The pattern formation in orientation and ocular dominance columns is one of the most investigated problems in the brain. From a known cortical structure, we build spinlike Hamiltonian models with long-range interactions of the Mexican hat type. These Hamiltonian models allow a coherent interpretation of the diverse phenomena in the visual map formation with the help of relaxation dynamics of spin systems. In particular, we explain various phenomena of self-organization in orientation and ocular dominance map formation including the pinwheel annihilation and its dependency on the columnar wave vector and boundary conditions.

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In animal experiments, the observed orientation preference (OP) and ocular dominance (OD) columns in the visual cortex of the brain show various pattern types. Here, we show that the different visual map formations in various species are due to the crossover behavior in anisotropic systems composed of orientational and scalar components such as easy-plane Heisenberg models. We predict the transition boundary between different pattern types with the anisotropy as a main bifurcation parameter, which is consistent with experimental observations. PACS numbers: 75.10.Hk, 89.75.Fb The highly ordered structure in the mammalian visual cortex has attracted much attention from theoretical neurobiologists and has been thoroughly studied with the expectation of providing the basis for neural dynamics and computational models. Though most models of the visual map formation are based on common postulates, such as Hebbian synapses, connections or competitions between neighboring neurons and synaptic normalization, there are quite a number of successful models with unique mechanisms [1,2]. The Hamiltonian models with spin variables were proposed for the visual map formation with a striking analogy with the physical systems, such as magnetism [3,4]. Recently, the characteristics of visual maps are systematically explored through the statistical properties of vortices in magnetism [5]. The spin-like Hamiltonian models represent essential ingredients of neural interactions in the visual map formation without paying much attention to particular neural control mechanisms and can be shown to exhibit common statistical properties of the vortex formation as in other development models [6].As vast experimental data on visual maps are accumulated, the various patterns in visual cortices of different animals have drawn much interest with the expectation of testing different neural models experimentally [7,8,9,10,11,12,13]. The observed visual patterns can be classified as at least three different types. In macaque monkeys, the OD columns form parallel bands of regular spacing with relatively few branching points that are mainly oriented perpendicular to area boundaries [7]. The degree of OD segregation is strong and the typical average spacing of OD, Λ OD , is larger than that of the OP columns, Λ OP (Λ OD > Λ OP ) [8]. In cats and ferrets, OD columns form an array of beaded bands exhibiting only a weak tendency of elongation perpendicular to area boundaries [9,10,11]. The degree of OD segregation is intermediate with Λ OD < Λ OP [12]. In the case of tree shrews, the OD segregation is very weak or absent, while more stripe-like patterns are observed in extensive OP columnar regions with low densities of orientation centers [13].Some experimental works show that the OP and OD patterns are structurally correlated. The singular points, so-called pinwheels, in OP columns tend to align with the centers of OD bands and the iso-orientation contours intersect the borders of OD bands at right or steep angles [8]. Such correlations between two ...