An Architectural Mechanism for 
Direction-tuned Cortical Simple Cells: 
The Role of Mutual Inhibition 
Silvio P. Saba/ini 
silvio@dibe.unige.it 
Fabio Solari 
fabio@dibe.unige.it 
Giacomo M. Bisio 
bisio@dibe.unige.it 
Department of Biophysical and Electronic Engineering 
PSPC Research Group 
Genova, 1-16145, Italy 
Abstract 
A linear architectural model of cortical simple cells is presented. 
The model evidences how mutual inhibition, occurring through 
synaptic coupling functions asymmetrically distributed in space, 
can be a possible basis for a wide variety of spario-temporal simple 
cell response properties, including direction selectivity and velocity 
tuning. While spatial asymmetries are included explicitly in the 
structure of the inhibitory interconnections, temporal asymmetries 
originate from the specific mutual inhibition scheme considered. 
Extensive simulations supporting the model are reported. 
1 INTRODUCTION 
One of the most distinctive features of striate cortex neurons is their combined 
selectivity for stimulus orientation and the direction of motion. The majority of 
simple cells, indeed, responds better to sinusoidal gratings that are moving in one 
direction than to the opposite one, exhibiting also a narrower velocity tuning with 
respect to that of geniculate cells. Recent theoretical and neurophysiological stud- 
ies [1] [2] pointed out that the initial stage of direction selectivity can be related 
to the linear space-time receptive field structure of simple cells. A large class of 
simple cells has a very specific space-time behavior in which the spatial phase of 
the receptive field changes gradually as a function of time. This results in receptive 
field profiles that are tilted in the space-time domain. To account for the origin 
of this particular spario-temporal inseparability, numerous models have been pro- 
posed postulating the existence of structural asymmetries of the geniculo-cortical 
projections both in the temporal and in the spatial domains (for a review, see [3] 
An Architectural Mechanism for Direction-tuned Cortical Simple Cells 105 
Layer 2 ? 
eo 
ge.niculate 
nput 
el 'kla m/kld 
I 
eo 
(b) 
Figure 1: (a) A schematic neural circuitry for the mutual inhibition; (b) equivalent 
block diagram representation. 
[4]). Among them, feed-forward inhibition along the non-preferred direction, and 
the combination of lagged and non-lagged geniculate inputs to the cortex have been 
commonly suggested as the major mechanisms. 
In this paper, within a linear field theory framework, we propose and analyse an 
architectural model for dynamic receptive field formation, based on intracortical 
interactions occurring through asymmetric mutual inhibition schemes. 
2 MODELING INTRACORTICAL PROCESSING 
The computational characteristics of each neuron are not independent of the ones 
of other neurons laying in the same layer, rather, they are often the consequence of 
a collective behavior of neighboring cells. To understand how intracortical circuits 
may affect the response properties of simple cells one can study their structure 
and function at many levels of organization, from subcellular, driven primarily by 
biophysical data, to systemic, driven by functional considerations. In this study, we 
present a model at the intermediate abstraction level to combine both functional 
and neurophysiological descriptions into an analytic model of cortical simple cells. 
2.1 STRUCTURE OF THE MODEL 
Following a linear neural field approach [5] [6], we regard visual cortex as a con- 
tinuous distribution of neurons and synapses. Accordingly, the geniculo-cortical 
pathway is modeled by a multi-layer network interconnected through feed-forward 
and feedback connections, both inter- and intra-layers. Each location on the corti- 
cal plane represents a homogeneous population of cells, and connections represent 
average interactions among populations. Such connections can be modeled by spa- 
tial coupling functions which represent the spread of the synaptic influence of a 
106 S. P. Sabatini, E Solari and G. M. Bisio 
population on its neighbors, as mediated by local axonal and dendritic fields. From 
an architectural point of view, we assume the superposition of feed-forward (i.e., 
geniculate) and intracortical contributions which arise from inhibitory pools whose 
activity is also primed by a geniculate excitatory drive. A schematic diagram show- 
ing the "building blocks" of the model is depicted in Fig. 1. The dynamics of each 
population is modeled as first-order low-pass filters characterized by time constants 
r's. For the sake of simplicity, we restrict our analysis to 1-D case, assuming that 
such direction is orthogonal to the preferred direction of the receptive field [7]. This 
1-D model would produce spatio-temporal results that are directly compared with 
the spatio-temporal plots usually obtained when an optimal stimulus is moved along 
the direction orthogonal to the preferred direction of the receptive field. 
Geniculate contributions eo(x, t) are modeled directly by a spatiotemporal convo- 
lution of the visual input s(x, t) and a separable kernel ho(x, t) characteriaed in the 
spatial domain by a Gaussian shape with spatial extent 0-o and, in the temporal 
domain, by a first-order temporal low-pass filter with time constant r0. The out- 
put e(x, t) of the inhibitory neuron population results from the mutual inhibitory 
scheme through spatially organized pre- and post-synaptic sites, modeled by the 
kernels k,(x- ) and ka(x- ), respectively: 
dcx(i/3't) (2, t) + / kia(x )[e0(-,t) bm(-,t)]d (1) 
rx dt = -el - - 
i. 
m(x,t) = (2) 
where the function re(x, t) describes the spario-temporal mutual inhibitory inter- 
actions, and b is the inhibition strength. The layer 2 cortical excitation e2(x, t) is 
the result of feed-forward contributions collected (k2a) from the inhibitory loop, at 
axonal synaptic sites, and the geniculate input (eo(x, t)). To focus the attention on 
the inhibitory loop, in the following we assume a one-to-one mapping from layer 1 
to layer 2, i.e., ka(x - ) = 5(x - ), consequently: 
= + t) (a) 
r dt 
where r and r2 are the time constants associated to layer 1 and layer 2, respectively. 
2.2 AVERAGE INTRACORTICAL CONNECTIVITY 
When assessing the role of intracortical circuits on the receptive field properties 
of cortical cells, one important issue concerns the spatial localization of inhibitory 
and excitatory influences. In a previous work [8] we evidenced how the steady- 
state solution of Eqs. (1)-(3) can give rise to highly structured Gabor-like re- 
ceptive field profiles, when inhibition arises from laterally distributed clusters of 
cells. In this case, the effective intrinsic kernel k(x), defined as k(x- ) de__f 
f f kxa(-x',-')kxa(x- x', - ')dx'd', can be modeled as the sum of two Gaus- 
sians symmetrically offset with respect to the target cell (see Fig. 2)' 
( w, ) 
1 w exp[-(x - d)2/20-] + exp[-(x + d2)2/20-1 . 
]l(;r) --  k, 0-1 0'2 
(4) 
This work is aimed to investigate how spatial asymmetries in the intracortical cou- 
pling function lead to non-separable space-time interactions within the resulting 
discharge field of the simple cells. To this end, we varied systematically the ge- 
ometrical parameters (0-, w, d) of the inhibitory kernel to consider three different 
An Architectural Mechanism for Direction-tuned Cortical Simple Cells 107 
Figure 2: The basic inhibitory kernel used k(x -). The cell in the center receives 
inhibitory contributions from laterally distributed clusters of cells. The asymmetric 
kernels used in the model derive from this basic kernel by systematic variations of 
its geometrical parameters (see Table 1). 
types of asymmetries: (1) different spatial spread of inhibition (i.e., tr y tr2); (2) 
different amount of inhibition (w y w2); (3) different spatial offset (d y d2). A 
more rigorous treatment should take care also of the continuous distortion of the 
topographic map [9]. In our analysis this would result in a continuous deformation 
of the inhibitory kernel, but for the small distances within which inhibition occurs, 
the approximation of a uniform mapping produces only a negligible error. 
Architectural parameters were determined from reliable measured values of recep- 
tive fields of simple cells [10] [11]. Concerning the spatial domain, we fixed the size 
(tr0) of the initial receptive field (due to geniculate contributions) for an "average" 
cortical simple cell with a resultant discharge field of ~ 2; and we adjusted, ac- 
cordingly, the parameters of the inhibitory kernel in order to account for spatial 
interactions only within the receptive field. 
Considering the temporal domain, one should distinguish the time constant r, 
caused by network interactions, from the time constants r0 and r2 caused by tem- 
poral integration at a single cell membrane. In any case, throughout all simulations, 
we fixed r0 and r2 to 20ms, whereas we'varied r in the range 2 - lOOms. 
3 RESULTS 
Since visual cortex is visuotopicMly organized, a direct correspondence exists be- 
tween the spatial organization of intracortical connections and the resulting recep- 
tive field topography. Therefore, the dependence of cortical surface activity e2(x, t) 
on the visual input s(x,t) can be formulated as e2(x,t) = h(x,t)  s(x,t), where 
the symbol  indicates a spatio-temporal convolution, and h(x,t) is the equiva- 
lent receptive field interpreted as the spatio-temporal distribution of the signs of 
all the effects of cortical interactions. In this context, h(x,t) reflects the whole 
spatio-temporal couplings and not only the direct neuroanatomicM connectivity. 
To test the efficacy of the various inhibitory schemes, we used a drifting sine wave 
grating s(x,t) = Ccos[27r(fx :k ftt)] where C is the contrast, f and ft are the 
spatial and temporal frequency, respectively. The direction selectivity index (DSI) 
and the optimal velocity (vopt) obtained from the various inhibitory kernels of Fig.2 
are summarized in Table 1, for different values of r and b. The direction selectivity 
index is defined as D$I = Rr-R"r where Rp is the maximum response amplitude 
R,+ Rn, ' 
for preferred direction, and Rap is the maximum amplitude for non-preferred di- 
rection. The optimal velocity is defined as ffpt/fz, where fz is chosen to match 
the spatial structure of the receptive field, and ffpt is the frequency which elicits 
the maximum cell's response. As expected, increasing the parameter b enhances 
the effects of inhibition, thus resulting in larger DSI and higher optimal velocities. 
However, for stability reason, b should 'remain below a theshold value strictly re- 
108 S. P. Sabatini, E Solari and G. M. Bisio 
Table 1: 
r=2 r =10 r =20 r =100 
11 II DSX I II DSI I ovt II DSI I II DSI I II 
0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.60 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.80 0.08 1.82 0.24 1.82 0.00 0.00 0.00 0.00 
0.91 0.17 1.82 0.34 1.82 0.00 0.00 0.00 0.00 
0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.85 0.04 1.82 0.17 1.82 0.25 1.82 0.00 0.00 
1.00 0.06 1.82 0.19 1.82 0.28 1.82 0.00 0.00 
1.30 0.07 1.82 0.28 1.82 0.37 1.82 0.00 0.00 
0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
1.00 0.00 0.00 0.00 0.00 0.09 1.82 0.16 1.82 
5.00 0.02 1.82 0.05 1.82 0.09 1.82 0.39 3.64 
9.00 0.01 1.82 0.03 1.82 0.06 1.82 0.38 3.64 
0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.50 0.06 2.07 0.20 2.07 0.32 2.07 0.00 0.00 
0.60 0.06 2.07 0.39 4.14 0.40 2.07 0.00 0.00 
0.72 0.07 2.00 0.66 6.00 0.65 4.00 0.00 0.00 
0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.60 0.00 00.0 0.00 0.00 0.00 0.00 0.00 0.00 
0.80 0.08 1.82 0.23 1.82 0.00 0.00 0.00 0.00 
0.88 0.14 1.82 0.26 1.82 0.00 0.00 0.00 0.00 
0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
0.85 0.04 1.82 0.16 1.82 0.23 1.82 0.00 0.00 
1.00 0.05 1.82 0.18 1.82 0.26 1.82 0.00 0.00 
1.33 0.06 1.82 0.26 1.82 0.35 1.82 0.00 0.00 
ASY-1A 
ASY-1B 
ASY-2A 
ASY-2B 
ASY-3A 
ASY-3B 
lated to the inhibitory kernel considered. Moreover, we observe that, except for 
ASY-2A, the strongest direction selectivity can be obtained when the intracortical 
time constant rv has values in the range of 10 - 20 ms, i.e., comparable to r0 and r2. 
Larger values of r would result, indeed, in a recrudescence of the velocity low-pass 
behavior. For each asymmetry, Figs. 3 show the direction tuning curves and the x-t 
plots, respectively, for the best cases considered (cf. bold-faced values in Table 1). 
We have evidenced that appreciable DSI can be obtained when inhibition arises 
from cortical sites at different distance from the target cell (i.e., ASY-2B, d - d2). 
In such conditions we obtained a DSI as high as 0.66 and an optima] velocity up 
to ~ 6/s, as could be inferred also from the spatio-temporal plot which present a 
marked motion-type (i.e., oriented) non-separability (see Fig. 3ASY-2S). 
4 DISCUSSION AND CONCLUSIONS 
As anticipated in the Introduction, direction selectivity mechanisms usually relies 
upon asymmetric alteration of the spatial and temporal response characteristics of 
the geniculate input, which are presumably mediated by intracortical circuits. In 
the architectural model presented in this study, spatial asymmetries were included 
explicitly in the extension of the inhibitory interconnections, but no explicit asym- 
metric temporal mechanisms were introduced. It is worth evidencing how temporal 
asymmetries originate from the specific mutual inhibition scheme considered, which 
operates, regarding temporal domain, like a quadrature model [12] [13]. This can 
An Architectural Mechanism for Direction-tuned Cortical Simple Cells 109 
o 
7 
o 
o 
7 
1.0 
0.8 
0.6 
0.4 
0.2 
O0 
0.1 
1 10 100 
Velocity deg/s 
0.8 
0.6 
0.4 
0.2 ......... ,,,, 
0.0 
0.1 1 10 100 
Velocity deg/s 
4OO 
space (deg) 2 
4OO 
0.8 
0.6 
0.4 
0.2 
0.0 
0.1 
IB' 
I 10 100 
Velocity deg/s 
N 0.4 
0 
 0.2 
0 
Z 0.0 
0,1 
4OO 
I 10 100 
Velocity deg/s 
0 space(deg) 2 
4OO 
 1.0 
o 
m ' 0.8 
N 0.4 
E 0.2 
o 
Z 0.0 
0.1 I 10 100 
Velocity deg/s 
1.0 
0.8 
0.6 
0.4 
0.2 
0,0 
0.1 
I 10 100 
Velocity deg/s 
4OO 
0 space(deg) 2 
4OO 
space(deg) 2 0 space(deg) 2 0 space(deg) 2 
Figure 3: Results from the model related to the bold-typed values indicated in 
Table 1, for each asymmetry considered. (Top) direction tuning curve; (Bottom) 
spatio-temporal plots. We can evidence a marked direction tuning for ASY-2B, i.e., 
when inhibition arises from two differentially offset Gaussians 
be inferred by the examination of the equivalent transfer function H(w=, we) in the 
Fourier domain: 
Ho(w,wt) 
1 + jwtv + bK(w) 
+ jwtrl 
1 ) 
1 + jwtv + bK(w)' 
(5) 
where upper case letters indicate Fourier transforms, and j is the complex variable. 
The terms in parentheses in Eq. (5) can be interpreted as the sum of temporal com- 
ponents that are approximately arranged in temporal quadrature. Furthermore, one 
can observe that a direct monosynaptic influence (el) from the inhibitory neurons 
of layer i to the excitatory cells of layer 2, would result in the cancellation of the 
quadrature component in Eq. (5). 
Further improvement of the model should take into account also transmission delays 
between spatiMly separated interacting cells, theshold non-linearities, and ON-OFF 
interactions. 
110 S. P Sabatini, E Solari and G. M. Bisio 
Acknowledgements 
This research was partially supported by CEC-Esprit CORMORANT 8503, and by 
MURST 40%-60%. 
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