Systematic inhomogeneity

Not all properties of orientation maps are universal. Indeed, distinct map designs are consistent with the observed pinwheel statistics. For instance, the tree shrew differs from the other species by its often stripe-like and thus pinwheel sparse orga-nization of orientation columns along the V1/V2 border region and at the caudal pole. Fig. 4.11a exemplifies this by showing the map of local pinwheel density ρ(x) for the tree shrew orientation map from Fig. 4.1c (see Methods). Local densi-tiesρ(x)varied from below 1 at the V1/V2 border to values larger than 4 in more central representations. Densities varied comparably in tree shrew and ferret maps (p = 0.28, permutation test, Fig. 4.11b) and exhibited indistinguishable distribu-tions across all maps (Fig. 4.11c). However, in the tree shrew, pinwheel-sparse and pinwheel-rich regions covaried in maps from different individuals, whereas in the ferret, such regions were less related in different animals. This is revealed by the maps of systematic variation of local pinwheel densityρ_syst(x) for the tree shrew (Fig. 4.11d) and the ferret (Fig. 4.11e) calculated by superimposing and averag-ing maps of local pinwheel densityρ(x)from different animals. In the tree shrew, pinwheel densities were systematically lower than average along the V1/V2 bor-der and at the caudal pole (ρ_syst(x) < 2.5), and, compensating for this, higher than average in the representation of near central eccentricities (ρ_syst(x) > 3.5).

The histogram ofρ_syst(x) was significantly wider for the tree shrew (Fig. 4.11f, b, p = 0.005, permutation test). In contrast, distributions of nearest neighbor dis-tances of pinwheels were very similar in the two species (Fig. 4.11g, h) and also in the galago (data not shown). The histograms of these distances are statistics sen-sitive to the local arrangement of pinwheels indicating that despite the different systematic organization, local map properties are remarkably similar in the two species.

Figure 4.11: Species specific properties of pinwheel densities. a, Map of local pinwheel density ρ(x)(grey scale coded, see Methods) with pinwheel locations superimposed (blue diamonds) for the orientation map from Fig. 4.1c (mean pinwheel density,ρ = 3.08; SD, 0.63). b, SDs of local pinwheel densityρ(x)(left; mean over hemispheres; black bars indicate s.e.m.) and of systematic variation of local pinwheel densityρ_syst(x)(right; from N=26 hemispheres; fer-ret, mean and s.e.m. calculated from10⁴ subsets (N=26) randomly selected out of the N=82 hemispheres). c, Histograms of local pinwheel densitiesρ(x)from all hemispheres. d, e, Maps of systematic variation of local pinwheel densityρ_syst(x) = hρ(x)i for the tree shrew (d) and ferret (e) calculated by superimposing and averaging mapsρ(x)from all hemispheres at corre-sponding cortical locations (see Methods; blue scale coded; contour line drawn at average). f, Histograms ofρ_syst(x)(ferret, average histogram for subsets from b, right). g, h, Distributions of nearest neighbor distancesh_totbetween pinwheels of arbitrary topological charge (g), and of distancesh++,−−between pinwheels of equal (dashed) andh+−,−+of opposite charge (solid) (h).

4.5 Conclusion

We found that several properties of visual cortical orientation maps are universal in the ferret, tree shrew and galago. Relative to the typical distance of columns which is specific for a given species, pinwheels occurred with a constant average density.

Furthermore, in the three species, pinwheels exhibited an equal degree of spatial variation and a very similar statistics of nearest neighbor distances. This appears in contrast to the substantial differences reported for other columnar systems such as ocular dominance which is organized into regular blobs in the galago [110], in domains of rather irregular shape and size in the ferret [102], and in laminae in the tree shrew [42]. The universal organization of orientation columns is reminis-cent of many examples of complex systems where structures within a particular universality class are selected by self-organization despite differences in details of the microscopic dynamics [23]. In the following chapter, we therefore analyze the pinwheel statistics in solutions of a model formalizing the idea of self-organization

of orientation columns in the visual cortex.

Eyes are round.

(Claudia Benze)

5.1 Motivation

What mechanism can explain the observed universality of pinwheel statistics in the visual cortex? A common hypothesis states that random forces might form the structure of orientation maps. However, this hypothesis is hard to reconcile with our observation that pinwheel densities cannot be reproduced by random maps with second order statistics identical to real maps. Also genetic factors seem unlikely taking into account that the investigated species are separated for more than 30 million years of evolution. And sensory instruction, if prevailing, would rather counteract a universal organization given the substantial differences in their upstream visual systems in the different animals. An alternative explanation is of-fered by self-organization of cortical circuitry. Self-organization has been observed to robustly produce large scale structures in various complex systems. Often, the class of pattern emerging depends on fundamental system properties such as sym-metries rather than on the system specific details. Its formation can therefore often be described by an effective model incorporating only these properties. The robust occurrence of a universal pinwheel statistics suggests that pinwheel formation may be explained by a simple model neglecting many details of cortical connectivity.

In analogy to the theory of complex systems we therefore adopted a generalized Swift-Hohenberg model of the form Eq. (2.5) with nonlinearities derived from key features of visual cortical organization. The model includes the effects of long-range intracortical connections between columns of similar orientation preference.

We study the average pinwheel density, the count statistics of pinwheels, and their nearest neighbor distributions for solutions of the generalized Swift-Hohen-berg model (2.5) close to the bifurcation point. In the limit of infinite interaction range, we calculate the average pinwheel density analytically. The results are checked by numerical simulations away from the bifurcation point. We find that for a realistic range of nonlocal interaction, the model reproduces the observed pinwheel statistics quantitatively.

5.2 Methods