1 Department of Neuroscience, Georgetown University Medical Center, Washington, DC 20007, USA, 2 Present address: Department of Computer Science and Electrical Engineering, OGI School of Science and Engineering, Oregon Health and Science University, Beaverton, OR 97006, USA
Address correspondence to Geoffrey J. Goodhill, Department of Neuroscience, Georgetown University Medical Center, Washington, DC 20007, USA.
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Abstract |
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Key Words: computational model ocular dominance map orientation map visual cortex
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Introduction |
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Several computational models have made important contributions with regard to understanding the structure of OD and OR maps (reviewed in Erwin et al., 1995; Swindale, 1996
; Carreira-Perpiñán and Goodhill, 2002
). In particular, low dimensional or feature space models (Durbin and Mitchison, 1990
; Goodhill and Willshaw, 1990
; Obermayer et al., 1992
; Goodhill et al., 1997
; Swindale and Bauer, 1998
; Goodhill and Cimponeriu, 2000
; Carreira-Perpiñán and Goodhill, 2004
), generate maps that closely reproduce the structure of OR and OD maps (Erwin et al., 1995
; Swindale, 1996
). However, the application of these models has been limited up to now to only one or two features (OD and/or OR) in addition to VF. This naturally raises three questions: (i) can such models still successfully reproduce map structure when multiple features are considered; (ii) what quantitative predictions do these models make in this case; and (iii) what light does the behavior of models in this case shed on the biological mechanisms underlying map formation. In this paper we address these questions by simulating the combined development of VF, OD, OR, DR and SF using the elastic net model (Durbin and Willshaw, 1987
). This is a feature space model which works by minimizing an objective function that explicitly trades off coverage versus continuity constraints (Swindale, 1996
). In addition, we show that in this multiple map case the elastic net can also reproduce recent experimental results regarding monocular deprivation (Crair et al., 1997
) and single orientation rearing (Sengpiel et al., 1999
). Taken together, our results extend the range of phenomena to which computational modeling of map formation in visual cortex has been applied. Most importantly, they provide strong additional support for the hypothesis that the computational principles underlying algorithms such as the elastic net capture the essential biological principles underlying map formation in the real cortex.
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Materials and Methods |
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The elastic net (Durbin and Willshaw, 1987; Durbin and Mitchison, 1990
; Carreira-Perpiñán and Goodhill, 2002
) produces maps that minimize a tradeoff E =
C + (ß/2)R between coverage C of the stimulus space and continuity R of the cortical representation. The coverage term is defined as follows:
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![]() | (2) |
The Stimulus Space
The stimulus space consists of feature points evenly spaced along orthogonal feature dimensions. For visual field (VFx, VFy) the stimulus values (training set) are a grid of Nx x Ny points in the rectangle [0, 1] x [0, 1]. Orientation preference and selectivity (OR, ORr) are represented by adding dimensions of NOR x 1 points in [(/2), (
/2)] x {rOR}, where OR is conventionally represented by two variables in polar coordinates, with NOR values uniformly arranged on a ring of radius rOR. Direction preference and selectivity (DR, DRr) are represented by adding a dimension of points {OR (
/2),OR + (
/2)} on a ring of radius rDR, as in Swindale and Bauer (1998)
. The ocular dominance (OD) dimension has NOD values in [lOD, lOD]. Spatial frequency (SF) is represented by NSF values in [lSF, lSF]. Consistent with several experimental reports we take NSF = 2, representing high and low frequencies (Hübener et al., 1997
; Shoham et al., 1997
; Kim et al., 1999
). [It should be noted that others have suggested the representation of SF may be more continuous (Silverman et al., 1989
; Everson et al., 1998
; Issa et al., 2000
)].
Training Regime
We trained nets with the following configuration. Training set: Nx = Ny = 20, NOR = 6, NDR = 2, NOD = 2, NSF = 2 (a total of 19 200 training points); lOD = 0.06 (lOD = 0.08 for strabismus), lSF = 0.06, rOR = 0.08, and rDR = 0.08. Elastic net: M = 128 x 128 = 16 384 centroids; = 1, ß = 10; nonperiodic boundary conditions. The net was started from a random initial starting configuration with some global topographic bias; where results have been averaged over multiple simulations this corresponds to taking different random initial starting configurations for the net. As is common in the elastic net, the minimization of the energy is interleaved with decreasing (annealing) K. We annealed K from a starting point of 0.2 with a rate of 0.9925 to the point at which the maps have just arisen (K
0.03). An efficient minimization method based on Cholesky factorization was used (Carreira-Perpiñán and Goodhill, 2004
). The simulations were performed using custom software written in Matlab.
Monocular Deprivation and Singleorientation Rearing
For the default case (no deprivation) the coverage term was 1 for all stimulus points xn. However, in order to implement deprivation of some stimulus (e.g. OD),
was redefined as a vector with N components, where the value of the component
n represents the relative strength with which the stimulus point xn is represented in the input. Thus, for monocular deprivation we took
n = depOD
(0, 1) for each xn matching the deprived eye. In order to implement enhancement of OR for single orientation rearing, we took
n = depOR > 1 for each xn matching the enhanced angle of OR. In order to capture the experimental observation that SF maps consist of patches of low frequency in a broader expanse of high frequency (Shoham et al., 1997
), one value of SF was deprived with depSF = 0.5 for all simulations.
Column Width
The power of the discrete Fourier transform of the angle maps was roughly isotropic and concentrated around a ring. We summarized it by the mean wavelength (mean column width), averaged over all directions.
Crossing Angles
We disregarded the points lying 5 pixels or less from the boundaries of the net (a thin inner stripe framing the maps in e.g. Fig. 1) to eliminate border effects. At each pixel, we computed the angle between the gradient vectors of pairs of dimensions (see below) and mapped it to [0°, 90°], thus obtaining the angle between contours for the dimensions. Each such angle counted with a weight proportional to the product of the gradient moduli of the dimensions, e.g. ||OR|| x ||
OD|| for OR and OD, so that pixels lying in areas of either constant OR or constant OD (i.e. away from borders of OR or contours of OD) were effectively removed from the histogram. This is because, in an area where either map is nearly constant, the gradient vector is negligible in modulus and its direction basically arbitrary; unlike along borders, where the gradient vector is large and its direction well defined. We refer to the graphs we plot as histograms, even though in a strict sense they are not.
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An OR singularity, or pinwheel, is defined positive if the orientation angle increases in a clockwise direction around the pinwheel and negative if anticlockwise. Pinwheels were automatically located in the OR maps as follows. First, the winding number of each pixel in the OR map was estimated by summing the increments of OR angle (in [(/2), (
/2)]) along a closed path (a square of radius 1 pixel centered in the pixel considered) in a clockwise direction and dividing the result by 2
; this results in 0 for non-pinwheel points and +1/2 (1/2) for pixels at or adjacent to a positive (negative) pinwheel, respectively. The exact pinwheel location was obtained by grouping clusters of nonzero winding number and computing their centers of mass. To quantify the layout of pinwheels with respect to OD or SF columns, we computed, for every pinwheel, the distribution of distances of a given pinwheel to its closest OD or SF border. We prefer this to calculating distances to OD centers since the center of an OD/SF column is generally not well defined, except for columns which are translations of each other, and becomes difficult to use with columns of changing width, forks, islands or other complex shapes. In contrast, the OD/SF borders are well defined in all these cases. To compute the distances between a pinwheel and its closest OD or SF border, we represented the OD/SF borders by a finely spaced collection of points (the contours generated by Matlab for the OD/SF map); the said distance was then given by the point in this collection closest to the given pinwheel. We considered that a pinwheel lay on an OD/SF border if the distance between the two was 1 pixel or less.
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Results |
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In a feature space model such as the elastic net, feature dimensions are represented as spatial dimensions in a Euclidean space. Thus there are, for instance, two dimensions for VF, and one for OD. Since OR is periodic it is usually represented by a ring in two dimensions, making five dimensions. To this we added two dimensions representing DR (also periodic), and one representing SF, making eight dimensions in total. The precise distribution of feature points in each of these dimensions is described in Materials and Methods.
Figure 1 shows an initial simulation of VF and OR, and the effect of adding dimensions representing DR, OD and SF respectively in subsequent simulations. The top row corresponds to OR simulated just with VF, with subsequent rows showing the addition of DR, OD and SF respectively. It is apparent that adding dimensions does not change the qualitative character of maps, but does tend to reduce their wavelengths in the model (quantified in Fig. 2). Looking at the last row of Figure 1, it is apparent that the general structure of all four of the OR, DR, OD and SF maps have a good correspondence, at least qualitatively, with experimental data (Shmuel and Grinvald, 1996; Weliky et al., 1996
; Shoham et al., 1997
; Hübener et al., 1997
; Kim et al., 1999
). (Simulations started with different random seeds gave maps with similar global characteristics.) Also in Figure 1 is inset a joint contour map for all four columnar systems, to show the extent to which they intersect at right angles, and the relation of OR pinwheels to OD and SF columns (these properties are quantitatively analyzed in Fig. 3). OR, OD and SF all show strong orthogonality to each other, as observed experimentally (Hübener et al., 1997
).
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Figure 3 shows a quantitative analysis of the effect of adding feature dimensions on the distribution of interpinwheel distances, distances of pinwheels to OD and SF borders, and intersection angles between columns. As in Figure 1 successive rows show the cases of OR simulated alone (i.e. with VF but without DR, OD and SF); OR and DR together; OR, DR and OD; and finally all four feature dimensions (OR, DR, OD, SF). The first column shows histograms of distance between pairs of pinwheels that are nearest neighbors. As in the experimental data (Müller et al., 2000), singularities repel each other compared to a random distribution, though the repulsion weakens as more maps are added. For OR/VF simulated alone, the percentage of nearest-neighbor pinwheels that are of the same sign is 18.7 ± 1.3%, which matches well with experimentally determined values of 19.7 ± 1.5% (ferret) and 21.4 ± 2.0% (cat) (Müller et al., 2000
). However, in the simulations this rises to 30.9 ± 1.1% when all features are simulated.
The second column of Figure 3 shows histograms of OR pinwheel distance to OD border and SF border (the OD case is shown for the simulations with and without SF). Distances are normalized by mean wavelength of the OD/SF map respectively, so that a normalized distance of 0.25 roughly corresponds to the center of the OD/SF columns (we prefer to use distance of pinwheels to column borders rather than column centers since borders have a less ambiguous definition than centers; see Materials and Methods). Pinwheels tend to lie away from the borders of OD and SF, and the histogram of pinwheel to border distance reliably has a strong peak at slightly less than 0.2 of mean wavelength. Adding SF reduces this distance slightly for OD, i.e. moves OR pinwheels slightly closer to OD borders. It is hard to make a precise quantitative comparison with experimental data due to ambiguities in the definition of distance to column center, but at least qualitatively there is a good match (Bartfeld and Grinvald, 1992; Obermayer and Blasdel, 1993
; Hübener et al., 1997
).
The third column of Figure 3 shows intersection angles for map pairs. This shows the tendency to orthogonality for all combinations except OR and DR, where the intersection angles are low. It is clear that the shape of the intersection angles histogram for each pair of maps is very robust to the presence of additional feature dimensions. In order to make a direct comparison with numerical values quoted experimentally, Table 1 shows mean intersection angles (scalar average) for the simulations. It can be seen that adding features causes maps to intersect at very slightly shallower mean angles. The mean angle for OD/OR reported by Hübener et al. (1997) was 51.7 ± 0.8°, which appears statistically indistinguishable from our value of 52.5 ± 0.2° when all maps are simulated together. Our value for SF/OR is 52.0 ± 0.2°; that found by Hübener et al. (1997)
was 49.7 ± 0.8°, which they did not report as significantly different from their OD/OR value. It should be noted though that such means are only a very crude way of characterizing these angle distributions, and that 90° is the modal angle of intersection.
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Strabismus has previously been modeled in the elastic net by increasing the spacing of points along the OD dimension (Goodhill and Löwel, 1995). However, the effect on OR, DR ad SF maps of strabismus in the model was not investigated. We therefore repeated the simulations described above for the strabismic case (see Materials and Methods). Although as previously reported, OD columns become wider (Goodhill and Löwel, 1995
), no significant differences were found in the interrelationships between maps reported above for the non-strabismic case (data not shown). This is consistent with the experimental finding that strabismus does not greatly alter the relationships between OR and OD maps (Löwel et al., 1998
).
Monocular Deprivation and Single-orientation Rearing
In addition to the normal development simulated above, we were also interested to examine abnormal development in response to visual deprivation. In particular we focused on the experimentally well-characterized phenomenon of monocular deprivation (MD) (reviewed in Hubel and Wiesel, 1977; Katz and Shatz, 1996
), and the more recently discovered phenomenon that single-orientation rearing (SOR) leads to an over-representation of that orientation in the OR map (Sengpiel et al., 1999
). In each case, deprivation was modeled by changing the strength of the coverage term in the elastic net energy function relative to the continuity term (see Materials and Methods). In effect, we decreased the influence of feature points representing inputs in the deprived eye for the cortex relative to the open eye for MD, and increased the influence of feature points representing the single orientation for the cortex relative to other orientations for SOR.
Goodhill and Willshaw (1994) modeled MD in the elastic net using the same method of reduced influence for a feature space consisting only of VF and OD. However, they examined only constant MD existing throughout development. Figure 5 shows a more complete analysis of the effect of MD with varying-length time windows and start times for the deprivation (here OR was simulated in addition to OD for analysis of the relationship between OR and OD see Fig. 6). It can be seen that, analogously to experimental data (reviewed in Hubel and Wiesel, 1977
; Katz and Shatz, 1996
), OD deprivation has significant effects only in a critical period located shortly after the moment the OD map starts to arise but before it is fully developed, with OD deprivation either before or after having little effect.
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Discussion |
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The adult arrangement of maps in visual cortex has also been analyzed from a wiring optimization perspective (Koulakov and Chklovskii, 2001, 2002
; Chklovskii and Koulakov, 2004
). Although this approach does not address the development of maps, and also has not so far included topography as a variable to be mapped, it can successfully reproduce some of the experimental findings we have addressed here with the elastic net. In fact recent work has shown both a formal link between wiring optimization, Mexican-hat lateral interactions and the elastic net continuity term (Carreira-Perpiñán and Goodhill, 2004
), and that subtle differences in patterns of intracortical connectivity can have a strong influence on the relationships between individual maps.
The only other study to have examined the development of multiple columnar systems in visual cortex using a feature space model is that of Swindale (2000). Here the development of up to nine feature dimensions in addition to VF was simulated using Kohonen's algorithm. However, the approach was more abstract than ours in that all feature dimensions were binary, and were not interpreted in terms of specific visual features. Swindale found that the structure of each map individually, including its wavelength, was fairly robust to additional maps, but that the relationships between maps (specifically intersection angles) changed in a gradual way as more maps were added. These results differ somewhat from our own results: we found wavelengths to decrease as maps are added, yet more robustness of intersection angles as maps are added. We did, however, find a shift in the location of pinwheels relative to OD columns with the addition of the SF dimension [as no periodic dimensions were simulated in Swindale (2000)
the question of the location of pinwheels relative to other maps did not arise in that case]. The reason for the difference in results between these two studies is unclear, but it could be a manifestation of the fact that, although the elastic net and Kohonen algorithms are similar in flavor, they can behave differently in fine details.
What light do our results shed on the biological mechanisms underlying map formation? The elastic net represents a particular mathematical instantiation of the hypothesis that visual cortical maps are the result of an optimization process. In particular, the elastic net attempts to jointly optimize both coverage, the degree to which all input features are uniformly represented, and continuity, the degree to which the spatial representation of features is smooth in some sense (see Materials and Methods and Carreira-Perpiñán and Goodhill, 2002). It was already known that such optimization hypotheses are sufficient to generate the detailed structure of OR and OD maps under normal conditions (Erwin et al., 1995
; Swindale, 1996
). Our results now show that optimization principles based on the coveragecontinuity trade-off are also sufficient to reproduce a large array of additional data, both when additional maps are considered, and when particular types of visual deprivation are introduced.
How might the visual cortex actually implement such optimization principles? One way the elastic net equations can be interpreted biologically is in terms of a Hebbian process: the simple gradient descent learning rule for optimizing the elastic net objective function is equivalent to an activity-dependent strengthening of connections between the input feature that is presented at each instant and the neurons that respond most strongly to that feature. In contrast, evidence is now mounting that at least some properties of visual map formation are activity-independent (e.g. Katz and Crowley, 2002). However, the elastic net equations can be interpreted in other ways, and in fact the algorithm originally grew out of an activity-independent model that relied on the matching of molecular cues (Willshaw and von der Malsburg, 1979
). The fact that such optimization models work so well, and are now the only developmental models available for reproducing the wide array of map properties addressed in this paper, suggest that the optimization hypothesis is key to understanding visual cortex, independent of details of how precisely it might be implemented.
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Acknowledgments |
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