INTERACTION AMONG OCULARITY, 
RETINOTOPY AND ON-CENTER/OFF- 
CENTER PATHWAYS DURING 
DEVELOPMENT 
Shigeru Tanaka 
Fundamental Research Laboratories, NEC Corporation, 
34 Miyukigaoka, Tsukuba, Ibaraki 305, Japan 
ABSTRACT 
The development of projections from the retinas to the cortex is 
mathematically analyzed according to the previously proposed 
thermodynamic formulation of the self-organization of neural networks. 
Three types of submodality included in the visual afferent pathways are 
assumed in two models: model (A), in which the ocularity and retinotopy 
are considered separately, and model (B), in which on-center/off-center 
pathways are considered in addition to ocularity and retinotopy. Model (A) 
shows striped ocular dominance spatial patterns and, in ocular dominance 
histograms, reveals a dip in the binocular bin. Model (B) displays 
spatially modulated irregular patterns and shows single-peak behavior in 
the histograms. When we compare the simulated results with the observed 
results, it is evident that the ocular dominance spatial patterns and 
histograms for models (A) and (B) agree very closely with those seen in 
monkeys and cats. 
1 INTRODUCTION 
A recent experimental study has revealed that spatial patterns of ocular dominance columns 
(ODCs) observed by autoradiography and profiles of the ocular dominance histogram 
(ODH) obtained by electrophysiological experiments differ greatly between monkeys and 
cats. ODCs for cats in the tangential section appear as beaded patterns with an irregularly 
fluctuating bandwidth (Anderson, Olavarria and Van Sluyters 1988); ODCs for monkeys are 
likely to be straight parallel stripes (Hubel, Wiesel and LeVay, 1977). The typical ODH for 
cats has a single peak in the middle of the ocular dominance corresponding to balanced 
response in ocularity (Wiesel and Hubel, 1974). In contrast to this, the ODH for monkeys 
has a dip in the middle of the ocular dominance (Hubel and Wiesel, 1963). Furthermore, 
neurons in the input layer of the cat's primary visual cortex exhibit orientation selectivity, 
while those of the monkey do not. 
Through these comparisons, we can observe distinct differences in the anatomical and 
physiological properties of neural projections from the retinas to the visual cortex in 
monkeys and cats. To obtain a better understanding of these differences, theoretical analyses 
of interactions among ocularity, retinotopy and on-center/off-center pathways during visual 
18 
Interaction Among Ocularity, Retinotopy and On-center/Off-center Pathways 19 
cortical development were performed with computer simulation based on the previously 
proposed thermodynamic formulation of the self-organization of neural networks (Tanaka, 
1990). 
Two models for the development of the visual afferent pathways are assumed: model (A), in 
which the development of ocular dominance and retinotopic order is taken into account, and 
model (B), in which the development of on-center/off-center pathway terminals is 
considered in addition to ocular dominance and retinotopic order. 
2 MODEL DESCRIPTION 
The synaptic connection density of afferent fibers from the lateral geniculate nucleus CLGN) 
in a local equilibrium state is represented by the Potts spin variables o3,,,'s because of their 
strong winner-take-all process (Tanaka, 1990). The following function ({ 03..,}) gives the 
distribution of the Potts spins in equilibrium: 
zre,( {,,,u})= lexp(-H({&tt}) ) (1) 
Z T 
with Z =  exp( H({ ,,,tt}) ) (2) 
{q.,.=.o} T 
The Hamiltonian H in the argument of the exponential function in (1) and (2) determines 
the behavior of this spin system at the effective temperature T, where H is given by 
-=-Y_,Y_, Y_, v*? ' ,-,.   o) 
Function v.V. c. represents the interaction between synapses at positions j and j' in layer 4 
of the primary visual cortex; function F.'.; represents the correlation in activity between 
LGN neurons at positions k and k' of cell types/.t and/.t'. The set B i represents a group 
of LGN neurons which can project their axons to the position j in the visual cortex; 
therefore, the magnitude of this set is related to the extent of afferent terminal arborization 
in the cortex ^. 
Taking the above formulation into consideration, we have only to discuss the 
thermodynamics in the Potts spin system described by the HamiltonJan H at the 
temperature T in order to discuss the activity-dependent self-organization of afferent neural 
connections during development. 
Next, let us discuss more specific descriptions on the modeling of the visual afferent 
pathways. We will assume that the LGN serves only as a relay nucleus and that the signal 
is transferred from the retina to the cortex as if they were directly connected. Therefore, the 
LGN R 
correlation function F.'.; can be treated as that in the retinas F.;.;. This function is 
V, R , 
given by using the lateral interaction function in the retina : and the correlation function 
20 Tanaka 
of stimuli to RGCs G ,;n,' in the following: 
kl,k2 
For simplicity, the stimuli are treated as white noise: 
Now, we can obtain two models for the formation of afferent synapfic connections between 
the retinas and the primary visual cortex: model (A), in which ocularity and retinotopy are 
taken into account: 
/{left, right}, K=[ 1 r] 
rl 1 
where r (0 g r g 1 ) is the correlation of activity between the left and right retinas; and 
model (B), in which on-center and off-center pathways are added to model (A): 
{(left, on-center), (left, off-center), (fight, on-center), (fight, off-center)} , 
1 F1 +/'2 F1 F1 
r +r2 1 r r 
K= 
r r 1 n 
r r n +r2 1 
where r2 (- 1 g r2 g 1 ) is the correlation of activity between the on-center and off-center 
RGCs in the same retina when there is no correlation between different retinas. A negative 
value of r2 means out-of-phase firings between on-center and off-center neurons. 
3 COMPUTER SIMULATION 
Computer simulations were carried out according to the Metropolis algorithm (Metropolis, 
1953; Tanaka, 1991). A square panel consisting of 80x80 grids was assumed to be the 
input layer of the primary visual cortex, where the length of one grid is denoted by a. The 
Potts spin is assigned to each grid. Free boundary conditions were adopted on the border of 
the panel. One square panel of 20><20 grids was assumed to be a retina for each submorality 
/z. The length of one grid is given as 4a so that the edges for the square model cortex and 
model retinas are of the same length. 
The following form was adopted for the interactions V[;; 's (v= VC or R ): 
(8) 
Interaction Among Ocularity, Retinotopy and On-center/Off-center Pathways 21 
All results reported in this paper were obtained with parameters whose values are a.s 
follows: qVC = 1.0, qVC/, = 5.0, 3.vc = 0.15, 3.vc/,, = 1.0, qR, = l, 3.Ra = 0.5, 
;Rm = 1.0, ;^ = 1.6, a = 0.1, T = 0.001, rl = 0, and r2 = - 0.2. It is assumed that qR, = 0 
for model (A) while qRm = 0.5 for model (B). By considering that the receptive field (RF) 
of an RGC at position k is represented by #V; ;, RGCs for model (A) and (B) have low- 
pass and high-pass filtering properties, respectively. Monte Carlo simulation for model (A) 
was carried out for 200,000 steps; that for model (B) was done for 760,000 steps. 
L 
(a) (b) (c) 
R 
L 
(d) (e) (O 
R 
(g) 
Fig. 1 Simulated results of synapfic terminal and neuronal distributions and ocular 
dominance histograms for models (A) and (B). 
22 Tanaka 
4 RESULTS AND DISCUSSIONS 
The distributions of synaptic terminals and neurons, and ocular dominance histograms are 
shown in Fig. 1, where (a), Co) and (c) were obtained from model (A); (d), (e), (0 and (g) 
were obtained from model (B). The spatial distribution of synaptic terminals originating 
from the left or fight retina (Figs. la and ld) is a counterpart of an autoradiograph of the 
ODC by the eye-injection of radiolabeled amino acid. The bandwidth of the simulated ODC 
(Fig. la) is almost constant as well as the observed bandwidth for monkeys (Hubel and 
Wiesel, 1974). The distribution of ocularity in synaptic terminals shown in Fig. ld is 
irregular in that the periodicity seen in Fig. la disappears even though a patchy pattern can 
be seen. This pattern is quite similar to the ODC for cats (Anderson, Olavarria and Van 
Sluyters 1988). 
By calculating the convolution of the synaptic connections oi..,'s with the cortical 
interaction function V vc, 
i.i ' the ocular dominance in response of cortical cells to monocular 
stimulation and the spatial pattern of the ocular dominance in activity (Figs. lb and le) 
were obtained. Neurons specifically responding to stimuli presented in the right and left 
eyes are, respectively, in the black and white domains. This pattern is a counterpart of an 
electrophysiological pattern of the ODC. The distributions of ocularity in synaptic 
terminals correspond to those of ocular dominance in neuronal response to monocular 
stimulation (a to b; d to e in Fig. 1). This suggests that the borders of the autoradiographic 
ODC pattern coincide with those of the electrophysiological ODC pattern. This 
correspondence is not trivial since strong lateral inhibition exerts in the cortex. 
Reflecting the narrow transition areas between monocular domains in Fig. lb, a dip appears 
in the binocular bin in the corresponding ODH (Fig. lc). In contrast, the profile of the 
ODH (Fig. 113 has a single peak in the binocular bin since binocularly responsive neurons 
are distributed over the cortex (Fig. le). 
In model (B), on-center and off-center terminals are also segregated in the cortex in 
superposition to the ODC pattern (Fig. lg). No correlation can be seen between the spatial 
distribution of on-center/off-center terminals and the ODC pattern (Fig. ld). 
(a) CO) (c) 
Fig. 2 A visual stimulation pattern (a) and the distributions of active synaptic 
terminals in the cortex [Co) for model (A) and (c) for model (B)]. 
Figures 2b and 2c visualize spatial patterns of active synaptic terminals in the cortex for 
model (A) and model (B), when the light stimulus shown by Fig. ld is presented to both 
Interaction Among Ocularity, Retinotopy and On-center/Off-center Pathways 23 
retinas. A pattern similar to the stimulus appears in the cortex for model (A) (Fig. le). 
This supports the observation that retinotopic order is almost achieved. In other 
simulations for model (A), the retinotopic order in the final pattern was likely to be 
achieved when initial patterns were roughly ordered in retinotopy. In model (B), the 
retinotopic order seems to be broken at least in this system size even though the initial 
pattern has a well-ordered retinotopy (Fig. lc). There is a tendency for retinotopy to be 
harder to preserve in model (B) than in model (A). 
L R L R 
(a) (b) 
(c) (d) (e) 
Fig. 3 Representative receptive fields obtained from simulations. 
Model (A) reproduced only concentric RFs for both eyes. The dominant RFs of monocular 
neurons were of the on-center/off-surround type (right in Fig. 3a); the other RFs of the 
same neurons were of the type of the low-pass filter which has only the off response (left 
in Fig. 3a). In Model (B), RFs of cortical neurons generally had complex structures (Fig. 
3b). It can barely be recognized that the dominant RFs of monocular neurons showed 
simple-cell-like RFs. 
To determine why model (B) produced complex structures in RFs, another simulation of 
RF formation was carried out based on a model where retinotopy and on-center/off-center 
pathways are considered. Various types of RFs emerged in the cortex (bottom row in Fig. 
3). The difference in structures between Figures. 3c and 3e shows the difference in the 
orientation and the phase (the deviation of the on region from the RF center) in the simple- 
cell-like RFs. Fig. 3d shows an on-center concentric RF. Such nonoriented RFs were 
likely to appear in the vicinity of the singular points around which the orientation rotates 
by 180 degrees. 
3, ^ 
Simulations for model (A) with different values of parameters such as qVC, and qR 
were also carried out although the results are not visualized here. When qVC takes a small 
24 Tanaka 
value, the ODC bandwidth fluctuates). However large the fluctuation may be, the left-eye 
or right-eye dominant domains are well connected, and the pattern does not become an 
irregular beaded pattern as seen in the cat ODC. When afferent axonal arbors were widely 
spread in the cortex (;[^ >> 1), segregated ODC stripe patterns had only small fluctuation in 
the bandwidth. qRa, -0 corresponds to a monotonically decreasing function vR;k ' with 
respect to the radial distance d: When qRm was increased from zero, the number of 
monocular neurons was decreased. Therefore, the profile of the ODH changes from that in 
Fig. lc. 
In model (B), as the value of r2 became smaller, on-center and off-center terminals were 
more sharply segregated, and the average size of the ODC patches became smaller. The 
segregation of on-center and off-center terminals seems to interfere strongly with the 
development of the ODC and the retinotopic organization. This may be attributed to the 
competition between ocularity and on-center/off-center pathways. We have seen that only 
concentric or simple-cell-like RFs can be obtained (Fig. 3b) unless both the ocularity and 
the on-center/off-center pathways are taken into account in simulations. However, in model 
(B) in which the two types of submodality are treated, neurons have complex separated RF 
structures (Fig. 3b). This also seems to be due to the competition among the ocularity and 
the on-center/off-center pathways. The simulation of model (B) was performed with no 
correlation in activity between the left and right eyes r. This condition can be realized for 
binocularly deprived kittens (Tanaka, 1989). By considering this, we may conclude that the 
formation of normal RFs needs cooperative binocular input. 
In this research, we did not consider the effect of color-related cell types on ODC formation. 
Actually, there are varieties of single-opponent cells in the retina and LGN of monkeys 
such as four types of red-green opponent cells: a red on-center cell with a green inhibitory 
surround; a green on-center cell with a red inhibitory surround; a red off-center cell with a 
green excitatory surround; and a green off-center cell with a red excitatory surround. The 
correlation of activity between red on-center and green on-center cells or green off-center and 
red off-center cells may be positive in view of the fact that the spectral response functions 
between three photoreceptors overlap on the axis of the wavelength. However, the red on- 
center and green on-center cells antagonize the red off-center and green off-center cells, 
respectively. Therefore, the former two and latter two can be looked upon as the on-center 
and off-center cells seen in the retina of cats. This implies that the model for monkeys 
should be model (B); thereby, the ODC pattern for monkeys should be an irregular beaded 
pattern despite the fact that the ODC and ODH in model (A) resemble those for monkeys. 
To avoid such contradiction, the on-center and off-center cells must separately send their 
axons into different sublayers within layer 4Cl3, as seen in the visual cortex for Tree shrews 
(Fitzpatrick and Raczkowski, 1990). 
5 CONCLUSION 
In model (A), the ODC showed the striped pattern and the ODH revealed a dip in the 
binocular bin. In contrast to this, model (B) reproduced spatially modulated irregular ODC 
patterns and the single-peak behavior of the ODH. From comparison of these simulated 
results with experimental observations, it is evident that the ODCs and ODHs for models 
(A) and (B) agree very closely with those seen in monkeys and cats, respectively. Therefore, 
this leads to the conclusion that model (A) describes the development of the afferent fiber 
terminals of the primary visual cortex of monkeys, while model (B) describes that of the 
Interaction Among Ocularity, Retinotopy and On-center/Off-center Pathways 25 
cat. In fact, the assumption of the negative correlation (r2 < 0) between the on-center and 
off-center pathways in model (B) is consistent with the experiments on correlated activity 
between on-center and off-center RGCs for cats (Mastronarde, 1988). 
Finally, we predict the following with regard to afferent projections for cats and monkeys. 
[1] In the input layer of the visual cortex for cats, on-centex/off-center pathway terminals are 
segregated into patches, superposing the ocular dominance patterns. 
[2] In monkeys, the axons from on-center/off-center cells in the LGN terminate in different 
sublayers in layer 4CI of the primary visual cortex. 
Acknowledgment 
The author thanks Mr.Miyashita for his help in performing computer simulations of 
receptive field formation. 
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