A 
Model of Recurrent Interactions in 
Primary Visual Cortex 
Emanuel Todorov, Athanassios Siapas and David Somers 
Dept. of Brain and Cognitive Sciences 
E25-526, MIT, Cambridge, MA 02139 
Email: {emo,thanos,somers} @ai.mit.edu 
Abstract 
A general feature of the cerebral cortex is its massive intercon- 
nectivity - it has been estimated anatomically [19] that cortical 
neurons receive upwards of 5,000 synapses, the majority of which 
originate from other nearby cortical neurons. Numerous experi- 
ments in primary visual cortex (V1) have revealed strongly nonlin- 
ear interactions between stimulus elements which activate classical 
and non-classical receptive field regions. Recurrent cortical con- 
nections likely contribute substantially to these effects. However, 
most theories of visual processing have either assumed a feedfor- 
ward processing scheme [7], or have used recurrent interactions to 
account for isolated effects only [1, 16, 18]. Since nonlinear sys- 
tems cannot in general be taken apart and analyzed in pieces, it 
is not clear what one learns by building a recurrent model that 
only accounts for one, or very few phenomena. Here we develop 
a relatively simple model of recurrent interactions in V1, that re- 
flects major anatomical and physiological features of intracortical 
connectivity, and simultaneously accounts for a wide range of phe- 
nomena observed physiologically. All phenomena we address are 
strongly nonlinear, and cannot be explained by linear feedforward 
models. 
1 The Model 
We analyze the mean firing rates observed in oriented V1 cells in response to stimuli 
consisting of an inner circular grating and air outer annular grating. Mean responses 
of individual cells are modeled by single-valued "cellular" response functions, whose 
arguments are the mean firing rates of the cell's inputs and their maximal synaptic 
conductances. 
A Model of Recurrent Interactions in Primary Visual Cortex 119 
1.1 Neuronal model 
Each neuron is modeled as a single voltage compartment in which the membrane 
potential V is given by: 
C, dt 
gex(t)(Eex - V(t)) + ginh(t)(Einh -- V()) + 
gleak(Eleak -- V(t)) + gahf(t)(Eahp -- V(t)) 
where Cm is the membrane capacitance, E is the reversal potential for current x, 
and g is the conductance for that current. If the voltage exceeds a threshold Vs, 
a spike is generated, and afterhyperpolarizing currents are activated. The conduc- 
tances for excitatory and inhibitory currents are modeled as sums of a-functions, 
and the ahp conductance is modeled as a decaying exponential. The model consists 
of two distinct cell types, excitatory and inhibitory, with realistic cellular param- 
eters [13], similar to the ones used in [17]. To compute the response functions for 
the two cell types, we simulated one cell of each type, receiving excitatory and in- 
hibitory Poisson inputs. The synaptic strengths were held constant, while the rates 
of the excitatory and inhibitory inputs were varied independently. 
Although the driving forces for excitation and inhibition vary, we found that single 
cell responses can be accurately modeled if incoming excitation and inhibition are 
combined linearly, and the net input is passed through a response function that 
is approximately threshold-linear, with some smoothing around threshold. This is 
consistent with the results of intracellular experiments that show linear synaptic 
interactions in visual cortex[5]. Note that the cellular functions are not sigmoids, 
and thus saturating responses could be achieved only through network interactions. 
1.2 Cortical connectivity 
The visual cortex shares with many other cortical areas a similar pattern of intra- 
areal connections [12]. Excitatory cells make dense local projections, as well as 
long-range horizontal projections that usually contact cells with similar response 
properties. Inhibitory cells make only local projections, which are spread further in 
space than the local excitatory connections [10]. We assume that cells with similar 
response properties have a higher probability of connection, and that probability 
falls down with distance in "feature" space. For simplicity, we consider only two 
feature dimensions: orientation and RF center in visual space. Since we are dealing 
with stimuli with radial symmetry, one spatial dimension is sufficient. The extension 
to more dimensions, i.e. another spatial dimension, direction selectivity, ocularity, 
etc., is straightforward. 
We assume that the feature space is filled uniformly with excitatory and inhibitory 
cells. Rather than modeling individual cells, we model a grid of locations, and 
for each location we compute the mean firing rate of cells present there. The 
connectivity is defined by two projection kernels Kex, Kin (one for each presynaptic 
cell type) and weights Wee, Wei, Wie, Wii, corresponding to the number and strength 
of synapses made onto excitatory/inhibitory cells. The excitatory projection has 
sharper tuning in orientation space, and bigger spread in visual space (Figure 1). 
1.3 Visual stimuli and thalamocortical input 
The visual stimulus is defined by five parameters: diameter d of the inner grating 
(the outer is assumed infinite), log contrast cx,c2, and orientation 0x, 02 of each 
grating. The two gratings are always centered in the spatial center of the model. 
120 E. Todorov, A. Siapas and D. Somers 
Viu .1 
Yigure xitator (solid) and %nhibitor (dhed) connectivity kernels. 
orientation space 
Figure 2: LGN input for a stimulus with high contrast, orthogonal orientations of 
center and surround gratings. 
Each cortical cell receives LGN input which is the product of log contrast, orien- 
tation tuning, and convolution of the stinulus with a spatial receptive field. The 
LGN input computed in this way is multiplied by LCNe, LCNi, and sent to the 
cortical cells. Figure 2 shows an example of what the LGN input looks like. 
2 Results 
For given input LGN input, we computed the steady-state activity in cortex itera- 
tively (about 30 iteration were required). Since we studied the model for gradually 
changing stimulus parameters, it was possible to use the solution for one set of 
parameters as an initial guess for the next solution, which resulted in significant 
speedup. The results presented here are for the excitatory population, since i) it 
provides the output of the cortex; it) contains four times more cells than the in- 
hibitory population, and therefore is more likely to be recorded from. All results 
were obtained for the same set of parameters. 
2.1 Classical RF effects 
First we simulate responses to a central grating (ldeg diameter) for increasing log- 
contrast levels. It has been repeatedly observed that although LGN firing increases 
linearly with stimulus log-contrast, the contrast response functions observed in V1 
saturate, and may even supersaturate or (lec]ine[11, 2]. The most complete and re- 
cent model of that phenomena [6, 3] assumes shunting inhibition, which contradicts 
recent intracellular observations [4, 5]. Furthermore, that model cannot explain su- 
persaturation. Our model achieves saturatiolt (Figure 3A) for high contrast levels, 
A Model of Recurrent Interactions in Primary Visual Cortex 121 
80 
70 
60 
4o 
20 
10 
A) 
601 
B) 
o o 
Log Co ntraet 
Orientation 
Figure 3: Classical RF effects. A) Contrast response function for excitatory (solid) 
and inhibitory (dashed) cells. B) Orientation tuning for different contrast levels 
(solid); scaled LGN input (dashed). 
and can easily achieve supersaturation for different parameter setting. Instead of 
using shunting (divisive) inhibition, we suggest that inhibitory neurons have higher 
contrast thresholds than excitatory neurons, i.e. the direct LGN input to inhibitory 
cells is weaker. Note that we only need a subpopulation of inhibitory neurons with 
that property; the rest can have the same response threshold as the excitatory 
population. 
Another well-known property of V1 neurons is that their orientation tuning is 
roughly contrast-invariant[14]. The LGN input tuning is invariant, therefore this 
property is easily obtained in models where V1 responses are almost linear, rather 
than saturating [1, 16]. Achieving both contrast-invariant tuning and contrast sat- 
uration for the entire population (while singe cell feedforward response functions 
are non-saturating) is non-trivial. The problem is that invariant tuning requires the 
responses at all orientations saturate simultaneously. This is the case in our model 
(Figure 3B) - we found that the tuning (half width at half amplitude) varied within 
a 5deg range as contrast increased. 
2.2 Extraclassical RF effects 
Next we consider stimuli which include both a center and a surround grating. In the 
first set of simulations we held the diameter constant (at ldeg) and varied stimulus 
contrast and orientation. It has been observed [9, 15] that a high contrast iso- 
orientation surround stimulus facilitates responses to a low contrast, but suppresses 
responses to a high contrast center stimulus. T.his behavior is captured very well 
by our model (Figure 4A). The strong response to the surround stimulus alone is 
partially due to direct thalamic input (i.e. the thalamocortical projective field is 
larger than the classical receptive field of a V! cell). The response to an orthogonal 
surround is between the center and iso-orientation surround responses, as observed 
in [9]. 
Many neurons in V1 respond optimally to a center grating with a certain diameter, 
but their response decreases as the diameter increases (end-stopping). End-stopping 
in the model is shown in Figure 4B - responses to increasing grating diameter 
reach a peak and then decrease. In experiments it has been observed that the 
122 E. Todorov, A. Siapas and D. Somers 
60 
20 
10 
0 
A) B) 
Log Co ntrast 
6 I 
Diameter 
Figure 4: Extraclassical RF effects. A) Contrast response functions for center 
(solid), center q- iso-orientation surround (dashed), center q- orthogonal surround 
(dash-dot). B) Length tuning for 4 center log contrast levels (1, .75, .5, .4), response 
to surround of high contrast (dashed). 
border between the excitatory and inhibitory regions shifts outward (rightward) as 
stimulus contrast levels decline[8]. Our model also achieves this effect (for other 
parameter settings it can shift more downward than rightward). Note also that a 
center grating with maximal contrast reaches its peak response for a diameter value 
which is 3 times smaller than the diameter for which responses to a surround grating 
disappear. This is interesting because both the peak response to a central grating, 
and the first response to a surround grating can be used to define the extent of the 
classical RF - in this case clearly leading to very different definitions. This effect 
(shown in Figure 4B) has also been recently observed in V1 [15]. 
3 Population Modeling and Variability 
So far we have analyzed the population firing rates in the model, and compared them 
to physiological observations. Unfortunately, in many cases the limited sample size, 
or the variability in a given physiological experiment does not allow an accurate 
estimate of what the population response might be. In such cases researchers only 
describe individual cells, which are not necessarily representative. How can find- 
ings reported in this way be captured in a population model? The parameters of 
the model could be modified in order capture the behavior of individual cells on 
the population level; or, further subdivisions into neuronal subpopulations may be 
introduced explicitly. The approach we prefer is to increase the variance of the 
number of connections made across individual cells, while maintaining the same 
mean values. We consider variations in the amount of excitatory and inhibitory 
synapses that a particular cell receives from other cortical neurons. Note that the 
presynaptic origin of these inputs is still the "average" cortical cell , and is not 
chosen from a subpopulation with special response properties. 
The two examples we show have recently been reported in [15]. If we increase 
the cortical input (excitation by 100%, inhibition by 30%), we obtain a completely 
patch-suppressed cell, i.e. it does not respond to a center q- iso-orientation surround 
stimulus - Figure 5A. Figure 5B shows that this cell is an "orientation contrast 
detector", i.e. it responds well to Odeg center q- 90deg surround, and to 90deg 
A Model ofRecurrentlnteractions in Primary Visual Cortex 123 
center + Odeg surround. Interestingly, the cells with that property reported in [15] 
were all patch-suppressed. Note also that our cell has a supersaturating center 
response - we found that it always accompanies patch suppression in the model. 
A) B) 
70 35 
60 30 
50 25 / 
n_30 15 
o ol - 
Log Contrast Orientation Difference 
Figure 5: Orientation discontinuity detection for a strongly patch- suppressed cell. 
A) Contrast response functions for center (solid) and center + iso-orientation sur- 
round (dashed). B) Cell's response for Odeg c:nter, 0- 90deg surround (solid) and 
90deg center, 90- Odeg surround. 
A) B) 
8O 8O 
70 / 70 
60 /// 60 
-3o I o 
0  0 
10 / 10 
0 0 
Log ConArst 
S+O Con 
Figure 6: Nonlinear summation for a non patch-suppressed cell. A) Contrast re- 
sponse functions for center (solid) and center + iso-orientation surround (dashed). 
B) Cell's response for iso-orientation surround, orthogonal center, surround q- cen- 
ter, center only. 
The second example is a cell receiving 15% of the average cortical input. Not 
surprisingly, its contrast response function does not saturate - Figure 6A. However, 
this cell exhibits an interesting nonlinear property - it respond well to a combination 
of an iso-orientation surround q- orthogonal center, but does not respond to either 
stimulus alone (Figure 6B). It is not clear from [15] whether the cells with this 
property had saturating contrast response functions. 
124 E. Todorov, A. Siapas and D. Somers 
4 Conclusion 
We presented a model of recurrent computation in primary visual cortex that relies 
on a limited set of physiologically plausible mechanisms. In particular, we used 
the different cellular properties of excitatory and inhibitory neurons and the non- 
isotropic shape of lateral connections (Figure 1). Due to space limitations, we only 
presented simulation results, rather than analyzing the effects of specific parameters. 
A preliminary version of such analysis is given in [17] for local effects, and will be 
developed elsewhere for lateral interaction effects. 
Our goal here was to propose a framework for studying recurrent computation in 
V1, that is relatively simple, yet rich enough to simultaneously account for a wide 
range of physiological observations. Such a framework is needed if we are to analyse 
systematically the fundamental role of recurrent interactions in neocortex. 
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PART III 
THEORY 
