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Deriving connectivity patterns
in the primary visual cortex from
spontaneous neuronal activity
and feature maps
Barak Blumenfeld, Dmitri Bibitchkov, Shmuel Naaman,
Amiram Grinvald and Misha Tsodyks
Department of Neurobiology, Weizmann institute of Science, Rehovot, Israel
Abstract
Population activity across the surface of the primary visual cortex exhibits well-known
regular patterns. The location and the shape of activity patches depend on features of
the stimulus such as orientation. Recent studies have shown that activity patterns
generated spontaneously are similar to those evoked by different orientations of a
moving grating stimulus [Kenet, et. al., Nature 2003]. This suggests the existence of
intrinsic preferred states of the cortical network in this area of the brain. We deduce
possible connections in such a network from a set of single condition orientation maps
obtained by voltage-sensitive dye imaging. We assume the maps as attractor states of a
recurrent neural network and model the connectivity using a modified version of the
pseudo-inverse rule of the Hopfield network. The results suggest a local distancedependent Mexican-hat shaped connectivity. Long-range connections also exist and
depend mainly on the difference in orientation selectivity of the connected pixels. The
strength of connections correlates strongly with orientation selectivity of the neurons.
The dependence of the obtained synaptic weights on the distance between neurons
correlates with the pattern of correlations in the spontaneous activity, suggesting that
intrinsic connectivity in neuronal networks in this area of the brain underlies the activity
in both spontaneous and evoked regimes.
Experimental setup
A
Figure 1. Experimental setup for the voltage
sensitive dye optical imaging.
B
C
Figure 2. Orientation single
condition maps obtained by
voltage sensitive dye optical
imaging of a cat's area 17/18.
The activity was evoked by a
moving grating stimulus with an
orientation of (A) 0° (horizontal),
(B) 45°, and (C) 90° (vertical).
The direction of motion was
perpendicular to the stimulus
orientation.
Topology of intrinsic states
Evoked
PCA
p1
Spontaneous
Kohonen map
p2
180
20
135
(Mk p2)
10
0
90
-10
45
Templates
t1
t11 t 21
-20
-20
-10
0
10
(Mk p1)
20
0
Figure 3. Projections of 24 single condition orientation Figure 4. Kohonen algorithm performs a
maps Mk corresponding to orientations θk onto a plane topological mapping of spontaneous activity
spanned by the 1st two principle components p1 ,p2. The frames onto a set of templates on a ring. The
shapes of the learned templates resemble the
data is fitted by a circle (solid line).
evoked orientation maps .
n



z
x
Selectivity
 2 i k

M
x
 k e
k 1
If orientation maps form a perfect ring:



zx  ~ p1 x   ip2 x 
Spontaneous activity patterns
Spontaneous
Evoked
Spontaneous
180
135
90
35
0
A
B
Figure 5. Activity patterns obtained by
voltage sensitive dye optical imaging.
The pattern in (A) was evoked by a 0°
moving grating stimulus. It is very
similar to the spontaneous pattern (B).
Figure 6. Preferred orientation maps
calculated using evoked single
condition maps (A) and Kohonen
templates of spontaneous spontaneous
activity (B) [Kenet, et. al., 2003].
Network model with pseudo-inverse
connectivity
Recurrent neural network with functional maps
M 
N
k k 1
as attractors:
Network dynamics:  m
 x    m x  
W x, y  g m( y )  g m
T
W x, y    M x   Q   g M ( y )
y
Network connectivity:
1
k
k ,l
l
k ,l
Pattern correlation matrix:
Fixed points of dynamics:
Qk ,l   g M k ( x ) g M l ( x ) 
x
W x, y g M
k
( y )  M k ( x )
y
For a linear gain function, the connectivity results in a Hopfield network,
which stores two patterns corresponding to the principle components of
orientation maps:
W x, y   p1 x  p1  y   p2 x  p2  y 
Dependence of connectivity on
orientation selectivity
x 10-3
-3
2
1
0
-1
-2
A
Figure 9. Average synaptic weights as
a function of the difference between
preferred orientations of the pre- and
post synaptic neurons, for the pseudo
inverse connectivity .
B
5
4
3
2
1
0
-1
-2
-3
-4
x 10
2
1
0
-1
-2
C
Figure 10. Connectivity of individual
pixels. (A) Afferent synaptic weights of
the pixel marked by the yellow dot. (B) :
Activity pattern evoked by a 90º stimulus.
The yellow dot marks the same pixel as in
(A). (C) Afferent synaptic weights of
another pixel (close to a pinweel).
W x, y   | z ( x) |  | z ( y ) | cos2  ( x)   ( y )
Dependence of connectivity on
spatial separation
Figure 7. Average pixel-by-pixel correlation
coefficient of recorded spontaneous activity
as a function of distance between pixels on
the cortical surface. Solid line: fit using a
 r 2 / 1
r2 / 2
Mexican hat function C( r )  a1e
 a2e
Figure 8. Synaptic weights of the attractor
network as a function of distance between
pre- and post synaptic neuron, for the pseudo
inverse connectivity. The bin size was the
size of one pixel, which was ~50μm.
Network simulations
Simulation
Initial
Experiment
Stationary
A
B
Simulation
Initial
D
C
Experiment
Stationary
E
F
Figure 5.2 Simulations of the pseudo inverse
connectivity model with random initial activity. Panels
(A),(D) show the initial random activity patterns for
two trails. Panels (B),(E) show the corrsponding
stationary activity patterns (t=300). Panels (C),(F)
show the corresponding evoked activity pattern: (C)
37.5º and (F) 112.5º. In all trails, the stationary activity
pattern was similar to one particular evoked pattern,
and was never a mixture of several patterns evoked by
different orientations. This property is to be attributed
to the non-linearity of the gain function. By considering
this type of simulation as a model for spontaneous
activity, we conclude that the pseudo-inverse
connectivity can indeed produce the typical activity
patterns spontaneously.
Conclusions
Primary visual cortex has intrinsic activity states that emerge both
spontaneously and due to visual stimulation and can originate from intracortical interactions in this area of the brain.
Intrinsic states corresponding to orientation maps lie on a ring
embedded into a high-dimensional space of neuronal activities.
Attractor neural network with pseudo-inverse connectivity is capable to
generate experimental activity patterns.
The strength of modelled connections depends on the degree of
selectivity of connected neurons and on the difference between their
preferred orientations.
References:
• Kenet T., Bibitchkov D., Tsodyks M. , Grinvald A. , Arieli A. (2003) Spontaneously
emerging cortical representations of visual attributes. Nature 425: 954-956
• Personnaz L., Guyon I.I., Dreyfus G. (1986) Collective computational properties of neural
networks: New learning mechanisms. PHYS. REV. A. Nov;34(5):4217-4228.