A comparison between a neural network model for 
the formation of brain maps and experimental data 
K. Obermayer 
Beckman-Institute 
University of Illinois 
Urbana, IL 61801 
K. Schulten 
Beckman-Institute 
University of Illinois 
Urbana, IL 61801 
(3.(3. Blasdel 
Harvard Medical School 
Harvard University 
Boston, MA 02115 
Abstract 
Recently, high resolution images of the simultaneous representation of 
orientation preference, orientation selectivity and ocular dominance have 
been obtained for large areas in monkey striate cortex by optical imaging 
[1-3]. These data allow for the first time a "local" as well as "global" 
description of the spatial patterns and provide strong evidence for corre- 
lations between orientation selectivity and ocular dominance. 
A quantitative analysis reveals that these correlations arise when a five- 
dimensional feature space (two dimensions for retinotopic space, one each 
for orientation preference, orientation specificity, and ocular dominance) is 
mapped into the two available dimensions of cortex while locally preserving 
topology. These results provide strong evidence for the concept of topology 
preserving maps which have been suggested as a basic design principle of 
striate cortex [4-7]. 
Monkey striate cortex contains a retinotopic map in which are embedded the highly 
repetitive patterns of orientation selectivity and ocular dominance. The retinotopic 
projection establishes a "global" order, while maps of variables describing other 
stimulus features, in particular line orientation and ocularity, dominate cortical 
organization locally. A large number of pattern models [8-12] as well as models 
of development [6,7,13-21] have been proposed to describe the spatial structure of 
these patterns and their development during ontogenesis. However, most models 
have not been compared with experimental data in detail. There are two reasons 
for this: (i) many model-studies were not elaborated enough to be experimentally 
testable and (ii) a sufficient amount of experimental data obtained from large areas 
of striate cortex was not available. 8J 
84 Obermayer, Schulten, and Blasdel 
Figure 1: Spatial pattern of orientation preference and ocular dominance in mon- 
key striate cortex (left) compared with predictions of the SOFM-model (right). Iso- 
orientation lines (gray) are drawn in intervals of 11.25  (left) and 18.0  (right), re- 
spectively. Black lines indicate the borders (ws(r-') = 0) of ocular dominance bands. 
The areas enclosed by black rectangles mark corresponding elements of organization 
in monkey striate cortex and in the simulation result (see text). Left: Data obtained 
from a 3.1mm x 4.2mm patch of the striate cortex of an adult macaque (macaca 
hemestrina) by optical imaging [1-3]. The region is located near the border with 
area 18, close to midline. Right: Model-map generated by the SOFM-algorithm. 
The figure displays a small section of a network of size N = d = 512. The param- 
eters of the simulation were: e = 0.02, h = 5, v. = 20.48, vax = 15.36, 9  10 ? 
iterations, with retinotopic initial conditions and Jeriodic boundary conditions. 
Orientation and ocular dominance columns in monkey 
striate cortex 
Recent advances in optical imaging [1-3,22,23] now make it possible to obtain high 
resolution images of the spatial pattern of orientation selectivity and ocular domi- 
nance from large cortical areas. Prima vista analysis of data from monkey striate 
cortex reveals that the spatial pattern of orientation preference and ocular domi- 
nance is continuous and highly repetitive across cortex. On a global scale orienta- 
tion preferences repeat along every direction of cortex with similar periods. Locally, 
orientation preferences are organized as parallel slabs (arrow 1, Fig. la) in linear 
zones, which start and end at singularities (arrow 2, Fig. la), point-like disconti- 
nuities, around which orientation preferences change by :t:180  in a pinwheel-like 
fashion. Both types of singularities appear in equal numbers (359:354 for maps 
obtained from four adult macaques) with a density of 5.5/mm 2 (for regions close to 
A Neural Network Model for the Formation of Brain Maps Compared with Experimental Data 85 
Fourier transforms 
correlation functions 
feature gradients 
Gabor transforms 
= E, exp(iO 
= < i(O v') 
= + 
+ + - 
Table 1: Quantitative measures used to characterize cortical maps. 
the midline). Figure la reveals that the iso-orientation lines cross ocular dominance 
bands at nearly right angles most of the time (region number 2) and that singular- 
ities tend to align with the centers of the ocular dominance bands (region number 
1). Where orientation preferences are organized as parallel slabs (region number 
2), the iso-orientation contours are often equally spaced and orientation preferences 
change linearly with distance. 
These results are confirmed by a quantitative analysis (see Table 1). For the 
following we denote cortical location by a two-dimensional vector r'. At each lo- 
cation we denote the (average) position of receptive field centroids in visual space 
by (wx(, w2(). Orientation selectivity is described by a two-dimensional vector 
(ws(r, w4(), whose length and direction code for orientation tuning strength and 
preferred orientation, respectively [1,10]. Ocular dominance is described by a real- 
valued function ws(r-'), which denotes the difference in response to stimuli presented 
to the left and right eye. Data acquisition and postprocessing are described in detail 
in [1-3]. 
A Fourier transform of the map of orientation preferences reveals a spectrum which 
is a nearly circular band (Fig. 2a), showing that orientation preferences repeat 
with similar periods in every direction in cortex. Neglecting the slight anisotropy 
in the experimental data , a power spectrum can be approximated by averaging 
amplitudes over all directions of the wave-vector (Fig. 2b, dots. The location of 
the peak corresponds to an average period A0 = 710pm -t- 50pm  and it's width to 
a coherence length of 820pm :t: 130pm. The coherence length indicates the typical 
distance over which orientation preferences can change linearly and corresponds 
to the average size of linear zones in Fig. la. The corresponding autocorrelation 
functions (Fig. 2c) have a Mexican hat shape. The minimum occurs near 300pm, 
which indicates that orientation preferences in regions separated by this distance 
tend to be orthogonal. In summary, the spatial pattern of orientation preference is 
characterized by local correlation and global "disorder". 
Along axes parallel to the ocular dominance slabs, orientation preferences repeat on 
average every 660pm 4- 40pro; perpendicular to the stripes every 840pro 4- 40pro. The slight 
horizontal elongation reflects the fact that iso-orientation slabs tend to connect the centers 
of ocular dominance hands. 
2All quantities regarding experimental data are averages over four animals, nml-nm4, 
unle stated otherwise. Error margins indicate standard deviations. 
86 
Obermayer, Schulten, and Blasdel 
a) 
 1 . 1.o 
1 .o  0.5 
& o.o %.o ,.o 
spatiff equency distce (noflized) 
(noflized) 
Figure 2: Fourier analysis and correlation functions of the orientation map in 
monkey striate cortex (animal nm2) compared with the predictions of the SOFM- 
model. Simulation results were taken from the data set described in Fig. 1, right. 
(a) Fourier spectra of nm2 (left) and simulation results (right). Each pixel repre 
sents one mode; location and gray value of the pixel indicate wave-vector and energy, 
respectively. (b) Approximate power spectrum (normalized) obtained by averaging 
the Fourier-spectra in (a) over all directions of the wave-vector. Peak frequency 
of 1.0 corresponds to 1.4/mm for nm2. (c) Correlation functions (normalized). A 
distance of 1.0 corresponds to 725/m for nm2. 
Local properties of the spatial patterns, as well as correlations between orientation 
preference and ocular dominance, can be quantitatively characterized using Gabor- 
Helstrom-transforms (see Table 1). If the radius tr of the Gaussian function in the 
Gabor-filter is smaller than the coherence length the Gabor-transform of any of the 
quantities wa(, w4( and ws( typically consists of two localized regions of high 
energy located on opposite sides of the origin. The length IFil of the vectors/i, 
i  [3,4, 5], which corresponds to the centroids of these regions, fluctuates around 
the characteristic wave-number 2r/A0 of this pattern, and its direction gives the 
normal to the ocular dominance bands and iso-orientation slabs at the location F, 
where the Gabor-transform was performed. 
A Neural Network Model for the Formation of Brain Maps Compared with Experimental Data 87 
nm I - nm4 theory 
Figure 3: Gabor-analysis of cortical maps. The percentage of map locations is 
plotted against the parameters sx and s2 (see text) for 3,421 locations randomly 
selected from the cortical maps of four monkeys, nml-nm4, (left) and for 1,755 
locations randomly selected from simulation results (right). Error bars indicate 
standard deviations. Simulation results were taken from the data set described in 
Fig. 1. rg was 150pm for the experimental data and 28 pixels for the SOFM-map. 
Results of this analysis are shown in Fig. 3 (left) for 3,434 samples selected randomly 
from data of four animals. The angle between k'a and 4 is represented along the Sl 
axis. Histograms at the back, where Sl = 0 , represent regions where iso-orientation 
lines are parallel. Histograms in the front, where sx = 90 , represent regions con- 
taining singularities. The intersection angle of iso-orientation slabs and ocular dom- 
inance bands is represented along the s2 axis. The proportion of sampled regions 
increases steadily with decreasing s. As s approaches zero, values accumulate 
at the right, where orientation and ocular dominance bands are orthogonal. Thus 
linear zones and singularities are important elements of cortical organization but 
linear zones (back rows) are the most prominent features in monkey striate cortex s. 
Where iso-orientation regions are organized as parallel slabs, orientation slabs in- 
tersect ocular dominance bands at nearly right angles (back and right corner of 
diagrams). 
2 Topology preserving maps 
Recently, topology preserving maps have been suggested as a basic design principle 
underlying these patterns and its was proposed that these maps are generated by 
simple and biologically plausible pattern formation processes [4,6,7]. In the following 
we will test these models against the recent experimental data. 
We consider a five-dimensional feature space V which is spanned by quantities de- 
scribing the most prominent receptive field properties of cortical cells: position 
of a receptive field in retinotopic space (v, v2), orientation preference and tuning 
strength (rs, v4), and ocular dominance (rs). If all combinations of these properties 
aData from area 17 of the cat indicate that in this species, although both elements are 
present, sing]larities are more important [23] 
88 Obermayer, Schulten, and Blasdel 
are represented in striate cortex, each point in this five-dimensional feature space 
is mapped onto one point on the two-dimensional cortical surface A. 
In order to generate these maps we employ the feature map (SOFM-) algorithm 
of Kohonen [15,16] which is known to generate topology preserving maps between 
spaces of different dimensionality [4,5] 4. The algorithm describes the development 
of these patterns as unsupervised learning, i.e. the features of the input patterns 
determine the features to be represented in the network [4]. Mathematically, the 
algorithm assignes feature vectors t(, which are points in the feature space, to 
cortical units F, which are points on the cortical surface. In our model the surface is 
divided into N x N small patches, units F, which are arranged on a two-dimensional 
lattice (network layer) with periodic boundary conditions (to avoid edge effects). 
The average receptive field properties of neurons located in each patch are char- 
acterized by the feature vector t( whose components (tj( are interpreted as 
receptive field properties of these neurons. The algorithm follows an iterative pro- 
cedure. At each step an input vector 0', which is of the same dimensionality as t( 
is chosen at random according to a probability distribution P(. Then the unit 
F whose feature vector t( is closest to the input pattern 6 is selected and the 
components (tj(r of its feature vector are changed according to the feature map 
learning rule [15,16], 
= exp - - - 
(1) 
P( was chosen to be constant within a cylindrical manifold in feature space, 
where v max and v ax 
s,4 are some real constants, and zero elsewhere. 
Figure 4 shows a typical map, a surface in feature space, generated by the SOFM- 
algorithm. For the sake of illustration the five-dimensional feature space is projected 
onto a three-dimensional subspace spanned by the coordinate-axes corresponding 
to retinotopic location (v and va) and ocular dominance (va). The locations of 
feature vectors assigned to the cortical units are indicated by the intersections of 
a grid in feature space. Preservation of topology requires that the feature vectors 
assigned to neighboring cortical units must locally have equal distance and must be 
arranged on a planar square lattice in feature space. Consequently, large changes 
in one feature, e.g. ocular dominance va, along a given direction on the network 
correlate with small changes of the other features, e.g. retinotopic location v and 
v:, along the same direction (crests and troughs of the waves in Fig. 4) and vice 
versa. Other correlations arise at points where the map exhibits maximal changes 
in two features. For example for retinotopic location (v) and ocular dominance 
(v) to vary at a maximal rate, the surface in Fig. 4 must be parallel to the (v, v)- 
plane. Obviously, at such points the directions of maximal change of retinotopic 
location and ocular dominance are orthogonal on the surface. 
In order to compare model predictions with experimental data the surface in the five- 
dimensional feature space has to be projected into the three-dimensional subspace 
4The exnct form of the algorithm is not es.,mntial, however. Algorithms ha.ed on similnr 
principle, e.g. the elastic net algorithm [6], predict similnr pntterns. 
A Neural Network Model for the Formation of Brain Maps Compared with Experimental Data 89 
Figure 4: Typical map gen- 
V2 erated by the SOFM-algorithm. 
The five-dimensional feature 
space is projected into the three- 
dimensional subspace spanned by 
the three coordinates (vl, v2 and 
rs). Locations of feature vectors 
V5 which are mapped to the units in 
the network are indicated by the 
intersections of a grid in feature 
space. Only every fourth vector 
is shown. 
spanned by orientation preferences (rs and v4) and ocular dominance (rs). This 
projection cannot be visualized easily because the surface completely fills space, 
intersecting itself multiple times. However, the same line of reasoning applies: (i) 
regions where orientation preferences change quickly, correlate with regions where 
ocular dominance changes slowly, and (ii) in regions where orientation preferences 
change most rapidly along one direction, ocular dominance has to change most 
rapidly along the orthogonal direction. Consequently we expect discontinuities of 
the orientation map to be located in the centers of the ocular dominance bands and 
iso-orientation slabs to intersect ocular dominance bands at steep angles. 
Figures 1, 2 and 3 show simulation results in comparison with experimental data. 
The algorithm generates all the prominent features of lateral cortical organization: 
singularities (arrow 1), linear zones (arrow 2), and parallel ocular dominance bands. 
Singularities are aligned with the centers of ocular dominance bands (region 1) and 
iso-orientation slabs intersect ocular dominance stripes at nearly right angles (region 
2). The shape of Fourier- and power-spectra as well as of the correlation functions 
agrees quantitatively with the experimental data (see Fig. 2). Isotropic spectra 
are the result of the invariance of eqs. (1) and (2) under rotation with respect to 
cortical coordinates F; global disorder and singularities are a consequence of their 
invariance under translation. The emergence of singularities can also be under- 
stood from an entropy argument. Since dimension reducing maps, which exhibit 
these features, have increased entropy, they are generated with higher probability. 
Correlations between orientation preference and ocular dominance, however, follow 
from geometrical constraints and are inherent properties the topology preserving 
maps. 
3 Conclusions 
On the basis of our findings the following picture of orientation and ocular domi- 
nance columns in monkey striate cortex emerges. Orientation preferences are or- 
ganized into linear zones and singularities, but areas where iso-orientation regions 
form parallel slabs are apparent across most of the cortical surface. In linear zones, 
90 Obermayer, Schulten, and Blasdel 
iso-orientation slabs indeed intersect ocular dominance slabs at right angles as ini- 
tially suggested by Hubel and Wiesel [8]. Orientation preferences, however, are 
arranged in an orderly fashion only in regions 0.8mm in size, and the pattern is 
characterized by local correlation and global disorder. 
These patterns can be explained as the result of topology-preserving, dimension 
reducing maps. Local correlations follow from geometrical constraints and are a 
direct consequence of the principle of dimension reduction. Global disorder and 
singularities are consistent with this principle but reflect their generation by a local 
and stochastic self-organizing process. 
Acknowledgement s 
The authors would like to thank H. Ritter for fruitful discussions and comments 
and the Boehringer-Ingelheim Fonds for financial support by a scholarship to K. 
O. This research has been supported by the National Science Foundation (grant 
numbers DIR 90-17051 and DIR 91-22522). Computer time on the Connection 
Machine CM-2 has been made available by the National Center for Supercomputer 
Applications at Urbana-Champaign funded by NSF. 
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