Link between orientation and retinotopic maps in primary visual cortex

Abstract

Maps representing the preference of neurons for the location and orientation of a stimulus on the visual field are a hallmark of primary visual cortex. It is not yet known how these maps develop and what function they play in visual processing. One hypothesis postulates that orientation maps are initially seeded by the spatial interference of ON- and OFF-center retinal receptive field mosaics. Here we show that such a mechanism predicts a link between the layout of orientation preferences around singularities of different signs and the cardinal axes of the retinotopic map. Moreover, we confirm the predicted relationship holds in tree shrew primary visual cortex. These findings provide additional support for the notion that spatially structured input from the retina may provide a blueprint for the early development of cortical maps and receptive fields. More broadly, it raises the possibility that spatially structured input from the periphery may shape the organization of primary sensory cortex of other modalities as well.

Sensory cortex is organized into vertical columns of neurons sharing similar preferences for stimulus properties (1, 2). The preference of neurons across the cortical surface can be represented as a map assigning each cortical site a set of preferred stimulus parameters. Two salient structures are normally observed in primary visual cortex of higher mammals: a retinotopic map, which assigns each location on the cortical surface a point in visual space, and an orientation map, which assigns each cortical site a preferred stimulus orientation (3–7). How these maps are established during development, how they relate to each other, and what function they play in normal visual processing remain fundamental, open questions in visual neuroscience (8–13).

Building on earlier work (14, 15), we have recently proposed that orientation maps may be established initially by the interference pattern of ON- and OFF-center, quasi-periodic retinal mosaics (16). Normal visual experience and activity-dependent synaptic learning can subsequently help maintain and refine this initial organization. Some of the model’s predictions take the form of relationships between different maps, such as those for orientation, spatial frequency, and visual space (8). Under normal developmental conditions orientation maps in the adult are very similar to the earliest ones one can measure (17). Thus, we reasoned robust map properties predicted by the seeding mechanism might be detectable in the adult. Indeed, previous work has shown that the predicted hexagonal structure of the interference pattern is reflected in the organization of the orientation map (16, 18). Here we use the model to derive a peculiar prediction relating orientation and retinotopic maps and confirm that it holds in tree shrew primary visual cortex.

Results

Spatial Interference Model.

We first consider an ideal version of the model, which helps develop an intuition for how the link between the maps come about. Suppose the receptive fields of ON-center and OFF-center retinal ganglion cells (RGCs) of a given class lie at the vertices of a perfect hexagonal grid (16). When two such patterns, having slightly different periodicities and orientations, are superimposed, the result is a periodic interference pattern (Fig. 1A). The period of the interference pattern depends on the ratio between periods of the component lattices and their relative orientation (16). An important property of the interference pattern is that the nearest neighbor of an ON-center cell is an OFF-center cell and vice versa. Thus, one may consider the interference pattern as composed of ON/OFF pairs or dipoles (15, 16). Our hypothesis is that in the early stages of development cortical cells have inputs dominated by individual dipoles that generate orientation-tuned receptive fields with side-by-side subregions of opposite sign (8, 16, 19) (Fig. 1A, Lower). Orientation dipoles systematically change their orientation over space, generating a blueprint for the orientation map (Fig. 1 B and C).

Fig. 1.

Moiré interference of retinal mosaics predicts a link between retinotopic and orientation maps. (A) (Upper) Two hexagonal lattices representing ON- (red) and OFF-center (blue) ganglion cell receptive fields generate an interference pattern that can be described in terms of ON/OFF dipoles. The relative angle between the two mosaics in this example is zero. (Lower) A cortical cell with input dominated by a dipole has a receptive field with side-by-side subregions of opposite sign and can be tuned for orientation. The preferred orientation is orthogonal to the line joining the receptive field centers of the ON/OFF that define the dipole. (B) (Upper) The orientation of dipoles in the interference pattern, indicated by the orientation of short line segments, changes over space, generating a blueprint for an orientation map. The model generates orientation singularities of opposite signs, some with chirality −2 (blue squares) and others with chirality +1 (red squares). (Lower) The organization of orientation preferences around negative (Left) and positive (Right) singularities. (C) Pseudocolor representation of the orientation map resulting from the dipole organization in B along with the structure of positive and negative singularities. The dashed black circle represents the size of the neighborhood used in subsequent analyses of local angle distributions (the neighborhood radius was 0.3 of the period of the orientation map, which represents 75% of the mean nearest-neighbor distance to the closest singularity). (D) Overlaying a retinotopic axis on an orientation singularity allows the assignment of two angles to each point within its neighborhood. One represents the displacement angle with respect to the horizontal meridian () and the other represents its preferred orientation angle (). A positive singularity is used in this example. (E) The local organization of dipole orientation around a positive singularity is dominated by three sets of cocircular arcs (white contours), reflecting the organization in B, Lower Right. The pseudocolor image shows the absolute value of the angular difference at each location within the neighborhood. Along three directions the difference is large (red areas), but most of the area is dominated by small differences (blue areas). As a result, the distribution of angular differences shows a peak at zero (F). (G) The same predicted relationship holds when the distributions of the angular differences are calculated using the full model, which samples the RGC mosaics with an isotropic Gaussian function to derive the shape of receptive fields at each location. (H) The distribution of mean angles across singularities shows that positive singularities ought to have a resultant near 0°, whereas negative singularities ought to have a resultant at ±90°. Black bar represents the scale for the circular distribution and red bar represents the scale for the resultant vector.

Predicted Link Between Orientation and Retinotopic Maps.

The model generates an orientation map with a global structure that depends on the relative orientation between ON and OFF mosaics (16) (Fig. S1). We concentrate our discussion on the case when the relative orientation is zero. Other values of the relative orientation produce similar results (SI Methods).

When the relative angle between the RGC mosaics is zero, the resulting dipole orientations align along concentric circles centered on negative orientation singularities, with winding number −2, where orientation rotates clockwise two full cycles as one circumvents the singularity in a clockwise direction (Fig. 1B, Lower Left dipole pattern). Positive singularities, with winding number +1, where orientation rotates counterclockwise one cycle as one circumvents the singularity in a clockwise direction, are located equidistant from neighboring negative singularities (Fig. 1B, Lower Right dipole pattern). Both types of singularities occur at the vertices of hexagonal lattices (Fig. 1B, red and blue squares). The density of +1 singularities is twice that of −2 singularities, making the average topological sign over large areas equal to zero. A simulation of the full model based on such input gives rise to a smooth orientation map that reflects the basic organization of the dipoles (Fig. 1C) (16).

Consider now a local coordinate system aligned with the main retinotopic axes at the center of an orientation singularity (Fig. 1D). Each cortical site within its neighborhood can be assigned two angles, one representing the preferred orientation of the cortical column at that location, , and the other representing its angular displacement with respect to the horizontal meridian, . In what follows, we first focus our analysis on the distribution of angular differences (modulus 180°) at orientation singularities and consider their joint distribution later.

The model predicts that angular-difference distributions at singularities of opposite signs should have disparate shapes. In the neighborhood of negative singularities, where a cocircular organization of preferred dipole orientations dominates the local organization (Fig. 1B, Lower Left), the two angles are near orthogonal to each other. Thus, one expects the distribution of the angular differences to peak at . In contrast, in the neighborhood of positive singularities, the local orientation of the dipoles is dominated by three concentric groups of arcs (Fig. 1 B, Lower Right, and E, white contours). The angular difference in this case changes within the neighborhood. Along three directions joining the center of the negative singularity with those of neighboring positive singularities, the angles are orthogonal to each other (Fig. 1E, red regions). However, the area covered by these regions is smaller than those at intermediate locations where the angular difference is small (Fig. 1E, blue regions). As a result, the distribution of angular difference over the entire neighborhood shows a broad peak centered at 0° (Fig. 1F). These expectations, developed by considering the orientation of ON/OFF dipoles, are confirmed when we calculate the angular difference distributions derived from the actual receptive fields and orientation maps generated by the model (Fig. 1G). Thus, the theory predicts positive pinwheels ought to have a distribution of angular differences with a mean near zero, whereas negative pinwheels ought to have a mean near .

An alternative way to illustrate the prediction is by computing the mean angle of the angular difference distribution independently for each singularity in the map and subsequently forming a histogram of the resulting values for positive and negative pinwheels (Fig. 1H). The mean angle for negative singularities is expected to be at whereas for positive angles it is expected to be zero.

Confirmation of the Predicted Link in Tree Shrew V1.

We tested the predicted relationship by analyzing orientation maps from tree shrew primary visual cortex (20). In these maps, the boundary between V1 and V2 is clearly visible and represents the vertical meridian in the visual field (Fig. 2A). Moreover, the retinotopic map in tree shrews is near isotropic within the representation of central visual space (21, 22). This relationship allows us to estimate the alignment of the main retinotopic axes (Fig. 2A, coordinate system at the center of the orientation map) and to calculate the angular difference distributions at pinwheels of different signs. To avoid boundary effects, we restricted our analyses to regions of interest within the center of the orientation maps (Fig. S2).

Fig. 2.

Predicted link between retinotopic and orientation maps is confirmed in tree shrew primary visual cortex. (A) Sample orientation map in tree shrew visual cortex from the work of Mooser et al. (20). The V1/V2 boundary is clearly visible and marked with a dashed line. The orientation of the areal boundary serves to anchor the retinotopic axes within the center of the orientation map where the analyses are conducted. Note that the measurement of the axial angle must take into account the orientation of the positive axes of the retinotopy. (B) Theoretical, experimental, and control orientation maps. In each case we illustrate the orientation map, with detected pinwheels (+1 singularities in white and −1 in black). Note that in the model, which now includes positional noise, only −1 singularities are observed. (C) Distribution of average angular differences and their correlation coefficients (calculated by Matlab’s corrcoef function). The distribution of correlation coefficients from Monte Carlo simulations by randomly rotating the orientation maps is shown at the Inset, with the vertical red line marking the experimental value that achieves a significance level of 0.005. (D) Circular distribution and resultant of the mean angle across singularities of different signs (positive on the top and negative on the bottom). The average distributions for the control case are statistically indistinguishable from uniform.

Before embarking on the analysis of the experimental data it is necessary to verify that the predictions hold true in a more realistic version of the model and over a reasonable range of parameter values. This procedure is required because it became clear during the course of our studies that the location of orientation singularities can shift in the presence of noise (Fig. S3). For example, it turns out the −2 singularities are unstable and readily split into two −1 singularities in the presence of small amounts of RGC noise (Fig. 2B, Left). Our simulations show that, nevertheless, the predicted link holds and it is robust over a substantial range of noise and scaling factors (Fig. 2 C, Left, and D, Left and Fig. S4).

With the assurance that the prediction is robust to changes in parameters and noise levels, we proceeded to test whether the data showed evidence of the predicted link. Remarkably, analysis of the layout of preferred orientations around singularities ( positive and negative singularities in five maps) confirms the predicted link between retinotopic and orientation maps (Fig. 2 B–D, Center). As expected, the average distributions for positive and negative singularities peak at zero and and are significantly anticorrelated. Moreover, the distribution of the mean angles across singularities of opposite signs differs significantly from uniform and has resultants with the expected angles (Fig. 2D, Center).

As a control condition, we randomly rotated the five orientation maps with respect the retinotopic axis and repeated the calculations. Each realization of such control condition simulates the outcome of our measurements under the null hypothesis that there is no angular relationship between the retinotopic and orientation maps. In each simulated control experiment, we calculated the correlation coefficient between the distributions for the positive and negative singularities. Using the distribution of the resulting correlation coefficients (n = 1,000 simulations), the probability that the observed value of could have resulted by chance under the null hypothesis is (Fig. 2C, Center Inset). Thus, the data reject the null hypothesis that retinotopic and orientation maps are independent of each other. Finally, the average distributions in the control condition are, as expected from the randomization procedure, statistically indistinguishable from uniform (Fig. 2 C, Right, and D, Right).

To examine in more detail how the angular variables deviate from statistical independence we calculated at both types of singularities (Fig. S5). The model predicts that the deviation from independence at positive singularities, , must show three discrete positive peaks near the unity line and three discrete negative peaks at locations where the angles are offset by (Fig. 3A, Upper Left). In contrast, the prediction at negative singularities, , must show a more diffuse, band-like structure, with negative values aligned with the unity line and positive values aligned at locations where the angles are offset by (Fig. 3A, Lower Left).

Fig. 3.

Deviations from statistical independence. We studied the deviations from statistical independence of the angular variables, at positive and negative singularities. (A) Model predictions (Left), measured deviations (Center), and one realization of the control (Right). Dashed white lines indicate the unity line , whereas dashed black lines indicate the relationship . (B) Joint distribution of and under the null hypothesis that the angular variables are statistically independent. The data have similarity indexes of and , as indicated by the white circle. The probability of the vertically hatched area under the null hypothesis is <0.03, that of the horizontally hatched area is <0.05, and that of their intersection is <0.005.

Remarkably, the measured deviations in the data, and resemble their theoretical predictions very well (Fig. 3A, Center). To quantify this agreement and to assess its statistical significance we defined a similarity index by

where is the dot product between the 2D functions. The index is near zero if the argument is close to and near if the argument is close to We computed the similarity indexes for both the data and controls , each obtained by randomly rotating the five orientation maps in our dataset with respect to the retinotopic axis (one realization of a control is shown in Fig. 3A, Right). The control data yield a joint distribution of indexes, and , under the null hypothesis that the angular variables are statistically independent (Fig. 3B, heat map distribution). A perfect agreement between the data and the model would result in similarity indexes for the data falling on the (1, 0) coordinate point on this graph (Fig. 3B, red asterisk). The actual indexes are and , which places the data closer to the predicted (1, 0) than the cloud of control data points (Fig. 3B, white circle vs. heat map). The likelihood that the control set would generate a distribution at positive singularities closer to the model’s prediction than the one observed is (Fig. 3B, probability of the vertical hatched area); the likelihood that the same would be the case at negative singularities is (Fig. 3B, probability of horizontally hatched area); and finally, the likelihood the control would generate data in better agreement with the prediction, simultaneously for both positive and negative singularities, than what is observed is (Fig. 3B, probability of cross-hatched area). Thus, the degree of similarity between the data and the model is statistically significant.

Discussion

The present study unveiled a relationship between orientation singularities and the retinotopic map that had, until now, escaped detection. Remarkably, the maps are linked in a way consistent with the prediction of the moiré interference model, explaining the modes of the angular differences (Fig. 2) and the deviation from independence in their joint distributions (Fig. 3).

The theory provides a parsimonious explanation for a diverse set of phenomena. For example, it explains some key features of the statistics of monosynaptic connections between the thalamus and cortex (19). In the model, the sign rule (23), which refers to the tendency for ON/OFF-center inputs to connect to simple-cell subregions of the same sign, is explained by the limited overlap of inputs present in the retinal mosaics (figure 4A in ref. 19). The limited overlap of ON/OFF inputs is also consistent with the clustering of ON- and OFF-center afferents in layer 4 (24) and the finding that the average input from thalamic afferents is biased with a preferred orientation that matches that of the target cortical column (25, 26). The model also explains a tendency for simple-cell receptive fields to have odd-symmetric profiles (27–29), which is a consequence of their inputs being initially dominated by a single dipole that generates an odd-symmetric receptive field (Fig. 1A and figure 5 in ref. 16). Moreover, the model provides a simple explanation for the emergence of orientation columns. Namely, a set of neurons within a cortical column that share the same inputs will be biased toward the same preferred orientation. The global structure of the interference pattern accounts for the hexagonal symmetry of orientation maps (16, 18) and a tendency for cocircularity in their organization (30–34). Altogether, the ability of the model to account for these diverse findings lends support to the notion that spatially structured and limited input from the contralateral retina may seed receptive fields and the orientation map during the earliest stages of development (9, 10, 35–51).

One may be surprised that the organization of map structure visualized via optical imaging of signals in layer 2+3, even in species with different laminar architecture (16), would show remnants of a spatially structured retinal input to the cortex. However, we note that the model makes a general statement about the class of linear receptive fields that may be implemented at any one point in the visual field given a limited set of retinal inputs. So long as we restrict ourselves to a linear combination of the retinal signals, the particular anatomical organization of the early visual pathways is immaterial. The class of realizable receptive fields is determined by the structure of the RGC mosaics and the assumption that the input to the first neurons exhibiting orientation selectivity can be well approximated as a linear combination of signals from the retina.

Testing the relationship between the maps discussed here in other species is clearly important. We were limited in our study to the tree shrew because it was the only case that allowed us to estimate the orientation of the retinotopic axes from the orientation of the V1/V2 boundary that was visible in the orientation maps (Fig. 2A). Similar tests should be carried out in other species by careful imaging of both orientation and retinotopic maps.

Finally, we note that many important questions emerge in relation to the proposed scheme, which require further research. Are the retinal mosaics sufficiently regular to allow for the proposed spatial interference? What mosaic class is responsible for establishing the orientation map? How precise should the retinotopic map be? How are binocular receptive fields with matching orientation established? Although much remains to be explored, the simplicity and explanatory power of the moiré interference model provide a competing hypothesis that deserves to be seriously explored. More broadly, one may conjecture that spatially structured input from the periphery may also help establish the organization of primary sensory cortices in other modalities (52, 53). It would then be of interest to explore whether the model can be applied to explain the properties of other systems. If these ideas are confirmed, they could transform the way we view cortical maps, their development, and their function.

Methods

Data.

The dataset used in our analysis is the one first published in Mooser et al. (20). These orientation maps were generously shared with us by David Fitzpatrick (Max Planck Florida Institute, Jupiter, FL) and his colleagues. A detailed description of the experimental methods by which the maps were obtained can be found in the original study. The dataset consists of five orientation maps that included the boundary between V1/V2 within the imaging window. This dataset allowed us to estimate the alignment of the retinotopic axis (Fig. 2A). The region of interest for our analyses was restricted to the central region of the maps to avoid boundary effects (Fig. S2).

Model.

The moiré interference model simulated was the same as described previously (16). Briefly, RGC mosaics were simulated by adding various amounts of random displacement to each vertex of a hexagonal lattice that represents the position of ON- and OFF-center receptive fields. The centers of RGC receptive field position vectors are defined by

Here, represents the grid spacing for the OFF mosaic, represents the grid spacing for the ON mosaic, the matrix

represents the relative rotation between the ON and OFF mosaics, represents 2D, Gaussian (i.i.d.) noise with a SD , and are the vertices of an hexagonal grid,

The SD of the noise, , is conveniently expressed as a fraction of the grid spacing, . The noise-free, ideal model corresponds to . A random relative spatial shift between the two mosaics can be added. However, except for the particular case where the mosaics have the same period, this shift has no consequence, because a rotation and translation can be written as a rotation around a different center.

The period of the interference pattern on the retina is a multiple, , of the lattice period, . We call the scaling factor, and it is fully determined by the relative orientation and relative period of the component lattices and is given by (16)

In monkeys we can estimate the scaling factor required to match the average period orientation maps in the cortex to be ∼8 (16), which is the nominal value we use here.

The full, statistical wiring model includes a stochastic component that allows cells in the same cortical column to develop slightly different receptive fields (19), as both the probability of connection to an afferent and its synaptic strength are random variables. Here, we did not simulate the whole model but computed only the mean receptive field at each location. We have previously shown the mean receptive field can be computed efficiently as an isotropic Gaussian-weighted sum of the afferent input to the cortical column (8). The profile of the resulting receptive field was analyzed to estimate its preferred orientation and selectivity. Finally, the preferred orientation of well-tuned cells was spatially smoothed to generate the predicted orientation maps. The details of all these procedures are described in ref. 16.

Analysis of Orientation Maps.

The analysis of the maps was scaled according to the orientation map period in each case, which we denote by (16). As a preprocessing step we minimally smoothed the maps with a Gaussian kernel with . The orientation maps were represented as a complex field, and singularities were detected by the intersection of the zero crossings of its real and imaginary components following the method described by Kaschube et al. (9).

The neighborhood around each pinwheel used to compute the angular distributions was defined by a disk of radius . This radius was selected to avoid the boundaries of the neighborhood from becoming too close to adjacent pinwheels, as the average nearest-neighbor distance of pinwheels was . Data from all five maps had to be pooled together to obtain a significant number of pinwheels of both signs to reach a statistically significant result.

A circular v-test was used to test for uniformity in Figs. 1H and 2D, with the alternative being that the mean was zero for positive singularities and for negative singularities. When circular statistics were used, they were applied to twice the angle to map the orientation domain to the full circle.

Analysis of Joint Angular Distributions.

Let us denote by the deviation from independence measured for positive singularities in the ith orientation map, . Similarly, we denote by those measured for negative singularities. We combined the data by computing the average shape of the departures from statistical independence by _. Each individual realization of the control condition, obtained by randomly rotating the orientation maps, resulted in deviations that that were combined in exactly the same fashion.

Theoretical predictions were generated by the same procedure, averaging the deviation from independence, , of maps generated by the model. Here we simulated cases with a fixed scale factor of , relative rotation angles ranging from to in steps of 2° (yielding a total of five maps), and positional noise levels matching those of experimental mosaics (16). The simulated maps were large, providing a total of 455 positive and 459 negative singularities that contributed to the prediction. The restriction to rotation angles between and is a consequence of the model, which predicts that only small, relative angular rotations generate scaling factors that can match the period of the orientation maps in the cortex given the typical ratios of ON/OFF-center cell densities in retinal mosaics (figure 1C in ref. 16). Finally, the resulting 2D distributions were, in all cases, filtered with a Gaussian kernel having a SD of 25°.