The elastic net (Durbin and Willshaw 1987) can account for the development of both topography and ocular dominance in the mapping from the lateral geniculate nucleus to primary visual cortex (Goodhill and Willshaw 1990). Here it is further shown for this model that (1) the overall pattern of stripes produced is strongly influenced by the shape of the cortex: in particular, stripes with a global order similar to that seen biologically can be produced under appropriate conditions, and (2) the observed changes in stripe width associated with monocular deprivation are reproduced in the model

Bwel S.

David J. Willshaw

Geoffrey J. Goodhill

Hubel D. H.

Jones D. G.

Yuille A. L.

The elastic net [Durbin &amp; Willshaw 1987] can account for the development of both topography and ocular dominance in the mapping from the lateral geniculate nucleus to primary visual cortex [Goodhill &amp;Willshaw 1990]. Here it is further shown for this model that (a) the overall pattern of stripes produced is strongly influenced by the shape of the cortex: in particular stripes with a global order similar to that seen biologically can be produced under appropriate conditions, and (b) the observed changes in stripe width associated with monocular deprivation are reproduced in the model. 1 Introduction  Two well-documented phenomena associated with ocular dominance stripe formation in primary visual cortex are as follows. Firstly, for the macaque monkey there is a global order to the stripe pattern [LeVay et al 1985]. Secondly, the relative widths of left- and right-eye stripes can change following monocular deprivation in the cat [Shatz &amp; Stryker 1978] and in the macaque [Hubel et al 1977]..

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Elastic Net Model of Ocular Dominance: Overall Stripe Pattern and Monocular Deprivation

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Goodhill, G. J.

Willshaw, D. J.

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Communicated by Graeme Mitchison 
Elastic Net Model of Ocular Dominance: Overall Stripe 
Pattern and Monocular Deprivation 
Geoffrey J. Goodhill 
David J. Willshaw 
Centrefor Cognitive Science, University i f  EdinDiirgh, 
2 Biicc le i ih  Place, Edinbiirg/i E H 8  9LW, UK 
The elastic net (Durbin and Willshaw 1987) can account for the de- 
velopment of both topography and ocular dominance in the mapping 
from the lateral geniculate nucleus to primary visual cortex (Goodhill 
and Willshaw 1990). Here it is further shown for this model that (1) the 
overall pattern of stripes produced is strongly influenced by the shape 
of the cortex: in particular, stripes with a global order similar to that 
seen biologically can be produced under appropriate conditions, and 
(2) the observed changes in stripe width associated with monocular 
deprivation are reproduced in the model. 
1 Introduction 
Two well-documented phenomena associated with ocular dominance 
stripe formation in primary visual cortex are as follows. First, for the 
macaque monkey there is a global order to the stripe pattern (LeVay e ta / .  
1985). Second, the relative widths of left- and right-eye stripes can change 
following monocular deprivation in the cat (Shatz and Stryker 1978) and 
in the macaque (Hubel rt al. 1977). In this paper empirical results are 
presented for the elastic net model of ocular dominance, showing that it 
can reproduce both these phenomena under appropriate conditions. 
The elastic net is an algorithm for finding neighborhood-preserving 
mappings between spaces of different dimensionalities (Durbin and Will- 
shaw 1987). It was originally developed from a model for retinotopic 
map formation (Willshaw and von der Malsburg 1979), and has been 
applied to the problem of finding mappings that are both striped and 
topographic, such as the ocular dominance map (Goodhill and Willshaw 
1990). The algorithm finds a mapping between a "feature" space and a 
"cortical" space. For the topography and ocular dominance problem, the 
feature space consists of all positions in both eyes, and here we refer to 
these feature points as  LGN (lateral geniculate nucleus) units. Distances 
between LGN units in the space encode the "similarity" of units to each 
other, such that similar units lie close to each other. Similarity could be 
Ntwrnl Cottipiitation 6, 615-621 (1994) @ 1994 Massachusetts Institute of Technology 
616 Geoffrey J. Goodhill and David J. Willshaw 
interpreted as the degree to which the activity of units is correlated in 
the feature space (Yuille et al. 1991). 
Each LGN is represented as a two-dimensional sheet of points, and 
the two sheets lie atop one another separated by a small gap. The third 
dimension represents ocularity: distances in the ocularity dimension rep- 
resent similarities according to the same metric as in the other two dimen- 
sions. The images of cortical points under the mapping are envisaged 
as an elastic sheet moving in the feature space: these points are referred 
to as cortical units. For a fuller discussion of this formulation of the 
problem see Goodhill (1991), Yuille rt a/. (1991). 
Refer to the positions of LGN units as x, and cortical units as y,. The 
change in the position Ay, of each cortical unit at each time step is given 
by 
The first term is a matching term that represents the ”pull” of LGN units 
for cortical units, which is traded off with ratio o / . j k  against a regular- 
ization term representing a “tension” in the sheet, that is, a desire for 
neighboring cortical units to represent neighboring points in the feature 
space. N ( j )  refers to the set of points in the sheet that are neighboring 
to j, The “weights“ zu,, are defined as follows: 
where 
Over the course of a simulation, the scale parameter k is gradually re- 
duced, so that the matching term comes to dominate the regularization 
term. 
These equations can be interpreted as saying that each cortical unit has 
a gaussian receptive field at position y, in the feature space (Durbin and 
Mitchison 1990). The amount by which cortical unit j responds to input i 
at position x, is given by zu,,. The normalization of zu,, by the response of 
all other cortical units to input i implements a form of soft competition 
between cortical units. Although all cortical units are adapted toward 
input i, those that respond most strongly are adapted the most. The first 
term can therefore be seen as Hebbian. The second term says that cortical 
units are also adapted toward inputs that their neighbors respond to. 
It was shown in Goodhill and Willshaw (1990) that stripe width is 
controlled by the ratio of the separation of corresponding units between 
the LGN sheets to the separation of neighboring units within each LGN 
sheet. A simple analysis in one dimension suggests that stripe width in- 
creases linearly with this ratio (Goodhill 1991). A much deeper analysis 
Elastic Net Model of Ocular Dominance 617 
of stripe width in the elastic net model can be found in Dayan (1993), 
where the relative influences of the input similarities and the cortical 
interaction function are determined. Interpreting similarity as correla- 
tion, stripe width for the elastic net increases as the degree of correlation 
between the two eyes decreases. Similar behavior is found in the com- 
petitive model of Goodhill (1993), where the prediction was made that 
stripe width should thus be wider in strabismatic cats than normal cats. 
This prediction has recently been confirmed experimentally (Lowel and 
Singer 1993). 
2 Overall Stripe Pattern 
Naturally occurring ocular dominance stripes in the monkey exhibit 
global order: stripes remain roughly parallel over large distances along 
the dorsal-ventral axis. This result might not be expected from the op- 
eration of the purely local mechanisms that are present in most models. 
LeVay et al. (1985) provide a review of theoretical ideas regarding pos- 
sible sources of an anistropic ordering influence, and hence how such 
global order might arise. Such ideas include elongation of geniculocor- 
tical arborizations (von der Malsburg 1979), anisotropic growth of the 
cortex (Swindale 1980), and differences in the strength of two orthogo- 
nal gradients of adhesiveness (Fraser 1985). LeVay et al. (1985) suggest 
that the effect could be due to the shape of primary visual cortex as com- 
pared to the shape of the LGNs. The basic geometry of the mapping in 
the macaque is that of two roughly circular LGN layers projecting to a 
roughly elliptical region with a ratio of major to minor axes of 21. As- 
sume that the map is interdigitated into parallel stripes. Then, in order 
to fit both of the circular LGN regions into the elliptical region, each cir- 
cular region will be less "stretched" if stripes are formed running parallel 
to the short axis of the ellipse, compared to the case of stripes running 
parallel to the long axis. This hypothesis was tested for the elastic net 
algorithm by comparing the mapping formed between two disks of LGN 
units and a sheet of cortical units that is (1) a disk and (2) an elliptical 
region as described above. The dimensions of the cortex were chosen 
such that the number of units was about the same as the total in both 
LGNs. Results are shown in Figure 1. It can be seen that the shape of 
the cortex exerts a strong influence on the overall stripe pattern: for a 
circular boundary there is no preferred stripe orientation, whereas for an 
elliptical boundary stripes are indeed aligned with the short axis, as seen 
biologically. 
This boundary effect could in general provide another test of the ad- 
equacy of models for ocular dominance, for instance (Miller et d. 1989; 
Obermayer et al. 1992). It would not be expected to occur in models where 
there is no consideration of topography [such as Swindale (1980)], since 
61 8 Geoffrey J. Goodhill and David J .  Willshaw 
the effect relies on the undesirability of stretching, that is, deformations in 
topography in certain directions. The cortical shape effect hypothesized 
by LeVay d 01. (1985) has also recently been demonstrated in a different 
computational scheme by Jones ct d. (1991). They defined a particular 
cost function measuring topographic distortion, and then exactly mini- 
mized this for the mapping between two LGNs and an elliptical cortex. 
However, their result was obtained by a brute-force minimization of the 
cost function, without a particular biological mechanism in mind. 
Elastic Net Model of Ocular Dominance 619 
3 Monocular Deprivation 
If one eye is occluded or sewn shut during the critical period for ocular 
dominance development in the cat or monkey, it is found that substan- 
tially more of the cells in layer IV of primary visual cortex can be driven 
by the normal eye as compared to the deprived eye. The anatomical 
correlate of this is that ocular dominance stripes receiving input from the 
normal eye expand at the expense of the stripes from the deprived eye: 
however, stripe periodicity remains the same [see, e.g., for the cat Shatz 
and Stryker (1978) and for the macaque Hubel et al. (1977)l. 
It is the matching term in the elastic net which determines the amount 
of ”pull” each LGN unit exerts on the cortical sheet. In equation 1.1 each 
LGN unit exerts the same total pull. We modeled monocular deprivation 
by a procedure analogous to that used in other models [e.g., Miller ef al .  
(1989)l. The pull of all the units in one LGN was reduced by a fixed 
amount, corresponding to a decrease in competitive strength for that eye. 
This is most simply achieved by redefining the constant (P so that it has a 
different value for each eye. The amount of deprivation, or ”deprivation 
parameter,” was defined to be the ratio of 0 for the deprived eye to that 
for the normal eye. The variation of stripe pattern with the size of this 
deprivation parameter is shown for the elastic net algorithm in Figure 2. 
It can be seen that the deprived eye takes over less of the cortex than 
the normal eye: the deprived-eye stripes become thinner. However, the 
stripe periodicity remains the same as in the normal case. Note that we 
have only changed (r, all other parameters remaining the same. 
Figure 1: Facing page. The effect of the shape of the cortex on overall stripe 
pattern. Each LGN was a unit disk of approximately 1750 units arranged in a 
hexagonal array. The disks were separated by a gap of 0.08. A small random 
component was added to each 3-D position to prevent any artefacts that might 
arise from complete regularity. The cortical sheet contained approximately 3500 
units in a hexagonal array, bounded by either a circle (a) or an ellipse (b). 
Results are shown after 250 iterations: a slightly more efficient optimization 
procedure than steepest descent was used to reduce computation time [see, 
e.g., Durbin and Mitchison (1990); Goodhill (1991)l. The ocularity of cortical 
units is represented: each is colored white or black depending on which LGN 
is closer. All units have become monocular. In the circular boundary case (a) 
there is no preferred stripe direction, whereas in the elliptical boundary case (b) 
stripes tend to be aligned parallel to the short axis. Other parameters: = 0.2, 
/j = 2.0, initial value of k = 0.2, factor by which the annealing parameter k is 
reduced at each iteration = 0.99. Initial conditions were defined by assigning 
each cortical unit an arbitrary ocularity value, and a topographic position was 
choser? randomly from a uniform distribution within 0.5 (i.e., half the width of 
the LGN sheets) of its “ideal” location. 
620 Geoffrey J.  Goodhill and David J. Willshaw 
Figure 2: Effects of monocular deprivation. For the "black" eye, o = 0.2. (a) 
Deprivation parameter = 0.75. (b) Deprivation parameter = 0.5. All other 
parameters (including initial conditions) as in Figure l(b). Stripes representing 
the deprived eye become increasingly narrow as the size of the deprivation 
parameter is increased. 
Acknowledgments 
Funding: G. J. G. by an  SERC postgraduate studentship (at Sussex Uni- 
versity) and a Joint Councils Initiative in CogSci/HCI postdoctoral train- 
ing fellowship; D. J. W. by MRC Programme Grant PG9119632. We thank 
the anonymous referees for useful suggestions, and Martin Simmen for 
helpful discussions. 
Elastic Net Model of Ocular Dominance 621 
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~ ~~ ~ 
Received April 20, 1993; accepted November 17, 1993. 


Elastic net model of ocular dominance - overall stripe pattern and monocular deprivation

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