From a theoretical perspective, two hypotheses have been proposed to solve the problem of pinwheel stability. (1) Pinwheels are stabilized by interactions of different features under the constraint of coverage optimization (Wolf & Geisel, 1998). (2) The spatial layout of OPM is shaped by intrinsic processes such as long-range intracortical interactions (Wolf, 2005).
The first hypothesis is based on the observed spatial relationships between different feature maps in the visual cortex (Crair et al., 1997b,a; Hübener et al., 1997; Löwel et al., 1998; Müller et al., 2000). Previous studies revealed a tendency for pinwheel center singularities of OPMs to lie in the center regions of the ocular dominance columns(ODCs), and for iso-orientation bands to cross ocular domi-nance borders at right angles (Figure 1.5). A general framework of dimension reduction (Durbin & Mitchison, 1990) has been proposed for understanding cortical mappings that preserve neighborhood relations in the feature space. To avoid functionally ‘blind spots’ in the visual field, the concept of coverage uni-formity constrains the maps to fill the input space with near-uniform density while maintaining continuity. This leads to maps where rapid changes in one feature component are correlated with slow changes in other components. Nu-merical studies further suggested that strong ocular dominance segregation can slow down the process of pinwheel annihilation in developmental models (Wolf &
Geisel, 1998).
However, recent quantitative studies (Kaschube, 2005; Kaschube et al., 2006) found a universal pinwheel density in adult animals of several species despite substantially different organizations of ocular dominance columns. This appears inconsistent with theoretical prediction that suggested a correlation between the numbers of pinwheels per hypercolumn and the degree of ocular dominance seg-regation in different species (for discussion see Wolf & Geisel (1998)). This raises the question whether the OPMs are stabilized by interaction with other maps, or whether the ubiquitous pinwheel structure is selected by a universal pattern forming process independent of the number of mapped features.
1.6 An Overview of my Work
The objective of this project is to develop a methodology for analysis and simu-lations of cortical dynamics from the perspective of nonlinear dynamical systems and applying this method to comprehensively study the problem of pinwheel stability in models of interacting columnar patterns. The model behavior was characterized in different dynamical regimes, for various system sizes, feature
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E F C
Figure 1.5: Segregated ODCs and OPM in strabismic cat V1 (A,B) Ac-tivity patterns for the left(A) and right eye (B) are complementary. Regions highly activated by left eye (dark regions in A, outlined in white) are only weakly activated by the right eye (light grey regions in B). (C) Orientation preference map obtained in the same area. (D) Superposition of the angle map and the out-lined borders of adjacent ocular dominance columns (white contours in A). (E) Histograms of intersection angles between iso-orientation and ocular dominance columns in the same animal. Note that intersection angles between 75 and 90 are most abundant in the original data. (F) Data from shifted maps: iso-orientation contours of one animal superimposed with the ocular dominance borders of an-other animal. Note in addition that the histograms are always flat after shifting the maps (Data not shown here). Modified from Löwel et al. (1998).
space dimensionalities, and visual stimulus distributions.
The phase transition to spontaneous pattern formation of visual maps was characterized by a stability analysis of the Kohonen model. The bifurcation diagrams were obtained numerically by varying the control parameter σ, which corresponds to the size of co-activated cortical domains. Large scale simulations were then performed in different dynamical regimes identified from the bifurcation diagrams.
In simulations the model cortex was initialized with an unselective state and was trained by random stimuli drawn from Gaussian distributions. We first con-sidered the development of orientation preference interacting only with retino-topy. The dynamics of OPMs was found to rely on the size of the system. In small systems of one hypercolumn, a checkerboard pattern consisting of four pinwheels was always maintained. However, in the larger systems the initially pinwheel-rich patterns were typically unstable. The kinetics of pinwheel anni-hilation was quantified by the average numbers of pinwheels per hypercolumn, which decayed below 2 in various parameter regimes.
In simulations of high-dimensional feature space models, more feature dimen-sions were included to test whether pinwheel annihilation could be stopped as suggested by the dimension reduction framework. Intriguingly, only two active feature dimensions were represented close to the bifurcation threshold. The other feature dimensions were suppressed and became represented only beyond a sec-ondary bifurcation point. Beyond this secsec-ondary bifurcation point, the generated patterns of different feature maps were either pinwheel-free stripes or a repetitive checkerboard pattern of pinwheel crystals.
To establish the robustness of this behavior we further compared simula-tions performed with different stimulus statistics. Similar results were found for spherical uniform distributions and for products of angular variables with circu-lar uniform distributions. We conclude that the dynamics of OPMs is generally unstable in current developmental models of interacting columnar systems.
The first part of this thesis is organized as follows: In Chapter 2, we first describe the model of Kohonen’s self-organizing feature mapping. Then we re-address the question of pattern formation by a linear stability analysis (Wolf et al., 2000) and derive the initial growth rate and the maximum unstable wave-length (related to the column spacing) of the emerging pattern. Based on this mathematical analysis, we designed a precisely controlled numerical method in Chapter 3. In Chapter 4, we study the dynamics of OPM interacting only with a retinotopic map. More feature dimensions are included in Chapter 5 to track the development of OPMs coupled to other feature maps. Finally, simulation re-sults with stimuli of non-Gaussian distributions are described in Chapter 6. We discuss the main findings of our study in Chapter 7.