Pinwheel crystallization in a dimension reduction model of visual cortical development
© Keil and Wolf; licensee BioMed Central Ltd. 2009
Published: 13 July 2009
The primary visual cortex (V1) of higher mammals contains a topographic representation of visual space in which neighborhood-preserving maps of several variables describing visual features such as position in visual space, line orientation, movement direction, and ocularity are embedded . It has been hypothesized that the complex spatial layouts of these representations can be interpreted as ground states of a smooth mapping of a high-dimensional space of visual stimulus features to an effectively two dimensional array of neurons [2, 3]. Competitive Hebbian models of cortical development have been widely used to numerically study the properties of such mappings [2–5], but no analytical results about their ground states have been obtained so far. A classical example of such dimension reducing mappings is the Elastic Network Model (EN), which was proposed in .
Here we use a perturbative approach to compute the ground states of the EN for the joint mapping of two visual features: (i) position in visual space, represented in a retinotopic map and (ii) line orientation, represented in an orientation preference map (OPM). In this framework, the EN incooporates a mapping from a four-dimensional feature space to the twodimensional cortical sheet of neurons. We show that the dynamics of both feature representations can be treated within a general theory for the stability of OPMs . We find various ground states as a function of the lateral intracortical interactions and external stimulus distribution properties. However, in all parameter regimes, the grounds states of the Elastic Network Model are either stripe-like, or crystalline representation of the two visual features. We present a complete phase diagram of the model, summarizing pattern selection. Analytical predictions are confirmed by direct numerical simulations. Our results question previous studies (see  and references therein) concluding that the EN correctly reproduces the spatially aperiodic arrangement of visual cortical processing modules.
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