Mutual consistency of multiple visual feature maps constrains combined map topology

The primary visual cortex (V1) is the first cortical area that processes visual information relayed from the thalamus. The topologies permitted in joint ocular dominance (OD), orientation preference (OP), and direction preference (DP) maps in V1 are considered, with the aim of finding a maximally symmetric periodic case that can serve as the basis for perturbations toward natural realizations. It is shown that mutual consistency of the maps selects just two possible such lattice structures, and that one of these is much closer to experiment than the other. This comprises a hexagonal lattice of alternating positive and negative OP singularities, with each unit cell or hypercolumn containing four such singularities, each of which radiates three DP discontinuities that follow OP contours and end at OP singularities of opposite sign. Other DP discontinuities emanate at 90 degrees to the midpoints of the ones that link OP singularities, and cross OP contours perpendicularly. These features explain experimentally observed relationships between DP discontinuities and OP contours, including sudden approximately 90-degree changes of direction in the former.

Figure 1

Experimental feature maps in V1. (a) Combined OD-OP map, reprinted with permission from Ref. [25]. The bold black lines are the OD borders, and the gray curves are iso-orientation contours. (b) Combined OP-DP map of adult cat, adapted with permission from Ref. [9]. The OP and PWCs are represented by the thin black contours and the blue stars, respectively. Thick red lines are DP fractures. Some examples of DP fractures that intersect iso-orientation contours at steep angles are marked by black arrows.

Figure 2

Schematic of the combined OD-OP-DP map in one hypercolumn containing four PWCs. The thin solid black lines represent the borders of the left and right OD stripes, within each of which there are two PWCs with opposite signs, as shown by the heavy dots. Dashed lines represent OP contours, and the thick green contour lines and arrows show DP fractures and adjacent direction preferences.

Figure 3

Schematic of possible hypercolumn topologies formed from lattice of regular polygons. The blue and red dots indicating positive and negative PWCs, respectively. (a) Triangular lattice, showing that it is impossible for all neighboring PWCs to have opposite signs. (b) Square lattice. The gray and white stripes bounded by dashed lines show a pair of left and right OD stripes. The large green square outlines a hypercolumn containing four pinwheels. (c) Hexagonal lattice showing two possible hypercolumn arrangements as green parallelograms. Blue arrows indicate distances between positive PWCs parallel (${\lambda }_{\parallel }$) and perpendicular (${\lambda }_{\perp }$) to the OD stripe borders, respectively.

Figure 4

Fracture geometry near a positive PWC. (a) OP (blue) and DP (red) vs azimuth around the PWC showing one fracture as a vertical arrow. (b) OP-DP map corresponding to frame (a), with arrows indicating DP and colors showing OP. (c) As for (a) but with three equally spaced fractures. (d) OP-DP map corresponding to frame (c).

Figure 5

Simulated map showing the relationship between OP map and DP fractures. Reprinted with permission from Ref. [46]. The OP is plotted by blue contour lines and the DP fractures are in black.

Figure 6

Schematic of how full DP coverage is achieved between adjacent PWCs linked by a fracture. The solid black line represents the DP fracture, the black arrows show the DP, and the color bar indicates the OP in degrees. The white dashed lines indicate loci of zero DP selectivity. (a) One DP fracture per OP pinwheel. The left pinwheel covers ${180}^{\circ }$ of DP ranging from ${90}^{\circ }$ to $-{90}^{\circ }$, and the right pinwheel covers the rest ${180}^{\circ }$ of DP. (b) Three DP fractures per OP pinwheel. The left pinwheel covers DP ranging from ${90}^{\circ }$ to ${150}^{\circ }, -{30}^{\circ }$ to ${30}^{\circ }$, and $-{150}^{\circ }$ to $-{90}^{\circ }$, and the right pinwheel covers the remaining three sectors of DP.

Figure 7

Experimental map showing DP fractures (red) on OP contours (black), adapted with permission from Ref. [11]. The numbered sites indicate some places where DP fractures intersect OP iso-orientation contours at steep angle. The blue circles highlight some locations where DP fractures fade out around their midpoints.

Figure 8

Schematic of square lattice with one DP fracture per PWC. Shading indicates left and right OD stripes (which is which does not matter) and dots (filled or unfilled) indicate PWCs of alternating sign. The solid black line is an assumed DP fracture, while the dotted lines are possible fractures emanating from the lower left PWC in the central square hypercolumn highlighted in red.

Figure 9

Combined feature maps. (a) OP-OD map corresponding to the hexagonal lattice with vertical OD stripes as shown in Fig. 3. Positive and negative PWCs are marked by $+$ and $-$ signs. Black dotted lines denote the OD borders and the white lines outline one hypercolumn. The color bar indicated the orientation preference in degrees. (b) OP-DP map of one hexagon extracted from (a). OP is color coded and DP is shown by arrows. The white lines indicate DP fractures and the dashed lines indicate zero DP selectivity. Positive and negative PWCs are marked. (c) Examples of DP fractures that parallel OP contours (solid lines) and intersect OP contours (dashed lines) overlaid on colored OP contours. The color bar indicates the OP in degrees.

Figure 10

Power spectrum of the hexagonal lattice of OP shown in Fig. 9. The color bar indicates the squared amplitude.

Figure 11

OD, OP, and DP selectivity in one hypercolumn with PWCs located at the vertices and $a=1$ mm. The red lines outline one hyper column and the dashed lines are the OD border. Four positive and negative OP pinwheels are marked by the $+$ and $-$ signs. (a) OD selectivity map with black and white indicate left and right OD respectively. (b) OP selectivity map with zero selectivity around PWCs and high selectivity elsewhere. (c) DP selectivity map. Zero selectivity occurs along the DP fractures (i.e., hexagon sides) and the perpendicular bisectors of these fractures.