Development and Spatial Structure of Cortical 
Feature Maps: A Model Study 
K. Obermayer 
Beckman-Institute 
University of Illinois 
Urbana, IL 61801 
H. Ritter 
Technische Fakultiit 
Universit/it Bielefeld 
D-4800 Bielefeld 
K. Schulten 
Beckman-Institute 
University of Illinois 
Urbana, IL 61801 
Abstract 
Feature selective cells in the primary visual cortex of several species are or- 
ganized in hierarchical topographic maps of stimulus features like "position 
in visual space", "orientation" and "ocular dominance". In order to un- 
derstand and describe their spatial structure and their development, we in- 
vestigate a self-organizing neural network model based on the feature map 
algorithm. The model explains map formation as a dimension-reducing 
mapping from a high-dimensional feature space onto a two-dimensional 
lattice, such that "similarity" between features (or feature combinations) 
is translated into "spatial proximity" betveen the corresponding feature 
selective cells. The model is able to reproduce several aspects of the spatial 
structure of cortical maps in the visual cortex. 
1 Introduction 
Cortical maps are functionally defined structures of the cortex, vhich are charac- 
terized by an ordered spatial distribution of functionally specialized cells along the 
cortical surface. In the primary visual area(s) the response properties of these cells 
must be described by several independent features, and there is a strong tendency to 
map combinations of these features onto the cortical surface in a way that translates 
"similarity" into "spatial proximity" of the corresponding feature selective cells (see 
e.g. [1-6]). A neighborhood preserving mapping between a high-dimensional fea- 
ture space and the two dimensional cortical surface, however, cannot be achieved, so 
the spatial structure of these maps is a compromise, preserving some neighborhood 
relations at the expense of others. 
The compromise realized in the primary visual area(s) is a hierarchical represen- 
tation of features. The variation of the secondary features "preferred orientation", 
11 
12 Obermayer, Ritter, and Schulten 
"orientation specifity" and "ocular dominance" is highly repetitive across the pt/- 
mary map of retinal location, giving rise to a large number of small maps, each 
containing a complete representation of the full range of the secondary features. If 
the neighborhood relations in feature space are to be preserved and maps must be 
continuous, the spatial distributions of the secondary features "orientation prefer- 
ence", "orientation specifity" and "ocular dominance" can no longer be independent. 
Interestingly, [here is experimental evidence in the ma. ca.que that this is the case, 
namely, that regions with smooth change in one feature (e.g. "ocular dominance") 
correlate with regions of rapid change in another feature (e.g. "orientation") [7,8]. 
Preliminary results [9] indicate that these correlations may be a natural consequence 
of a dimension reducing mapping which preserves neighborhood relations. 
In a previous study, we investigated a model for the joint formation of a retino- 
topic projection and an orientation column system [10], which is based on the 
self-organizing feature map algorithm [11,12]. This algorithm generates a repre- 
sentation of a given manifold in feature space on a neural network with prespecified 
topology (in our case a two-dimensional sheet), such that the mapping is continous, 
smooth and neighborhood relations are preserved to a large extent.  The model 
has the advantage that its rules can be derived froin biologically plausible devel- 
opmental principles [15,16]. Therefore, it can be interpreted not only as a pattern 
model, which generates a representation of feature combinations subject to a set of 
constraints, but also as a pattern formation model, which describes an input driven 
developmental process. In this contribution ve will extend our previous work by the 
addition of another secondary feature, "ocular dominance" and we will concentrate 
on the hierarchical mapping of feature combinations as a function of the set of input 
patterns. 
2 Description of the Model 
In our model the cortical surface is divided into N x N small patches, units if, which 
are arranged on a two-dimensional lattice (network layer) with periodic boundary 
conditions (to avoid edge effects). The functional properties of neurons located in 
each patch are characterized by a feature vector Jr, which is associated with each 
unit , and whose components (r)k are interpreted as receptive field properties of 
these neurons. The feature vectors, Jr, as a function of unit locations F, describe the 
spatial distribution of feature selective cells over the cortical layer, i.e. the cortical 
map. 
To generate a representation of features along the network layer, we use the self- 
organizing feature map algorithm [1,2]. This algorithm follows an iterative proce- 
dure. At each step an input vector O', which is of the same dimensionality as Jr, 
is chosen at random according to a probability distribution P('). Then the unit 
', whose feature vector Jr is,closest to the input pattern 5', is selected and the 
components (r)k of it's feature vector are changed according to the feature map 
learning rule: 
,:(t + 1) = Jr(t) + e(t)hO',t)(5'-,:(t)) 
For other modelling approa.ches along these lines see [13,14]. 
Development and Spatial Structure of Cortical Feature Maps: A Model Study 13 
where h(r Z,t), the neighborhood function, is given by: 
exp (-(rl - 2 2 
= sl) /hl(t) -- -- 
3 Coding of Receptive Field Properties 
In the following ve describe the receptive field properties by the feature vector ,.- 
given by Z,.- = (xr-, Yr-, q,--cos(20r-), qr-sin(20,.-), z,.-) wilere (x, y,.-) denotes the 
position of the receptive field centers in visual space, (0') the preferred orientation, 
and (q,.-), (z) txvo quantities, which qualitatively can be interpreted as orientation 
specificity (see e.g. [17]) and ocular donfinance (see e.g. [18]). If qr is zero, then the 
units are unspecific for orientation; the larger q,.- becomes, the sharper the units are 
tuned. "Binocular" units are characterized by zr = 0, "monocular" units by a large 
positive or negative value of z,?. "Similarity" betveen receptive field properties is 
then given by the euclidean distance between the corresponding feature vectors. 
Tile components ,.- of the input vector  = (x, y, qcos(20), qsin(2O), z) describe 
stimulus features which should be represented by the cells in the cortical map. 
They denote position in the visual field (z, y), orientation , and two quantities q 
and z qualitatively describing pattern eccentricity and the distribution of activity 
betxveen both eyes, respectively. Round stimuli are characterized by q = 0 and the 
more eliptic a pattern is the larger is the value of q. A "binocular" stimulus is 
characterized by z = 0, while a "monocular" stinmlus is characterized by a large 
positive or negative value of z for "right eye" or "left eye" preferred, respectively. 
Input vectors were chosen with equal probability from the manifold 
V: {ffl x,  e [0, d]; q5 e[O, 7r]; q: qp.t; - 
i.e. all feature combinations characterized by a fixed value of q and Iz[ xvere selected 
equally often. If the model is interpreted from a developmental point of view, the 
manifold V describes properties of (subcortical) activity patterns, which drive map 
formation. The quantities d, qp,t and z.,at determine the feature combinations to 
be represented by the map. As we will see below, their values crucially influence 
the spatial structure of the feature map. 
4 Hierarchical Maps 
If qp,t and zp.t are smaller than a certain threshold then "orientation preference", 
"orientation selectivity" and "ocular dominance" are not represented in the map (i.e. 
qr: z,.-. = 0) but fluctuate around a stationary state of eq. (1), which corresponds 
to a perfect topographic representation of visual space. In this parameter regime, 
the requirement of a continous dimension-reducing map leads to the suppression of 
the additional features "orientation" and "ocular dominance". 
Let us consider an ensemble of networks, each characterized by a set {.} of feature 
vectors, and denote the time-dependent distribution function of this ensemble by 
14 Obermayer, Ritter, and Schulten 
$(,t). Following a method derived in [19], we can describe the time-development 
of $(tF, t) near the stationary state by the Fokker-Planck equation 
1o, (4) 
pmqn pmqn 
where the origin of $(.,t) was shifted to the stationary state {vt), using now the 
new argument variable g- = .- t. The eigenvalues of __ determine the stability 
of the stationary state, the topographic representation, while B_. and __D together 
govern size and time development of fluctuations < uFiu. j >. 
^ 
Let us define the Fourier modes ffi of the equilibrium deviations fit by ff = 
1/N .eik'ff '. For small values of qpat and Zrat the eigenvalues of B are all neg- 
ative, hence the topographic stationary state is stable. If qpat and Zrat are larger 
than 2 
 1 d min(o.nl, o-h2), (S) 
d min(o.nl o.n2), zth, - x/T  
qthres --  , -- 
however, the eigenvalues corresponding to the set of modes ff[ which are perpen- 
dicular to the (x, y)-plane and whose wave-vectors  are given by 
] -' -]-210'hi } if < (6) 
IEI = 2/0.. if o'hi '--o'h2 , ky -- 0 o'hi o.h2, 
become positive. For larger values of qpat and Zrat then, the topographic state 
becomes unstable and a "column system" forms. 
For an isotropic neighborhood function (O'hl -- O'h2 -' O'h) , the matrices /() and 
/5(') can be diagonalized simultaneously and the mean square amplitude of the 
fluctuations around the stationary state can be given in explicit form: 
d 2 (0.4k/4 + 1/12)exp(-o.k=/4) 
< u(/) >= a--o.]N2 exp(o.k2/4)- 1 + 
(7) 
e 2 d 2 exp(-o.k 2/4) 
< u].() >= a--o. N2 exp(o.k2/4)_ 1 
(8) 
 o.nqrt exp(o.k2/4)_ (N2qptk2)/(2d2) 
< u() >=< uu2() >= a-e   exp(-o.k2/4) 
2 e 2 2 exp(-o.k2/4) 
< >= - 
a- 2 o'zr' exp(o.k2/4) - (N2qtk2)/d 2 
(9) 
(lO) 
21n the derivation of the following formnlas several approximations ha.ve to be made. A 
comparison with numerical simulations, hmvever, demonstrate tha.t these approxima.tions are 
valid except, if the vahm qp. or zp is within a thw percent, of qa,-,, or za,.,..,, respectively. 
Details of these calculations will be published elsewhere 
Development and Spatial Structure of Cortical Feature Maps: A Model Study 15 
Figure 1: "Orientation preference" (a, left), "ocular dominance" (b, center) and 
locations of receptive field centers (c, right) as a function of unit loaction. Figure 
la displays an enlarged section of the "orientation map" only. Parameters of the 
simulation were: N = 256, d = 256, qrat = 12, Zpat = 12, a = 5, e = 0.02 
where Ull , u. denote the amplitude of fluctuations parallel and orthogonal to  
in the (x, y)-plane, uv , uv2 parallel to the orientation feature dimension and u, 
parallel to the ocular dominance feature dimension, respectively. 
Thus, for qr.t --* qt., or zr.t - zt., the mean square amplitudes of fluctuations 
diverge for the modes which become unstable at the threshold (the denominator of 
eqs. (9,10) approaches zero) and the relaxation time of these fluctuations goes to 
infinity (not shown). The fact that either a ring or two groups of modes become 
unstable is reflected in the spatial structure of the maps above threshold. 
For larger values of qpt and zt orientation and ocular dominance are represented 
by the network layer, i.e. feature values fluctuate around a stationary state which 
is characterized by a certain distribution of feature-selective cells. Figure 1 displays 
orientation preference qbr (Fig. la), ocular dominance z,.- (Fig. lb) and the locations 
(xr, Yr) of receptive field centers in visual space (Fig. lc) as a function of unit 
location f'. Each pixel of the images in Figs. la,b corresponds to a network unit 
Z Feature values are indicated by gray values: black --. white corresponds to an 
angle of 0  --. 180  (Fig. la) and to an ocular dominance value of 0 --* max (Fig. 
lb). White dots in Fig. la mark regions where units still completely unspecific for 
orientation are located ("foci"). In Fig. lc the receptive field center of every unit 
is marked by a dot. The centers of units which are neighbors in the network layer 
were connected by lines, which gives rise to the net-like structure. 
The overall preservation of the lattice topology, and the absence of any larger dis- 
continuities in Fig. lc, demonstrate that "position" plays the role of the primary 
stimulus variable and varies in a topographic fashion across the network layer. On 
a smaller length scale, however, numerous distortions are visible which are caused 
by the representation of the other features, "orientation" and "ocular dominance". 
The variation of these secondary features is highly repetitive and patterns strongly 
resembling orientation columns (Fig. lb) and ocular dominance stripes (Fig. lc) 
have formed. Note that regions unspecific for orientation as well as "binocular" 
regions exist in the final map, although these feature combinations were not present 
in the set of input patterns (3). They are correlated with regions of high magnitude 
of the "orientation" and "ocular dominance"-gradients, respectively (not shown). 
These structures are a consequence of the neighborhood preserving and dimension 
16 Obermayer, Ritter, and Schulten 
Figure 2: Two-dimensional Fourier spectra of the "orientation" (a, left) and "ocular 
dominance" (b, center) coordinates for the map shown in Fig. 1. c, right: Autocor- 
relation function of the feature coordinate wr3 for the map shown in Fig. 1. 
reducing mapping; they do not result from the requirement of representing this 
particular set of feature combinations. 3 
Figure 2a,b shows the tvo-dimensional Fourier spectra ibt7,occ = .ei'r'z' and 
tl,ori -- reik'q,,(cos(2c)r) + isin(2qbr)) for the "ocular dominance" (Fig. 2b) and 
"orientation" (Fig. 2a) coordinates, respectively. Each pixel corresponds to a single 
mode  and its brightness indicates the mean square amplitude [Sl 2 of the mode 
/. For an isotropic neighborhood function the orientation map is characterized 
by wave vectors from a ring shaped region in the Fourier domain (Fig. 2a), which 
becomes eccentric with increasing rral/aa2 (not shown) until the ring dissolves into 
two separate groups of modes. The phases (not shown) seem to be random, but 
we cannot exclude correlations completely. Figure 2c shows the autocorrelation 
function S33(8-') -- < W('_r) 3 W(')3 > as a function of the distance }' between cells 
in the network layer. The origin of the g'-pla. ne is located in the center of the image 
and the brightness indicates a positive (white), zero (medium gray) or negative 
(black) value of $33. The autocorrelation functions have a Mexican-hat form. The 
(negative) minimum is located at half the wavelength , associated with the the 
wave number of the modes with high amplitude in Fig. 2a. At this distance 
the response properties of the units are anticorrelated to some extent. If cells are 
separated by a distance larger than ,X, the response properties are uncorrelated. 
If qp,t and zp,t are large enough, the feature hierarchy observed in Figs. 1,2 breaks 
down and "preferred orientation" or "ocular dominance" plays the role of the pri- 
mary stimulus variable. Figure 3 displays orientation preference qbr (Fig. 3a) and 
ocular dominance z. (Fig. 3b) as a function of unit location Z There is only one 
continous region for each interval of "preferred orientation" and for each eye, but 
each of these regions now contains a representation of a large part of visual space. 
Consequently the position map shows multiple representations of visual space. 
Hierarchical maps are generated by the feature map algorithm whenever there is a 
hierarchy in the variances of the set of patterns along the various feature dimensions 
aln t, he cort,ex, however, cells nnspecific for orientat, ion seem t,o he import'ant' for vis.al 
proceqsing. To improve the descript,ion oft'he spatrial st,r.ct,ure of cort,ical maps, it' is neceqsary 
to include thee fhat'ure comhinat'ions int,o the set V of input' pat, t,erns (see [9]). 
Development and Spatial Structure of Cortical Feature Maps: A Model Study 17 
Figure 3: "Orientation preference" (a, left) and "ocular dominance" (b, center) as 
a function of unit loaction for a map generated using a large value of qpat and Zrat. 
Parameters were: N = 128, d = 128, qpat -- 2500, Zpat -' 2500, Oh -' 5, e -- 0.1 
(In our example a hierarchy in the magnitudes of d, qrat and Zpat). The features 
with the largest variance become the primary feature; the other features become 
secondary features, which are represented multiple times on the network layer. 
Acknowledgements 
The authors would like to thank the Boehringer-Ingelheim Fonds for financial sup- 
port by a scholarship to K. O. This research has been supported by the National 
Science Foundation (grant number 9017051). Computer time on the Connection 
Machine CM-2 has been made available by the National Center for Supercomputer 
Applications at Urbana-Champaign and the Pittsburgh Supercomputing Center 
both supported by the National Science Foundation. 
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