A Correlation-Based Model for
Ocular Dominance
Remus Osan and Bard Ermentrout
University of Pittsburgh
A new correlation-based model for ocular dominance is proposed in this paper. In
contrast to previous correlation-based models, intra-eye correlations remain positive.
Furthermore intracortical connections are strictly positive. Mathematical analysis and
numerical simulations are used to show that under certain regimes of the parameters used
in the model spatial patterns arise. The same ideas can be extended to take into account
the orientation columns phenomena.
Keywords: ocular dominance, orientation selectivity, pattern formation.
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Introduction
There is no clear functional significance for the ocular dominance topographic maps phenomena,
defined by the responsiveness of the neurons from the layer 4 of the visual cortex to the input from
left and right eye. However, the mechanism underlying their formation of has been subject of
intense research (see review papers of Miller 1995, Swindale 1996, Ooyen 2001). The experimental
findings on this area are subject to different interpretation. Some researchers favor the idea that the
topographical maps are not activity driven, as they are found at very early stages of development,
before exposure to visual stimuli (Crowley and Katz 1999, Ernst et all 2001). Others believe either
that activity is essential to formation of maps (Stryker and Harris 1986) or that the presence of
visual activity refines and modifies the initial coarse topographic maps (Goodhill and Richards
1999); moreover many of the models that adopt this view use combinations of Hebbian learning
with various constraints in their models (Miller 1995, Swindale 1996). It is also of interest to note
that models that don’t have a clear biological interpretation, such as the self-organized map and the
elastic net, (Goodhill 1993, Goodhill and Cimponeriu 2000) were used to produce topographic
maps that seem to fit the experimental data better than the models that rely on biological
assumptions.
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
The model
The visual cortex is modeled as a two-dimensional array of neurons that receive input from the
retinal input neurons. Connection weights corresponding to those inputs are used to characterize
the responsiveness to left or right eye. A cell that is more responsive to the left eye will have a left
connection weight much greater than a right connection weight. Only positive connection weights
are allowed in the inputs and the values encoding the ocular dominance always stay positive.
Moreover, the connection weights are bounded between 0 and 1. All interactions are local.
The equations characterizing the behavior of one individual cell are:
dwL
 F1  wL  (1   ) wR   1  wL 2 wR   F2  wL  (1   ) wR  wL
dt
(1)
dwR
 F1  wR  (1   ) wL   1  wL 2 wR   F2  wR  (1   ) wL  wR
dt
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
where  >  are parameters between 0 and 1, and F1 and F2 are sigmoid functions:
1
1
1
F1 ( x) 
, F2 ( x) 
, 1  2 
 1 ( x 1 )
  2 ( x 2 )
1 e
1 e
2
(2)
The parameter  is related to the correlation between inputs from the same eye, and the parameter 
is related to the correlation between inputs from different eyes. The term F1  w  (1   )w from
equations (1) is related to Hebbian learning, while F2  w  (1   )w corresponds to LTD synaptic
pruning. Note that  = ½ means that the pruning of the synapse is independent of which eye is
active. It is easy to see that these equations have a symmetrical fixed point wL = wR = ½. However,
if the initial state of the dynamical system is asymmetrical, that is, for example, wL < wR, then the
connection with the bigger weight, wR, will win over the connection with smaller weight, wL. This
is illustrated in figure 1.
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Figure 1. Competition for ocular dominance in one cell.
The two-dimensional array of connection weights is initialized with values close to the ½  small
random values. Up to now there is no interaction between cells so the asymptotic state of the
dynamical system corresponds to a random map. In order to have pattern formation we need to
have interaction terms between the neurons, so equations (1) become:
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah

 wL ( x, t )
 F1   J
t





 wR ( x, t )
 F1   J
t




 w w
( x  y )  wL (1 ) wR dy   1 L R

2


 






J
(
x

y
)


w

(
1


)
w

dy


L
R
2
 

F 
wL
(3)


 w w
( x  y )  wR (1 ) wL dy   1 L R

2


 






2  J ( x  y )  wR (1 ) wL dy 
 

F 
wR
where for the clarity we expressed equations (3) in the one dimensional domain. However, the
analysis can be performed on the two dimensional domain as well.
J+, J- are gaussian functions with different widths. J+ interaction function corresponds to small
range excitation, while J-corresponds to long range inhibition. In a region dominated by one type
of ocular dominance (left or right) the effect of these terms should result in recruitment of all cells
to that type of ocular dominance.
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A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Conditions leading to pattern formation
We analyze the conditions that would lead to pattern formation. We will use first order
perturbation theory. First of all, we need to linearize equations (3) around the fixed points,
wL  w0   uL , wR  w0   uR , obtaining the following equations:

 u L ( x, t )

 F1   J

t




u u

( x  y )  w0  (1  ) w0    u L  (1  ) u R dy   1 w0  L R

2






J
(
x

y
)


w

(
1


)
w




u

(
1


)
u


dy

 w0  wL
0
0
L
R
2
 


F 

 u R ( x, t )

 F1   J

t




 


(4)


u u
( x  y )  w0  (1  ) w0    u R  (1  ) u L dy   1 w0   L R

2








J
(
x

y
)


w

(
1


)
w



u

(
1


)
u

dy
 w0  wR
0
0
R
L
2 
 


F 
7


 

A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
We can rewrite equations (4) as:

 u L ( x, t )
 F1 ' w0 1  w0    J

t





( x  y )  u L  (1  ) u R dy  







J
(
x

y
)


u

(
1


)
u

dy

 
L
R
0
 

F2 ' w0  w 

 u R ( x, t )
 F1 ' w0 1  w0   J

t



F1 w0   u L 2 u R  
F2 w0  uL


( x  y )  u R  (1  ) u L dy  







J
(
x

y
)


u

(
1


)
u

dy
 
R
L
0 
 

F2 ' w0 w 
(5)
F1 w0   u L 2 u R  
F2 w0  uR
We want to write equations (5) in a simplified version, so we are going to define:
I

L
x  

 J ( x  y ) u L  y  dy,


I

R
x  
8


J
 ( x  y ) u R  y  dy,

(6)
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Now the equations (5) become:
 u L ( x, t )
 F1 ' w0 1  w0  I L x   1   F2 ' w0  w0 I L x   F 2w   F2 w0  u L x  
t
1
1   F ' w 1  w I
1
0
0

R
0
x   1   F2 ' w0 w0 I R x   F 2w  u R x 
1
0
(7)
 u R ( x, t )
 F1 ' w0 1  w0  I R x   1   F2 ' w0  w0 I R x    F 2w   F2 w0  u R x  
t
1
1   F ' w 1  w  I
1
0
0

L
0
x   1   F2 ' w0  w0 I L x   F 2w  uL x 
1
0
We are going to transform equations (7) in the Fourier space:
1
u L ( y) 
2
1
uR ( y) 
2

1
iky
a
(
k
)
e
dk
,
a
(
y
)

L
 L
2

1
iky
a
(
k
)
e
dk
,
a
(
y
)

R
 R
2
9

iky
u
(
y
)
e
dy
 L


iky
u
(
y
)
e
dy
R


(8)
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
We are going to make use of the auxiliary functions:
G  k  
1
2


iky


J
y
e
dy

(9)

where we used
1
G k  
2


J

x  y e

iky
1
dy 
2


iky


J
y
e
dy,


Equations (7) become now:


1   F ' w 1  w G k   1  F ' w  w G k     a
 a (k , t )
  F ' w 1  w G k    F ' w  w G k    F w a
t
1   F ' w 1  w G k   1  F ' w  w G k    a
 aL (k , t )
  F1 ' w0 1  w0 G  k   F2 ' w0  w0 G  k   F 2w   F2 w0  aL 
t
1

1
0

0
2

R
1
0
0

0
2
0
0

0

1
0
0
F1 w0

2

0
2
10
0
0


F1 w0
2
R
2
R
0
F1 w0
2
L

(10)
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
The perturbation around fixed points evolves in time like
u( k , t )  u0 e t
 A B


Where  is the largest eigenvalue of to the matrix:  B A
 F1 (k )

A   F1 ' ( w0 ) (1  w0 ) G (k )   F2 ' ( w0 ) w0 G (k )  
 F2 (k )
 2



(11)
F1 ( w0 )
B  (1   ) F1 ' ( w0 ) (1  w0 ) G (k )  (1   ) F2 ' ( w0 ) w0 G (k ) 
2

The eigenvectors of
 A B


 B A
are:
 1
 
 1
1
 
1

with:
asym  A  B; sym  A  B
asym  A  B  2  1F1 ' (w0 ) (1  w0 ) G  (k )  2  1F2 ' (w0 ) w0 G  (k )  F2 (k )
sym  A  B  F1 ' (w0 ) (1  w0 ) G  (k )  F2 ' (w0 ) w0 G  (k )  F1 (k )  F2 (k )
11
(12)
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
We want to find parameters such that asym is positive, for some values of k while sym is always
negative. In other word the symmetric stated decays as a function of time while the asymmetric
state grows, leading to ocular dominance patterns. This is possible in the frame of this model for
some range of the parameters. Close to the equilibrium point the fastest growing mode is one
corresponding to the maximum of asym as a function of k.
A graph of the eigenvalues
corresponding to the desired situation is presented in figure 2.

0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0.5
asym
-1
sym
-1.5
Figure 2. Graph of the eigenvalues asym and asym as a function of k.
12
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Numerical results
We were able to obtain ocular dominance patterns using the parameters obtained from the
mathematical analysis of the system. Two typical patterns obtained are presented in figure 3.
Figure 3. Two typical ocular dominance patterns obtained in this model: a) symmetric cortical
interaction functions, b) asymmetric cortical interaction functions. The images depict the difference
between wL and wR. The ocular dominance of a region is color-coded; the white regions are left
dominant while the black regions are right dominant.
13
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
Future research
We would like to adapt the ocular dominance model to obtain an orientation selectivity model
and, furthermore, to extend it into a joint model of ocular dominance and orientation selectivity
columns.
We would like to investigate to what extent the finite size effects influence the ocular dominance
patterns. This can be done by starting with the same initial random conditions both for a finite
size and periodic boundary conditions systems. After these two dynamical systems reach the
asymptotic stable state the difference in the ocular dominance patterns can be analyzed.
14
A Correlation-Based Model for Ocular Dominance, R. Osan and B. Ermentrout, SIAM conference on Applications of Dynamical Systems, May 20-24, 2001, Snowbird, Utah
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