62 
Centric Models of the Orientation Map in Primary Visual Cortex 
William Baxter 
Department of Computer Science, S.U.N.Y. at Buffalo, NY 14620 
Bruce Dow 
Department of Physiology, S.U.N.Y. at Buffalo, NY 14620 
Abstract 
In the visual cortex of the monkey the horizontal organization of the preferred 
orientations of orientation-selective cells follows two opposing rules: 1) neighbors tend 
to have similar orientation preferences, and 2) many different orientations are observed 
in a local region. Several orientation models which satisfy these constraints are found 
to differ in the spacing and the topological index of their singularities. Using the rate 
of orientation change as a measure, the models are compared to published experimental 
results. 
Introduction 
It has been known for some years that there exist orientation-sensitive neurons in 
the visual cortex of cats and monkeys . These cells react to highly specific patterns of 
light occurring in narrowly circumscribed regions of the visual field, i.e, the cell's 
receptive field. The best patterns for such cells are typically not diffuse levels of 
illumination, but elongated bars or edges oriented at specific angles. An individual cell 
responds maximally to a bar at a particular orientation, called the preferred orienta- 
tion. Its response declines as the bar or edge is rotated away from this preferred orien- 
tation. 
Orientation-sensitive cells have a highly regular organi?ation in primary cortex 3. 
Vertically, as an electrode proceeds into the depth of the cortex, the column of tissue 
contains cells that tend to have the same preferred orientation, at least in the upper 
layers. Horizontally, as an electrode progresses across the cortical surface, the preferred 
orientations change in a smooth, regular manner, so that the recorded orientations 
appear to rotate with distance. It is this horizontal structure we are concerned with, 
hereafter referred to as the orientation map. An orientation map is defined as a two- 
dimensional surface in which every point has associated with it a preferred orientation 
ranging from 0   -- 180 . In discrete versions, such as the army of cells in the cortex or 
the discrete simulations in this paper, the orientation map will be considered to be a 
sampled version of the underlying continuous surface. The investigations of this paper 
are confined to the upper layers of macaque striate cortex. 
Detailed knowledge of the two-dimensional layout of the orientation map has 
implications for the architecture, development, and function of the visual cortex. The 
organization of orientation-sensitive cells reflects, to some degree, the organization of 
intracortical connections in striate cortex. Plausible orientation maps can be generated 
by models with lateral connections that are uniformly exhibited by all cells in the 
layer 4, or by models which presume no specific intracortical connections, only 
appropriate patterns of afferent input 6. In this paper, we examine models in which 
intracortical connections produce the orientation map but the orientation-controlling 
circuitry is not displayed by all cells. Rather, it derives from localized "centers" which 
are distributed across the cortical surface with uniform spacing ?,8,9. 
American Institute of Physics 1988 
63 
The orientation map also represents a deformation in the retinotopy of primary 
visual cortex. Since the early sixties it has been known that V1 reflects a topographic 
map of the retina and hence the visual field . There is some global distortion of this 
mapping ,2,s, but generally spatial relations between points in the visual field are 
maintained on the cortical surface. This well-known phenomenon is only accurate for 
a medium-grain description of V1, however. At a finer cellular level there is consider- 
able scattering of receptive fields at a given cortical location TM. The notion of the hyper- 
column s proposes that such scattering permits each region of the visual field to be 
analyzed by a population of cells consisting of all the necessary orientations and with 
inputs from both eyes. A quantitative description of the orientation map will allow 
prediction of the distances between iso-orientation zones of a particular orientation, 
and suggest how much cortical machinery is being brought to bear on the analysis of a 
given feature at a given location in the visual field. 
Models of the Orientation Map 
Hubel and Wiesel's Parallel Stripe Model 
The classic model of the orientation map is the parallel stripe model first pub- 
lished by Hubel and Wiesel in 1972 s. This model has been reproduced several times in 
their publications 3,s.7 and appears in many textbooks. The model consists of a series of 
parallel slabs, one slab for each orientation, which are postulated to be orthogonal to 
the ocular dominance stripes. The model predicts that a microelectrode advancing 
tangentially (i.e., horizontally) through the 
orientations. The rate of change, which is 
determined by the angle of the electrode 
stripes. 
tissue should encounter steadily changing 
also called the orientation drift rate 8, is 
with respect to the array of orientation 
The parallel stripe model does not account for several phenomena reported in 
long tangential penetrations through striate cortex in macaque monkeys ?,9. First, as 
pointed out by Swindale 2, the model predicts that some penetrations will have flat or 
very low orientation drift rates over lateral distances of hundreds of micrometers. 
This is because an electrode advancing horizontally and perpendicular to the ocular 
dominance stripes (and therefore parallel to the orientation stripes) would be expected 
to remain within a single orientation column over a considerable distance with its 
orientation drift rate equal to zero. Such results have never been observed. Second, 
reversals in the direction of the orientation drift, from clockwise to counterclockwise 
or vice versa, are commonly seen, yet this phenomenon is not addressed by the parallel 
stripe model. Wavy stripes in the ocular dominace system  do not by themselves 
introduce reversals. Third, there should be a negative correlation between the orienta- 
tion drift rate and the ocularity "drift rate". That is, when orientation is changing 
rapidly, the electrode should be confined to a single ocular dominance stripe (low ocu- 
larity drift rate), whereas when ocularity is changing rapidly the electrode should be 
confined to a single orientation stripe (low orientation drift rate). This is clearly not 
evident in the recent studies of Livingstone and Hubel ? (see especially their figs. 3b, 
21 & 23), where both orientation and ocularity often have high drift rates in the same 
electrode track, i.e, they show a positive correlation. Anatomical studies with 2- 
deoxyglucose also fail to show that the orientation and ocular dominance column sys o 
tems are orthogonal 2. 
64 
Centric Models and the Topological Index 
Another model, proposed by Braitenberg and Braitenberg in 19797, has the orien- 
tations arrayed radially around centers like spokes in a wheel The centers are spaced at 
distances of about 0.Sram. This model produces reversals and also the sinusoidal pro- 
gressions frequently encountered in horizontal penetrations. However this approach 
suggests other possibilities, in fact an entire class of centtic models. The organizing 
centers form discontinuities in the otherwise smooth field of orientations. Different 
topological types of discontinuity are possible, characterized by their topological 
index 23. The topological index is a parameter computed by taking a path around a 
discontinuity and recording the rotation of the field elements (figure 1). The value of 
the index indicates the amount of rotation; the sign indicates the direction of rotation. 
An index of 1 signifies that the orientations rotate through 360; an index of 1/2 
signifies a 180  rotation. A positive index indicates that the orientations rotate in the 
same sense as a path taken around the singularity; a negative index indicates the 
reverse rotation. 
Topological singularities are stable under orthogonal transformations, so that if 
the field elements are each rotated 90  the index of the singularity remains unchanged. 
Thus a +1 singularity may have orientations radiating out from it like spokes from a 
wheel, or it may be at the center of a series of concentric circles. Only four types of 
discontinuities are considered here, +1, -1, +1/2, -1/2, since these are the most stable, i.e., 
their neighborhoods are characterized by smooth change. 
I I I I I I 
// I 
% 
/ i I I , 
/ / / \ \ 
+1 +1/2 -1 
figure 1. Topological singularities. A positive index indicates that the orientations rotate 
in the same direction as a path taken around the singularity; a negative index indicates 
the reverse rotation. Orientations rotate through 360  around +1 centers, 180  around 
/2 centers. 
Cytochrome Oxidase Puffs 
At topological singularities the change in orientation is discontinuous, which 
violates the structure of a smoothly changing orientation map; modeHers try to 
minimize discontinuities in their models in order to satisfy the smoothness constraint. 
Interestingly, in the upper layers of striate cortex of monkeys, zones with little or no 
orientation selectivity have been discovered. These zones are notable for their high 
cytochrome oxidase reactivity 24 and have been referred to as cytochrome oxidase puffs, 
dots, spots, patches or blob 17252627. We will refer to them as puffs. If the organizing 
centers of centtic models are located in the cytochrome oxidase puffs then the discon- 
tinuities in the orientation map are effectively eliminated (but see below). Braitenberg 
has indicated 2s that the +1 centers of his model should correspond to the puffs. Dow 
and Bauer proposed a model s with +1 and -1 centers in alternating puffs. Gotz proposed 
a similar model 9 with alternating +1/2 and -1/2 centers in the puffs. The last two 
models manage to eliminate all discontinuities from the interpuff zones, but they 
65 
assume a perfect rectangular lattice of cytochrome oxidase puffs. 
A Set of Centric Models 
There are two parameters for the models considered here. (1) Whether the posi- 
tive singularities are placed in every puff or in alternate puffs; and (2) whether the 
singularities are +l's or a/2's. This gives four centric models (figure 2): 
E1 : +1 centers in puffs, -1 centers in the interpuff zones. 
A1 : both +1 and -1 centers in the puffs, interdigitated in a checkerboard fashion. 
E1/2 : +1/2 centers in the puffs, -1/2 centers in the interpuff zones 
A1/2 : both +1/2 and -1/2 centers in the puffs, as in A1. 
The E1 model corresponds to the Braitenberg model transposed to a rectangular array 
rather than an hexagonal one, in accordance with the observed organization of the 
cytochrome oxidase regions 27. In fact, the rectangular version of the Braitenberg model 
is pictured in figure 49 of 27. The A1 model was originally proposed by Dow and 
Bauer s and is also pictured in an article by Mitchison 29. The A1/2 model was proposed 
by Gotz 9. It should be noted that the E1 and A1 models are the same model rotated 
and scaled a bit; the E1/2 and A1/2 have the same relationship. 
E1 
E 
A1 
A 
figure 2. The four centric models. Dark ellipses represent cytochrome oxidase puffs. 
Dots in interpuff zones of E1 & E1/2 indicate singularities at those points. 
66 
Simulo. tions 
Simulated horizontal electrode recordings were made in the four models to com- 
pare their orientation drift rates with those of published recordings. In the computer 
simulations (figure 2) the interpuff distances were chosen to correspond to histological 
measuremen 27. Puff centers are separated by 500 along their long axes, 350 along 
the short axes. The density of the arrays was chosen to approximate the sampling fre- 
quency observed in Hubel and Wiesel's horizontal electrode recording experiments 9, 
about 20 cells per millimeter. Therefore the cell density of the simulation arrays was 
about six times that shown in the figure. 
All of the models produce simulated electrode data that qualitatively resemble 
the published recording results, e.g., they contain reversals, and runs of constantly 
changing orientations. The orientation drift rate and number of reversals vary in the 
different models. 
The models of figure 2 are shown in perfectly rectangular arrays. Some impor- 
tant characteristics of the models, such as the absence of discontinuites in interpuff 
zones, are dependent on this regularity. However, the real arrangement of cytochrome 
oxidase puffs is somewhat irregular, as in Horton's figure 327. A small set of puffs from 
the parafoveal region of Horton's figure was enlarged and each of the centtic models 
was embedded in this irregular array. The E1 model and a typical simulal electrode 
track are shown in figure 3. Several problems are encountered when models developed 
in a regular lattice are implemented in the irregular lattice of the real system; the 
models have appreciably different properties. The -1 singularities in 1's interpuff 
zones have been reduced to -2s; the A1 and A1/2 models now have some interpuff 
discontinuities where before they had none. 
Quantitative Comparisons 
Measurement of the Orientation Drift Rate 
There are two sets of centtic models in the computer simulations: a set in the per- 
fectly rectangular array (figure 2) and a set in the irregular puff array (as in figure 3). 
At this point we can generate as many tracks in the simulation arrays as we wish. 
How can this information be compared to the published records? The orientation drift 
rate, or slope, is one basis for distinguishing between models. In real electrode tracks 
however, the data are rather noisy, perhaps from the measuring process or from 
inherent unevenness of the orientation map. The typical approach is to fit a straight 
line and use the slope of this line. Reversals in the tracks require that lines be fit piece- 
wise, the approach used by Hubel and Wiesel 9. Because of the unevenness of the data 
it is not always clear what constitutes a reversal. Livingstone and Hubel 7 report that 
the track in their figure 5 has only two reversals in 5 millimeters. Yet there seem to be 
numerous microreversals between the 1st and 3rd millimeter of their track. At what 
point is a change in slope considered a true reversal rather than just noise? 
The approach used here was to use a local slope measure and ignore the problem 
of reversals - this permitted the fast calculation of slope by computer. A single elec- 
trode track, usually several millimeters long, was assigned a single slope, the average 
of the derivative taken at each point of the track. Since these are discrete samples, the 
local derivative must be approximated by taking measurements over a small neighbor- 
hood. How large should this neighborhood be? If too small it will be susceptible to 
noise in the orientation measures, if too large it will "fiatten out" true reversals. Slope 
67 
E1 
rlrq 
figure 3. A centric model in a realistic puff array (from2?). A simulated electrode track 
and resulting data are shown. Only the E1 model is shown here, but other models 
were similarly embedded in this array. 
68 
measures using neighborhoods of several sizes were applied to six published horizontal 
electrode tracks from the foveal and parafoveal upper layers of macaque striate cortex: 
figures 5,6,7 from ?, figure 16 from 3, figure 1 from 3. A neighborhood of 0.1ram, 
which attempts to fit a line between virtually every pair of points, gave abnormally 
high slopes. Larger neighborhoods tended to give lower slopes, especially to those 
tracks which contained reversals. The smallest window that gave consistent measures 
for all six tracks was 0.2mm; therefore this window was chosen for comparisons 
between published data and the centric models. This measure gave an average slope of 
285 degrees per millimeter in the six published samples of track data, compared to 
Hubel & Wiesel's measure of 281 deg/mm for the penetrations in their 1974 pape r9. 
Slope measures of the centtic models 
The slope measure was applied to several thousand tracks at random locations 
and angles in the simulation arrays, and a slope was computed for each simulated elec- 
trode track. Average slopes of the models are shown in Table I. Generally, models 
with 1 centers have higher slopes than those with a/2 centers; models with centers 
in every puff have higher slopes than the alternate puff models. Thus E1 showed the 
highest orientation drift rate, A1/2 the lowest, with A1 and EV2 having intermediate 
rates. The E1 model, in both the rectangular and irregular arrays, produced the most 
realistic slope values. 
TABLE I Average slopes of the centric models 
Rectangular Irregular 
array array 
312 289 
216 216 
198 202 
117 144 
E1 
A1 
El/2 
A1/2 
Numbers are in degrees/mm. Slope measure (window = 0.2mm) applied 
to 6 published records yielded an average slope of 285 degrees/mm. 
Discussion 
Constraints on the Orientation Map 
Our original definition of the orientation map permits each cell to have an orien- 
tation preference whose angle is completely independent of its neighbors. But this is 
much too general. Looking at the results of tangential electrode penetrations, there are 
two striking constraints in the data. The first of these is reflected in the smoothness of 
the graphs. Orientation changes in a regular manner as the electrode moves horizon- 
tally through the upper layers: neighboring cells have similar orientation preferences. 
Discontinuities do occur but are rare. The other constraint is the fact that the orienta- 
tion is always changing with distance, although the rate of change may vary. 
Sequences of constant orientation are very rare and when they do occur they never 
carry on for any appreciable distance. This is one of the major reasons why the paral- 
lel stripe model is untenable. The two major constraints on the orientation map may 
be put informally as follows: 
69 
1. The smoothness constraint: neighboring points have similar orientation 
preferences. 
2. The heterogeneity constraint: all orientations should be represented 
within a small region of the cortical surface. 
This second constraint is a bit stronger than the data imply. The experimental results 
only show that the orientations change regularly with distance, not that all orienta- 
tions must be present within a region. But this constraint is important with respect to 
visual processing and the notion of hypercolumns 3. 
These are opposing constraints: the first tends to minimize the slope, or orienta- 
tion drift rate, while the second tends to maximize this rate. Thus the organization of 
the orientation map is analogous to physical systems that exhibit "frustration", that is, 
the elements must satisfy conflicting constraints 3. One of the properties of such sys- 
tems is that there are many near-optimal solutions, no one of which is significantly 
better than the others. As a result, there are many plausible orientation maps: any map 
that satisfies these two constraints will generate qualitatively plausible simulated elec- 
trode tracks. This points out the need for quantitative comparisons between models 
and experimental results. 
Centric models and the two constraints 
What are some possible mechanisms of the constraints that generate the orienta- 
tion map? Smoothness is a local property and could be attributed to the workings of 
individual cells. It seems to be a fundamental property of cortex that adjacent cells 
respond to similar stimuli. The heterogeneity requirement operates at a slightly larger 
scale, that of a hypercolumn rather than a minicolumn. While the first constraint may 
be modeled as a property of individual cells, the second constraint is distributed over a 
region of cells. How can such a collection of cells insure that its members cycle 
through all the required orientations? The topological singularities discussed earlier, by 
definition, include all orientations within a restricted region. By distributing these 
centers across the surface of the cortex, the heterogeneity constraint may be satisfied. In 
fact, the amount of orientation drift rate is a function of the density of this distribu- 
tion (i.e., more centers per unit area give higher drift rates). 
It has been noted that the E1 and the A1 organizations are the same topological 
model, but on different scales; the low drift rates of the A1 model may be increased by 
increasing the density of the +1 centers to that of the E1 model. The same relationship 
holds for the El/2 and A1/2 models. It is also possible to obtain realistic orientation drift 
rates by increasing the density of +1/2 centers, or by mixing +l's and +1/2's. However, 
these alternatives increase the number of interpuff singularities. And given the possible 
combinations of centers, it may be more than coincidental that a set of +1 centers at 
just the spacing of the cytochrome oxidase regions results in realistic orientation drift 
rates. 
Cortical Architecture and Types of Circuitry 
Thus far, we have not addressed the issue of how the preferred orientations are 
generated. The mechanism is presently unknown, but attempts to depict it have tradi- 
tionally been of a geometric nature, alluding to the dendritic morphologf '282. More 
recently, computer simulations have shown that orientation-sensitive units may be 
obtained from asymmetries in the receptive fields of afferents 6, or developed using 
7O 
simple Hebbian rules for altering synaptic weights 5. That is, given appropriate net- 
work parameters, orientation tuning arises an as inherent property of some neural net- 
works. Centtic models propose a quite different approach in which an originally 
untuned cell is "programmed" by a center located at some distance to respond to a 
specific orientation. So, for an individual cell, does orientation develop locally, or is it 
"imposed from without"? Both of these mechanisms may be in effect, acting synergisti- 
cally to produce the final orientation map. The map may spontaneously form on the 
embryonic cortex, but with cells that are nonspecific and broadly tuned. The organiza- 
tion imposed by the centers could have two effects on this incipient map. First, the 
additional influence from centers could "tighten up" the tuning curves, making the 
cells more specific. Second, the spacing of the centers specifies a distinct and uniform 
scale for the heterogeneity of the map. An unsupervised developing orientation map 
could have broad expanses of iso-orientation zones mixed with regions of rapidly 
changing orientations. The spacing of the puffs, hence the architecture of the cortex, 
insures that there is an appropriate variety of feature sensitive cells at each location. 
This has implications for cortical functioning: given the distances of lateral connec- 
tivity, for a cell of a given orientation, we can estimate how many other iso- 
orientation zones of that same orientation the cell may be communicating with. For a 
given orientation, the E1 model has twice as many iso-orientation zones per unit area 
as A1. 
Ever since the discovery of orientation-specific cells in visual cortex there have 
been attempts to relate the distribution of cell selectivities to architectural features of 
the cortex. Hubel and Wiesel originally suggested that the orientation slabs followed 
the organization of the ocular dominance slabs s. The Braitenbergs suggested in their 
original model ? that the centers might be identified with the giant cells of Meynert. 
Later centric models have identified the centers with the cytochrome oxidase regions, 
again relating the orientation map to the ocular dominance array, since the puffs them- 
selves are closely related to this array. 
While biologists have habitually related form to function, workers in machine 
vision have traditionally relied on general-purpose architectures to implement a 
variety of algorithms related to the processing of visual information 33. More recently, 
many computer scientists designing artificial vision systems have turned their atten- 
tion towards connectionist systems and neural networlm There is great interest in 
how the sensitivities to different features and how the selectivities to different values 
of those features may be embedded in the system architecture 34sa6. Linsker has pro- 
posed (this volume) that the development of feature spaces is a natural concomitance 
of layered networks, providing a generic organizing principle for networks. Our work 
deals with more specific cortical architectonics, but we are convinced that the study of 
the cortical layout of feature maps will provide important insights for the design of 
artificial systems. 
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