Mutual Consistency of Multiple Visual Feature Maps Constrains Combined Map Topology

To allow for maximal symmetry, we first approximate OD stripes as being parallel and of equal width, with a pair of such stripes in each hypercolumn. Each OD stripe must then contain two PWCs of opposite sign, located close to the midlines of the hypercolumns, each having PWCs of opposite sign as its nearest neighbors both within and outside the hypercolumn.

The arrangement of hypercolumns into a lattice is restricted by the requirement that PWCs alternate in sign. Only triangular, square, and hexagonal lattices of PWCs are topologically distinct in two dimensions (2D). Of these, the triangular lattice cannot satisfy the need for PWC signs to alternate, as seen in Fig. 3(A), a situation that is exactly analogous to frustration in a triangular lattice of antiferromagnetic spins [3, 56]. Therefore, we eliminate the triangular lattice from further consideration. On the other hand, the square and hexagonal lattices do allow PWC signs to alternate and a number of studies have used them to model various aspects of V1 structure, function, and activity, including during visual hallucinations [12, 15, 13, 54].

For the square and hexagonal topologies, we show the possible orientations of OD stripes as gray and white stripes in Fig. 3(B) and (C). Aside from rotations and translations that leave the system unchanged, there is only one topologically distinct possibility in the square case, with a hypercolumn outlined in the green square, which contains four pinwheels in Fig. 3(B), and OD bands as shaded and labeled. For the hexagonal case there are two topologically distinct OD-OP maps whose corresponding hypercolumns are shown as green parallelograms in Fig. 3(C), corresponding to horizontal and vertical OD stripes, respectively. In the horizontal case, PWCs lie exactly on the midline of the OD stripes shown, but these are narrow, each with a width of $a\sqrt{3}/2$, where $a$ is the PWC separation. In the vertical case, the width of each OD stripe is $3a/2$ and the PWCs are placed $a/4$ either side of the midline. Previous studies [4, 31, 44] revealed that PWCs tend to lie near the midlines of OD stripes. More precisely, the mean distance between the midline of the OD stripe and the nearest PWCs is around 100 $\mu$m in cat, which is around $1/5$ of the single OD stripe width [18]. Hence, the latter of the above alternatives better matches with the experiments, a point we consider further below. Additionally, the OP-OD topology of our hexagonal lattice also closely matches with previous findings [41, 48], which have predicted (also have justified on experimental OP maps) that columns with the same OP are approximately arranged in perturbed hexagonal lattice (i.e., not perfectly regular).

In both square and hexagonal cases, the distances between pinwheels with opposite signs are shorter than the distances between PWCs of the same sign, which matches with the experimental findings in [42]. Additionally, the OP-OD topologies in the two cases would not be affected if we stretch or compress the polygons in the vertical or horizontal direction.

We now discuss constraints on the combined OP-DP map both locally and across square and hexagonal lattices.

As noted above, OP for oriented edges strongly constrains DP, because only the component of motion perpendicular to an edge is detected, at least by simple cells in V1, so DP and OP angles are always perpendicular [35, 50, 53, 52, 57]. An important difference exists because OP only has a range of 180^∘ because rotation of an edge through this angle leaves it unchanged; in contrast, DP covers the full 360^∘ range of possible directions of motion of edges.

Figure 4(A) shows the variation of OP vs. azimuth around a positive PWC, starting at 0^∘ and increasing linearly to $180^{\circ}$, which is equivalent to 0^∘. (Note that all angles can be globally incremented by any fixed amount without changing our arguments.) If DP correspondingly starts at $90^{\circ}$ it reaches $270^{\circ}$ (i.e., covers half of the full DP cycle of $180^{\circ}$) after a complete circuit around the PWC, meaning that it is pointing in the opposite direction and there must be a discontinuity of 180^∘ at that point — a DP fracture — as shown in Fig. 4(B). Although the fracture could equally well occur at some other OP angle, its existence is inescapable. Figures 4(C) and (D) show another possibility, in which three fractures occur during the circuit around the PWC. In general, the total rotation of DP during a circuit of a PWC must be an integer multiple of 360^∘, so if there are $n$ fractures, we must have

where $m$ is an integer. This equation can only be satisfied for odd $n$, which confirms that at least one fracture must emanate from each PWC, consistent with statistics of experimental DP maps that show that more than $90\%$ of PWCs originate one or more observable DP fractures (most commonly 1 or 3), and most of these end at nearby PWCs [50, 52]. In addition, simulations of multiple feature maps [16, 40] also predict the existence of DP fractures originating from PWCs, as shown in Fig. 5. Moreover, this figure clearly shows that almost all DP fractures connect two neighboring PWCs.

We saw in Fig. 4 that DP only covers a 180^∘ range around any given PWC, which is also consistent with experiments [52, 57]. This means that the other PWC in each OD stripe within a hypercolumn must cover the remaining range to preserve symmetry and the definition of the hypercolumn itself. Together with the fact that the neighboring PWC is of opposite sign, this further constrains the local topology of the combined map to be of the form shown in Fig. 6(A) for a case of two PWCs linked by a fracture. We see that DP ranges from $90^{\circ}$ to $270^{\circ}$ (equivalent to $-90^{\circ}$) around the positive PWC at left, whereas the remaining range of $-90^{\circ}$ to $90^{\circ}$ is covered around the negative PWC at right. Likewise, the case of 3 DP fractures per PWC in Fig. 6(B) also covers the full range of DP, with each pinwheel covering half of the range. The left pinwheel covers three intervals of DP, from $90^{\circ}$ to $150^{\circ}$, $-30^{\circ}$ to $30^{\circ}$, and $-150^{\circ}$ to $-90^{\circ}$, while the remaining DP intervals are covered by the right pinwheel.

The topology seen in Fig. 6 implies the existence of important additional features of the DP map: at the midpoint (or some other point in a less idealized case) of the fracture, regions of opposite DP abut on the same side of the fracture. This can only occur if the DP selectivity vanishes at the midpoint, akin to the vanishing of OD selectivity at the boundary between L and R stripes and of OP selectivity at PWCs. A one-dimensional locus of vanishing DP selectivity must then radiate in both directions from the midpoint; in our idealized case, this will be a straight line, which is shown as white dotted line in Fig. 6, which represents a further DP fracture.

The new constraints noted in the previous paragraphs are consistent with experimental features seen in Fig. 7: (i) the radiation of DP fractures from PWCs, parallel to OP contours; (ii) the apparent “fading out” of many DP fractures between PWCs, examples of which are indicated by the blue circles, with some apparently resuming in much the same direction but opposite DP; and (iii) the sudden $\sim 90^{\circ}$ changes in direction of DP fractures, with many crossing OP contours at steep angles.

The simulated DP maps in Fig. 5 do not show the fading-out of fractures, a difference that may be due to the ability of simulations to determine DP reversals even where DP sensitivity is to low to be observed experimentally. However, simulated DP fractures do tend to follow OP contours, or to intersect them roughly perpendicularly, consistent with our predictions.

We now consider OP-DP maps when PWCs are arranged in a square or hexagonal lattice. In a square lattice the only way in which an odd number of fractures can emanate from each PWC without any ending between PWCs is for there to be one per PWC, but this cannot be achieved while preserving the fourfold symmetry of the lattice and very often three fractures are seen to emerge from a PWC [35, 53, 52], so we do not consider this case further.

The symmetry of the hexagonal lattice is consistent with there being three fractures emanating from each PWC and connecting it to its nearest neighbors, which are of opposite sign, as seen in Fig. 3(C). Topology alone does not force the fracture to parallel the OP contours, but the requirement that neighboring pinwheels cover all possible DPs means that fractures are directed toward one another along the same contour as they emerge from the PWCs, so contour following is favored. In the most symmetric case, DP selectivity would vanish along lines perpendicular to the midpoints of each edge of the hexagons, and would meet at the hexagon centers. Hence, we predict that the midpoints will have low sensitivity, consistent with the apparent “fading out” commonly seen experimentally. We also predicte the centers of hexagons to have low DP selectivity, but have not yet been able to locate any experimental results that would enable this prediction to be tested.

Having found that the hexagonal lattice is the only possible arrangement for our idealized OD-OP-DP map, we now consider how the OD bands relate to the OP-DP map. Figure 3(C) shows the two topologically distinct arrangements. We have already noted in Sec. IIIA that the case shown with vertical OD stripes has a better match with the spread of PWCs relative to the midlines of OD stripes than the case shown with horizontal OD stripes. We now further evaluate these alternatives against experimental data on PWC density and layout within OD stripes.

Let us denote the spacings between PWCs of the same sign as $\lambda_{\parallel}$ and $\lambda_{\perp}$ in directions parallel and perpendicular to the OD boundaries, respectively, as shown in Fig. 3(C). Results derived from the geometry of the two hexagonal cases are summarized in Table 1, together with corresponding experimental findings from macaque monkey [44]. Experimentally, $\lambda_{\perp}>\lambda_{\parallel}$, whereas both our idealized arrangements exhibit the reverse inequality. The experimental results thus correspond to the hypercolumns being compressed in the parallel direction in macaque relative to the idealized case, while leaving the topology unchanged. However, the vertical OD arrangement in Fig. 3(C) gives by far the better agreement with the observed $\lambda_{\perp}/\lambda_{\parallel}$, being only $26\pm 7$% smaller. Because it also agrees better with the spread of PWCs relative to the OD stripe midlines [18], as discussed above, we argue that it is the best option among the symmetric maps considered. For convenience below we write the ratio of the experimental value of $\lambda_{\parallel}/\lambda_{\perp}$ to our theoretical prediction of this quantity

The width of a single OD stripe equals $\lambda_{\perp}$. If we choose $a=2\lambda_{\perp}/3$, so that $\lambda_{\perp}$ matches the experimental width, and set $\lambda_{\parallel}/\lambda_{\perp}$ to 0.85 to accord with experiment, this gives the values of $a$ shown in the forth column in Table 2.

We next use our hexagonal array to predict the average pinwheel density and compare it with results from experiments. In our vertical OD arrangement, the pinwheel density is

where $a$ is the side length of the hexagon, and $\Delta$ is the ratio from Eq. (2), which is included here to accommodate the effect of real OP map being compressed in the direction parallel to the OD stripes. The value of $a$ in the sixth column of Table 2 is calculated from Eq. (3) with the experimental pinwheel density shown in the second column.

Several points are worth noting in Table 2: (i) the value for humans is the experimental one in a region at $5^{\circ}$ eccentricity (i.e., the angular distance from the center of the visual field) [59], and the pinwheel densities for other species are averaged values over the whole measurement area in each reference; (ii) cats and ferrets have patchy OD bands, rather than stripes; the OD width for these two animals is rather the average diameter of the OD patches; (iii) ocular dominance stripes are not found in tree shrew and mouse lemur, so the corresponding OD width does not exist.

By comparing the fifth and sixth columns of Table 2 we conclude that the estimates of $a$ from human, macaque and galago OD widths and PWC densities are similar in each case, although not always to within the indicated uncertainties. This implies that our hexagonal lattice is a reasonable semiquantitative approximation to the spatial structure of the experimental OP and OD maps in these three species. For ferret and cat the $a$ values estimated from OD stripe width are smaller than those estimated from pinwheel density. The possible reason for the mismatch is that the OD pattern in these animals appears as elongated patches with variable width and length, rather than true stripes. Although some patches are connected and form short parallel bands in a few millimeters, the overall OD pattern is more irregular than the striped OD arrangement in macaque [2, 18, 32, 36], so the poorer agreement is not surprising.

Figure 8(A) shows a color coded OD-OP map corresponding to the hexagonal lattice with vertical OD arrangement shown in Fig. 3(C). It is obtained by finding the OP angle $\phi_{1}$ around one PWC from [38]

where $(x_{c},y_{c})$ are the coordinates of the PWC, then reflecting the resulting pattern across axes of symmetry to obtain the OP near neighboring pinwheels. The PWCs are located at the vertices of each hexagon, six of which are marked $+$ (for positive pinwheels) and $-$ (for negative pinwheels). The white border outlines a hypercolumn with four pinwheels and the dashed lines are the OD borders. Figure 8(B) shows DP arrows overlaid on an OP map of one hexagon extracted from frame (A). DP reverses direction across fractures (white lines), and loci of zero DP selectivity are marked by dashed white lines. Figure 8(C) shows DP discontinuities overlaid on OP contours to parallel the experimental results in Fig. 7 (i.e., the numbered sites). This emphasizes the tendency for discontinuities to either follow OP contours or to intersect them at 90^∘.

Equation (3) gives the PWC density in terms of the PWC separation in the hexagonal lattice. Some previous works argued for a universal PWC density in aperiodic arrangements of PWCs, expressed in terms of the mean periodicity $\Lambda$ of OP, independent of species [22, 33, 34], with

Equations (3) and (5) are equivalent if

however, $\Lambda$ is the mean periodicity of OP, so we should estimate it for our hexagonal lattice. We estimate $\Lambda$ via Fourier analysis of the hexagonal lattice in Fig. 8(A) [33, 46, 44]. Firstly, we compute the power spectrum of the hexagonal lattice as the squared amplitude of its Fourier transform, and the resulting spectrum is shown in Fig. 9. It has six peaks arranged in hexagon and the magnitude of the corresponding $\mathbf{k}$ is

As we mentioned earlier, the experimental map corresponds to compressing the hexagonal lattice by a factor of $\Delta$ [i.e., Eq. (2)] in the vertical direction. In order to take this factor into account, the components of $\mathbf{k}$ in the Fourier domain are stretched by a factor of $1/\Delta$ in the vertical direction. If we denote the stretched $|\mathbf{k}|$ by $k$, the top and bottom $\mathbf{k}$ modes in Fig. 9 have squared wave number

and the other four modes have squared wave number

The averaged value of $k^{2}$ is then

The mean periodicity of OP is evaluated via

The estimate of $\Lambda$ from Eq. (6) is only 15% smaller than this estimate, which is acceptable, given that we are dealing with a specific regular lattice, whereas the expression in Eq. (5) was postulated for irregular ones. Moreover, the experimentally measured pinwheel densities had deviations of up to $\sim 10\%$ from $\pi/\Lambda^{2}$, which further supports consistency with Eqs (6) and (13). We use the above equation to compute $a$ from the experimental $\Lambda$ listed in the fourth column of Table 2. The resulting $a$ values in the last column are similar to the ones obtained from pinwheel density, implying that the hexagonal lattice with vertical OD stripes is also semiquantitatively consistent with these data.

As noted above, the OD, OP, and DP selectivities vary with cortical location and relate to the OD-OP-DP map topology. The main features are:

(i) OD selectivity peaks at the midlines of OD stripes and falls off towards the borders [18, 49]. If we define a net OD selectivity (right minus left) $\nu_{OD}$, normalized to unit maximum, we can approximate $\nu_{OD}$ for illustrative purposes as being sinusoidal with extremums located at the centers of the OD bands:

where $a$ is the side length of the hexagon. Figure 10(A) plots the OD selectivity in one hexagon. Black and white represent left and right OD selectivity, respectively, and the dashed line is the OD border, where $\nu_{OD}=0$ and the overall response is binocular. The centers of OD stripes are located at $x=\pm 3a/4$.

(ii) OP selectivity is low around PWCs, and initially increases roughly linearly with radial distance before saturating after 100-200 $\mu$m [5, 44, 51]. We thus approximate the OP selectivity $\nu_{OP}$ for illustrative purposes as

where the $\mathbf{r}_{i}$ are the vertices of the hexagon, $r_{0}=80\ \mu$m parametrizes the low OP selectivity around each PWC [39, 43, 47, 52]. The resulting map in Fig. 10(B) shows that OP selectivity shows is relatively uniform away from PWCs. It is clear from the figure that the low selectivity regions are around the six pinwheel centers. Furthermore, the selectivity increases as one moving away from the center and reaches maximum in the center area of the hexagon, where OP varies smoothly within this region.

(iii) DP selectivity is low at fractures and reverses sharply across these fractures [35, 50, 52]. Furthermore, DP selectivity is zero along perpendiculars to the midpoints of DP fractures, as was discussed in Sec. III.2. We thus approximate the DP selectivity $\nu_{DP}$ as

for illustrative purposes where ${\bf u}_{i}$ and ${\bf m}_{j}$ are unit vectors that point from the center of the hexagon toward the three vertices located at $(1/2,\sqrt{3}/2)$, $(1,0)$, and $(1/2,-\sqrt{3}/2)$ and the three edge midpoints at $(0,\sqrt{3}/2)$, $(3/4,\sqrt{3}/4)$, and $(3/4,-\sqrt{3}/4)$. We take the absolute value of the dot product to ensure positive selectivity. This gives the DP selectivity map shown in Fig. 10(C), which we have normalized to unit maximum. A few features worth noting are: (i) the DP fractures at hexagon sides have zero selectivity where the DP reverses, consistent with the low selectivity observed experimentally at these fractures [52, 57], given that neurons have nonzero size and measurements have limited spatial resolution; and (ii) the selectivity is also zero at the midlines that are perpendicular to the hexagon sides, so we predict that it should be low in the center of the hexagon where OP sensitivity is high, but we are not aware of any published experiments on this point.