Visual Cortex Circuitry and Orientation 
Tuning 
Trevor Mundel 
Department of Neurology 
University of Chicago 
Chicago, IL 60637 
mundel@math.uchicago.edu 
Alexander Dimitrov 
Department of Mathematics 
University of Chicago 
Chicago, IL 60637 
a-dimitrov@uchlcago.edu 
Jack D. Cowan 
Departments of Mathematics and Neurology 
University of Chicago 
Chicago, IL 60637 
cowan@math.uchicago.edu 
Abstract 
A simple mathematical model for the large-scale circuitry of pri- 
mary visual cortex is introduced. It is shown that a basic cor- 
tical architecture of recurrent local excitation and lateral inhi- 
bition can account quantitatively for such properties as orien- 
tation tuning. The model can also account for such local ef- 
fects as cross-orientation suppression. It is also shown that non- 
local state-dependent coupling between similar orientation patches, 
when added to the model, can satisfactorily reproduce such ef- 
fects as non-local iso-orientation suppression, and non-local cross- 
orientation enhancement. Following this an account is given of per- 
ceptual phenomena involving object segmentation, such as "pop- 
out", and the direct and indirect tilt illusions. 
i INTRODUCTION 
The edge detection mechanism in the primate visual cortex (V1) involves at least 
two fairly well characterized circuits. There is a local circuit operating at sub- 
hypercolumn dimensions comprising strong orientation specific recurrent excitation 
and weakly orientation specific inhibition. The other circuit operates between hy- 
percolumns, connecting cells with similar orientation preferences separated by sev- 
eral millimetres of cortical tissue. The horizontal connections which mediate this 
888 T. Mundel, A. Dimitrov and J. D. Cowan 
circuit have been extensively studied. These connections are ideally structured to 
provide local cortical processes with information about the global nature of stim- 
uli. Thus they have been invoked to explain a wide variety of context dependent 
visual processing. A good example of this is the tilt illusion (TI), where surround 
stimulation causes a misperception of the angle of tilt of a grating. 
The interaction between such local and long-range circuits has also been inves- 
tigated. Typically these experiments involve the separate stimulation of a cells 
receptive field (the classical receptive field or "center") and the immediate region 
outside the receptive field (the non-classical receptive field or "surround"). In the 
first part of this work we present a simple model of cortical center-surround inter- 
action. Despite the simplicity of the model we are able to quantitatively reproduce 
many experimental findings. We then apply the model to the TI. We are able to 
reproduce the principle features of both the direct and indirect TI with the model. 
2 PRINCIPLES OF CORTICAL OPERATION 
Recent work with voltage-sensitive dyes (Blasdel, 1992) augments the early work of 
Hubel  Wiesel (1962) which indicated that clusters of cortical neurons correspond- 
ing to cortical columns have similar orientation preferences. An examination of local 
field potentials (Victor et al., 1994) which represent potentials averaged over corti- 
cal volumes containing many hundreds of cells show orientation preferences. These 
considerations suggest that the appropriate units for an analysis of orientation se- 
lectivity are the localized clusters of neurons preferring the same orientation. This 
choice of a population model immediately simplifies both analysis and computation 
with the model. For brevity we will refer to elements or edge detectors, however 
these are to be understood as referring to localized populations of neurons with a 
common orientation preference. We view the cortex as a lattice of hypercolumns, 
in which each hypercolumn comprises a continuum of iso-orientation patches dis- 
tinguished by their preferred orientation b. All space coordinates refer to distances 
between hypercolumn centers. The population model we adopt throughout this 
work is a simplified form of the Wilson-Cowan equations. 
2.1 LOCAL MODEL 
Our local model is a ring (b = -90to + 90 ) of coupled iso-orientation patches and 
inhibitors with the following characteristics 
 Weakly tuned orientation biased inputs to V1. These may arise either from 
slight orientation biases of lateral geniculate nucleus (LGN) neurons or from 
converging thalamocortical afferents 
 Sharply tuned (space constant 7.5 ) recurrent excitation between iso- 
orientation populations 
 Broadly tuned inhibition to all iso-orientation populations with a cut-off 
of inhibition interactions at between 45  and 60  separation 
The principle constraint is that of a critical balance between excitatory and in- 
hibitory currents. Recent theoretical studies (Tsodyks  Sejnowski 1995; Vreeswijk 
 Sompolinsky 1996) have focused on this condition as an explanation for certain 
features of the dynamics of natural neuronal assemblies. These features include the 
irregular temporal firing patterns of cortical neurons, the sensitivity of neuronal 
assemblies in vivo to small fluctuations in total synaptic input and the distribu- 
tion of firing rates in cortical networks which is markedly skewed towards low mean 
Visual Cortex Circuitry and Orientation Tuning 889 
rates. Vreeswijk & Sompolinsky demonstrate that such a balance emerges naturally 
in certain large networks of excitatory and inhibitory populations. We implement 
this critical balance by explicitly tuning the strength of connection weights between 
excitatory and inhibitory populations so that the system state is subcritical to a 
bifurcation point with respect to the relative strength of excitation/inhibition. 
2.2 HORIZONTAL CONNECTIONS 
We distinguish three potential patterns of horizontal connectivity 
 connections between edge detectors along an axis parallel to the detectors 
preferred orientation (visuotopic connection) 
 connections along an axis orthogonal to the detectors preferred orientation, 
with or without visuotopic connections 
 radially symmetric connection to all detectors of the same orientation in 
surrounding hypercolumns 
Recent experimental work in the tree shrew (Fitzpatrick et al., 1996) and prelim- 
inary work in the macaque (Blasdel, personal communication) indicate that vi- 
suotopic connection is the predominant pattern of long-range connectivity. This 
connectivity pattern allows for the following reduction in dimension of the problem 
for certain experimental conditions. 
Consider the following experiment. A particular hypercolumn designated the "cen- 
ter" is stimulated with a grating at orientation  resulting in a response from the 
b-edge detector. The region outside the receptive area of this hypercolumn (in the 
"surround") is also stimulated with a grating at some uniform orientation q result- 
ing in responses from -edge detectors at each hypercolumn in the surround. In 
order to study the interactions between center and surround, then to first order ap- 
proximation only the center hypercolumn and interaction with the surround along 
the  visuotopic axis (defined by the center) and the  visuotopic axis (once again 
defined by the center) need be considered. In fact, except when  =  the effect 
of the center on the surround will be negligible in view of the modulatory nature 
of the horizontal connections detailed above. Thus we can reduce the problem (a 
priori three dimensional - one angle and two space dimensions) to two dimensions 
(one angle and one space dimension) with respect to a fixed center. This reduc- 
tion is the key to providing a simple analysis of complex neurophysiological and 
psychophysical data. 
3 RESULTS 
3.1 CENTER-SURROUND INTERACTIONS 
Although, we have modeled the state-dependence of the horizontal connections, 
many of the center-surround experiments we wish to model have not taken this 
dependence explicitly into account. In general the surround has been found to 
be suppresslye on the center, which accords with the fact that the center is usually 
stimulated with high contrast stimuli. A typical example of the surround suppressive 
effect is shown in figure 1. 
The basic finding is that stimulation in the surround of a visual cortical cell's re- 
ceptive field generally results in a suppression of the cell's tuning response that 
is maximal for surround stimulation at the orientation of the cell's peak tuning 
890 T. Mundel, A. Dirnitrov and J. D. Cowan 
 2o 
[ 60 30 0 +30 +60 +90 
 ORIENTATION OF BAR 
 " 
-30 0 ,30 +60 +90 
ORIIrrATION OF GRATING (deg) 
Figure 1: Non-local effects on orientation tuning - experimental data. Response to 
constant center stimulation at 15  and surround stimulation at angles [-90  , 90  ] 
(open circles), Local tuning curve (filled circles). Redrawn from Blakemore and 
Tobin (1972) 
response and falls off with stimulation at other orientations in a characteristic man- 
ner. Further examples of surround suppression can be found in the paper of Sillito 
et al. (1995). Figure 2 depicts simulations in which long-range connections to lo- 
cal inhibitory populations are strong compared to connections to local excitatory 
populations. 
These experiments and simulations appear to conflict with the consistent experimen- 
tal finding that stimulating a hypercolumn with an orthogonal stimulus suppresses 
the response to the original stimulus. 
The relevant results can be summarised as follows: cross-orientation suppression 
(with orthogonal gratings) originates within the receptive field of most cells exam- 
ined and is a consistent finding in both complex and simple cells. The degree of 
suppression depends linearly on the size of the orthogonal grating up to a critical 
dimension which is smaller than the classical receptive field dimension. It is possible 
to suppress a response to the baseline firing rate by either increasing the size or the 
contrast of the orthogonal grating. 
The model outlined earlier can account for all these observations, and similar mea- 
surements recently described by Sillitto et. al. (1995), in a strikingly simple fash- 
ion in the setting of single mode bifurcations. Orthogonal inputs are of the form 
a/211 + cos2s( - 0)] + b/211 q- cos2s( - 0 q- 90)], where a and b are ampli- 
tudes with a  b and   [-90 , 90]. By simple trigonometry this simplifies to 
(a + b)/2 + (a - b)/2 cos 2s( - b0) Thus the input of amplitude b reduces the ampli- 
tude of the orthogonal input and hence gives rise to a smaller response. This is then 
the mechanism by which local double orthogonal stimulation leads to suppression. 
The center-surround case is different in that the orthogonal input originates from 
the horizontal connections and (in the suppressive setting) is input primarily to 
the orthogonal inhibitory population. It can be shown rigorously that for small 
amplitude stimuli this is equivalent to an orthogonal input to the excitatory popu- 
lation with opposite sign. Thus we have a total input (a + b)/211 + cos(2s( - 0)] 
where b arises from the horizontal input and hence increases the amplitude of the 
fundamental component of the input. 
Visual Cortex Circuitry and Orientation Tuning 891 
1  
i0.5 
-90 -60 -30 0 30 60 90 
B 
-go -60 -30 0 30 60 go 
3 
:0.5 
C 
D 
-go -60 -30 0 30 60 90 -go -60 -30 0 30 60 90 
Figure 2: Non-local effects on orientation tuning. (o-o-o) = Center response to pre- 
ferred orientation at different surround orientations, (x-x-x) = Center orientation 
tuning without surround stimulation, (--) = Center response to surround stimula- 
tion alone. A - response of population with 20  orientation preference. B, C and D 
- response of populations with 5  orientation preference. 
More realistically, multiple modes may bifurcate simultaneously, though in the small 
amplitude linear regime where orientation preference is determined these can all be 
treated separately in the manner detailed above and lead to the same result. 
3.2 POPOUT AND GAIN CONTROL 
"Stimulus" dependent horizontal connections have recently been used to model a 
possible physiological correlate for the psychophysical phenomena of "pop-out" 
and enhancement (Stemruler, Usher & Niebur, 1995). In this model the horizontal 
input to excitatory neurons is via sodium channels which are dependent in the 
conventional manner on the differential between the membrane voltage and the 
sodium equilibrium potential. For such sodium channels the synaptic currents are 
attenuated as the membrane depolarizes towards the sodium equilibrium potential. 
This effect is opposite to that observed for the sodium channels mediating the 
horizontal input (Hirsch & Gilbert, 1991) which show increased synaptic currents as 
the membrane is alepolarized. Thus although this model reproduces the phenomena 
described above it does not do so on the basis of the known physiology. We have 
confirmed with our model that weak stimulus enhancement and strong stimulus 
pop-out can be modelled with a variety of formulations for the excitatory neuron 
horizontal input, including a formulation which attenuates with increasing activity 
as in the model of Stemruler et. al. 
It is interesting to note that one overall effect of the horizontal connections is to act 
as a type of gain control uniformizing the local response over a range of stimulus 
strengths. This gain control function has been discussed by Somers, Nelson & Sur 
(1994). 
3.3 THE TILT ILLUSION 
The tilt illusion (TI) is one of the basic orientation-based visual illusion. A TI 
occurs when viewing a test line against an inducing grating of uniformly oriented 
lines with an angle of 0 between the orientation of the test line and the inducing 
892 T. Mundel, A. Dirnitrov and J. D. Cowan 
grating. Two components of the TI have been described (Wenderoth & Johnstone, 
1988), the direct TI where the test line appears to be repelled by the grating- 
the orientation differential appears increased, and the indirect TI where the test 
line appears attracted to the orientation of the grating-the orientation differential 
appears decreased. Figure 3 depicts a typical plot of magnitude of the tilt effect 
versus the angle differential between inducing grating and test line reproduced from 
Wenderoth & Beh (1977). The TI thus provides compelling evidence that local 
k'ucir, g figure tm 
Figure 3: Direct (positive) and indirect (negative) tilt effects 
detection of edges is dependent on information from more distant points in the 
visual field. It is generally believed, that the direct TI is due to lateral inhibition 
between cortical neurones (Wenderoth & Johnstone, 1988: Carpenter & Blakemore, 
1973). It has been postulated that the indirect TI occurs at a higher level of visual 
processing. We show here that both the direct and indirect TI are a consequence 
of the lateral and local connections in our model. 
1 
0 
-0.5 
A 
_ f 
10 30 50 70 90 
-90 -60 -30 0 30 60 90 
1 
c 
10 30 50 70 90 
eo 
-90 -60 -30 0 30 60 90 
Figure 4: Model simulations of the tilt effect (A and C). B and D show the corre- 
sponding kernels mediating long-range interactions. Solid lines indicate the absolute 
kernel and dashed lines indicate the effective kernel 
In figure 4 we give examples of the TI obtained from the model system. The 
effective kernels for long-range interactions are obtained by filtering the absolute 
kernels with the local filter which has a band-pass characteristic. It is this effective 
kernel which determines the tilt effect in keeping with our simulations and analysis 
which show that orientation preference is determined at the small amplitude linear 
stage of system development. 
Visual Cortex Circuitry and Orientation Tuning 893 
4 SUMMARY AND DISCUSSION 
We have shown that a very simple center-surround organization, operating in the 
orientation domain can successfully account for a wide range of neurophysiological 
and psychophysical phenomena, all involving the effects of visual context on the 
responses of assemblies of spiking neurons. We expect to be able to show that 
such an organization can be seen in many parts of the cortex, and that it plays an 
important role in many forms of information processing in the brain. 
Acknowledgements 
Supported in part by Grant  96-24 from the James S. McDonnell Foundation. 
References 
C. Blakemore and E.A. Tobin, Lateral Inhibition Between Orientation Detectors in 
the Cat's Visual Cortex, Exp. Brain Res., 15,439-440, (1972) 
G.G. Blasdel, Orientation selectivity, preference, and continuity in monkey striate 
cortex, J. Neurosci. 12 No 8, 3139-3161 (1992) 
R.H.S. Carpenter and C. Blakemore, Interactions between orientations in human 
vision, Expl. Brain. Res., 18, 287-303, (1973) 
G.C. DeAngelis, J.G. Robson, I. Ohzawa and R.D. Freeman, Organization of Sup- 
pression in Receptive Fields of Neurons in Cat Visual Cortex, J. Neurophysiol., 68 
No 1, 144-163, (1992) 
D. Fitzpatrick, The Functional Organization of Local Circuits in Visual Cortex: 
Insights from the Study of Tree Shrew Striate Cortex, Cerebral Cortex 6, 329-341, 
(1996) 
J.D. Hirsch and C.D. Gilbert, Synaptic physiology of horizontal connections in the 
cat's visual cortex, J. Neurosci., 11, 1800-1809, (1991) 
A.M. Sillito, K.L. Grieve, H.E. Jones, J. Cudeiro & J. Davis, Visual cortical mech- 
anisms detecting focal orientation discontinuities, Nature, 378,492-496, (1995) 
D.C. Somers, S. Nelson and M. Sur, Effects of long-range connections on gain 
control in an emergent model of visual cortical orientation selectivity, Soc. Neu- 
rosci.,20, 646.7, (1994) 
M. Stemmler, M. Usher and E. Niebur, Lateral Interactions in Primary Visual 
Cortex: A Model Bridging Physiology and Psychophysics, Science, 269, 1877-1880, 
(1995) 
M.V. Tsodyks and T. Sejnowski, Rapid state switching in balanced cortical network 
models, Network, 6 No 2, 111-124, (1995) 
J.D. Victor, K. Purpura, E. Katz and B. Max), Population encoding of spatial 
frequency, orientation and color in macaque V1, J. Neurophysiol., 72 No 5, (1994) 
C. Vreeswijk and H. Sompolinsky, Chaos in neuronal networks with balanced exci- 
tatory and inhibitory activity. Science 274, 1724-1726, (1996) 
P. Wenderoth and H. Beh, Component analysis of orientation illusions, Perception, 
6 57-75, (1977) 
P. Wenderoth and S. Johnstone, The different mechanisms of the direct and indirect 
tilt illusions, Vision Res., 28 No 2, 301-312, (1988) 
