displayed two outputs to give an idea of the attained precision. A low value for variance, here expressed in the unit given by the square of the expectation, corresponds to a field whose hypercolumns have relatively constant size across the resulting orientation map; the results displayed here show that a very regular hypercolumnar organization is quite compatible with stochastic modeling, and is a direct consequence of the spectral thinness condition found in models. Moreover, the horizontal slope at zero shows that as regards the global properties of quasi-periodic maps, there is very little difference between a theoretically ideal monochromaticity and a more realistic (and more model-independent) spectral thinness
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Fig4: Variance is a decreasing function of spectral thinness. This is a plot of the variance of the random variable recording the number of times a given orientation is present on a straight line segment of fixed length. We considered here Invariant Gaussian Random Fields with uniform power spectra, and plotted the variance as a function of spectral width. For each percentage of the mean wavenumber, we displayed two outputs to give an idea of the attained precision. A low value for variance, here expressed in the unit given by the square of the expectation, corresponds to a field whose hypercolumns have relatively constant size across the resulting orientation map; the results displayed here show that a very regular hypercolumnar organization is quite compatible with stochastic modeling, and is a direct consequence of the spectral thinness condition found in models. Moreover, the horizontal slope at zero shows that as regards the global properties of quasi-periodic maps, there is very little difference between a theoretically ideal monochromaticity and a more realistic (and more model-independent) spectral thinness

Mentions: In order to keep the numerical errors from masking the “exact” effect of thickening the spectrum, we forced the software to optimize its calculation strategy (adaptive Monte-Carlo integration), detecting oscillations in the integrand and adapting the sampling requirements, and we extended evaluation time beyond the usual limits (by dropping the in-built restrictions on the recursion depths). When the difference between successive evaluations was tamed, this yielded the variance curve displayed on Fig. 4. Fig. 4


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

Afgoustidis A - J Math Neurosci (2015)

Variance is a decreasing function of spectral thinness. This is a plot of the variance of the random variable recording the number of times a given orientation is present on a straight line segment of fixed length. We considered here Invariant Gaussian Random Fields with uniform power spectra, and plotted the variance as a function of spectral width. For each percentage of the mean wavenumber, we displayed two outputs to give an idea of the attained precision. A low value for variance, here expressed in the unit given by the square of the expectation, corresponds to a field whose hypercolumns have relatively constant size across the resulting orientation map; the results displayed here show that a very regular hypercolumnar organization is quite compatible with stochastic modeling, and is a direct consequence of the spectral thinness condition found in models. Moreover, the horizontal slope at zero shows that as regards the global properties of quasi-periodic maps, there is very little difference between a theoretically ideal monochromaticity and a more realistic (and more model-independent) spectral thinness
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig4: Variance is a decreasing function of spectral thinness. This is a plot of the variance of the random variable recording the number of times a given orientation is present on a straight line segment of fixed length. We considered here Invariant Gaussian Random Fields with uniform power spectra, and plotted the variance as a function of spectral width. For each percentage of the mean wavenumber, we displayed two outputs to give an idea of the attained precision. A low value for variance, here expressed in the unit given by the square of the expectation, corresponds to a field whose hypercolumns have relatively constant size across the resulting orientation map; the results displayed here show that a very regular hypercolumnar organization is quite compatible with stochastic modeling, and is a direct consequence of the spectral thinness condition found in models. Moreover, the horizontal slope at zero shows that as regards the global properties of quasi-periodic maps, there is very little difference between a theoretically ideal monochromaticity and a more realistic (and more model-independent) spectral thinness
Mentions: In order to keep the numerical errors from masking the “exact” effect of thickening the spectrum, we forced the software to optimize its calculation strategy (adaptive Monte-Carlo integration), detecting oscillations in the integrand and adapting the sampling requirements, and we extended evaluation time beyond the usual limits (by dropping the in-built restrictions on the recursion depths). When the difference between successive evaluations was tamed, this yielded the variance curve displayed on Fig. 4. Fig. 4

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