Coordinated Optimization of Visual Cortical Maps (II) Numerical Studies

Abstract

In the juvenile brain, the synaptic architecture of the visual cortex remains in a state of flux for months after the natural onset of vision and the initial emergence of feature selectivity in visual cortical neurons. It is an attractive hypothesis that visual cortical architecture is shaped during this extended period of juvenile plasticity by the coordinated optimization of multiple visual cortical maps such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we introduced a class of analytically tractable coordinated optimization models and solved representative examples, in which a spatially complex organization of the OP map is induced by interactions between the maps. We found that these solutions near symmetry breaking threshold predict a highly ordered map layout. Here we examine the time course of the convergence towards attractor states and optima of these models. In particular, we determine the timescales on which map optimization takes place and how these timescales can be compared to those of visual cortical development and plasticity. We also assess whether our models exhibit biologically more realistic, spatially irregular solutions at a finite distance from threshold, when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. We show that, although maps typically undergo substantial rearrangement, no other solutions than pinwheel crystals and stripes dominate in the emerging layouts. Pinwheel crystallization takes place on a rather short timescale and can also occur for detuned wavelengths of different maps. Our numerical results thus support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the architecture of the visual cortex. We discuss several alternative scenarios that may improve the agreement between model solutions and biological observations.

Random Wiring, Ganglion Cell Mosaics, and the Functional Architecture of the Visual Cortex

“…Interestingly, an identical layout of iso-orientation domains was constructed by Braitenberg et al [ 91 ] based on an the idea that orientation preference is generated by discrete centers of inhibition in V1. It was also found by Reich et al to solve a symmetry defined class of models for the self-organization of iso-orientation domains [ 92 , 93 ]. Fig 3C shows the differences in preferred angle between the unfiltered domain layout of the Moiré interference pattern and its low frequency contribution, together with a histogram of the differences.…”

“…D The standard deviation of pinwheel density estimates for increasing subregion size. Red line shows a power law fit to the experimental data ( γ = 0.5, [ 23 ]), purple line indicates a fit to the perfectly ordered hexagonal pinwheel arrangement ( γ = 0.75, [ 23 , 92 , 93 ]). E Nearest neighbor (NN) distances for pinwheels irrespective of topological charge.…”

“…Blue and purple lines indicate disorder strengths shown on the left. The white line marks the theoretically predicted distance of NN pinwheels (2/3Λ c ) for vanishing disorder [ 92 , 93 ]. F same as E for pinwheels of equal charge, data green curve.…”

“…A variety of self-organization models on all levels of biological detail have been shown to generate orientation domains with hexagonal arrangements, e.g. [ 14 , 92 , 93 , 103 , 104 , 106 , 107 ] (notably including the earliest theory for the self-organization of orientation preference by von der Malsburg in 1973). Thus, hexagonal order, even if present, would not provide specific evidence in favor of Moiré interference between RGCs.…”

Monochromaticity of Orientation Maps in V1 Implies Minimum Variance for Hypercolumn Size

“…Quite strikingly, it has been observed [ 30 , 33 ] that one needs only add a spectral thinness condition to obtain maps that seem to have the right qualitative (a hyper columnar, quasi-periodic organization) and quantitative properties (a value of π for pinwheel density). These mathematical features stand out among theoretical models for orientation maps as producing a nice quasi-periodicity, with roughly repetitive “hypercolumns” of about the same size that have the same structure, as opposed to a strictly periodic crystal-like arrangement (see [ 14 , 21 ], compare [ 34 , 35 ]). The aim of this short note is to clarify the importance of this spectral thinness condition for getting a quasi-periodic “hypercolumnar” arrangement on the one hand, a pinwheel density of π on the other.…”