477 
A COMPUTATIONALLY ROBUST 
ANATOMICAL MODEL FOR RETINAL 
DIRECTIONAL SELECTIVITY 
Norberto M. Grzywacz 
Center Biol. Inf. Processing 
MIT, E25-201 
Cambridge, MA 02139 
Franklin R. Amthor 
Dept. Psychol. 
Univ. Alabama Birmingham 
Birmingham, AL 35294 
ABSTRACT 
We analyze a mathematical model for retinal directionally selective 
cells based on recent electrophysiological data, and show that its 
computation of motion direction is robust against noise and speed. 
INTRODUCTION 
Directionally selective retinal ganglion cells discriminate direction of visual motion 
relatively independently of speed (Amthor and Grzywacz, 1988a) and with high 
contrast sensitivity (Grzywacz, Arethor, and Mistler, 1989). These cells respond 
well to motion in the "preferred" direction, but respond poorly to motion in the 
opposite, nnll, direction. 
There is an increasing amount of experimental work devoted to these cells. 
Three findings are particularly relevant for this paper: 1- An inhibitory process 
asymmetric to every point of the receptive field underlies the directional selectivity 
of ON-OFF ganglion cells of the rabbit retina (Barlow and Levick, 1965). This 
distributed inhibition allows small motions anywhere in the receptive field center 
to elicit directionally selective responses. 2- The dendritic tree of directionally 
selective ganglion cells is highly branched and most of its dendrites are very fine 
(Amthor, Oyster and Takahashi, 1984; Arethor, Takahashi, and Oyster, 1988). 3- 
The distributions of excitatory and inhibitory synapses along these cells' dendritic 
tree appear to overlap. (Famiglietti, 1985). 
Our own recent experiments elucidated some of the spatiotemporal properties of 
these cells' receptive field. In contrast to excitation, which is transient with stimulus, 
the inhibition is sustained and might arise from sustained amacrine cells (Arethor 
and Grzywacz, 1988a). Its spatial distribution is wide, extending to the borders 
of the receptive field center (Grzywacz and Arethor, 1988). Finally, the inhibition 
seems to be mediated by a high-gain shunting, not hyperpolarizing, synapse, that 
is, a synapse whose reversal potential is close to cell's resting potential (Arethor 
and Grzywacz, 1989). 
478 Grzywacz and Arethor 
In spite of this large amount of experimentM work, theoretical efforts to put 
these pieces of evidence into a single framework have been virtually inexistent. 
We propose a directional selectivity model based on our recent data on the 
inhibition's spatiotemporal and nonlinear properties. This model, which is an elab- 
oration of the model of Torre and Poggio (1978), seems to account for several 
phenomena related to retinal directional selectivity. 
THE MODEL 
Figure 1 illustrates the new model for retinal directional selectivity. In this model, 
a stimulns moving in the nnll direction progressively activates receptive field regions 
linked to synapses feeding progressively more distal dendrites Of the ganglion cells. 
Every point in the receptive field center activates adjacent excitatory a/td inhibitory 
synapses. The inhibitory synapses are assumed to canse shunting inhibition. (We 
also formulated a pre-ganglionic version of this model, which however, is outside 
the scope of this paper). 
NULL 
)ENDRITE 
DENDRITE 
FIGURE 1. The new model for retinal directional selectivity. 
This model is different than that of Poggio and Koch (1987), where the motion axis is 
represented as a sequence of activation of different dendrites. Furthermore, in their 
model, the inhibitory synapses must be closer to the soma than the excitatory ones. 
(However, our model is similar a model proposed, and argued against, elsewhere 
(Koch, Poggio, and Torre, 1982). 
An advantage of our model is that it accounts for the large inhibitory areas to 
most points of the receptive field (Grzywacz and Arethor, 1988). Also, in the new 
model, the distributions of the excitatory and inhibitory synapses overlap along the 
dendritic tree, as suggested (Famiglietti, 1985). Finally, the dendritic tree of ON- 
OFF directionally selective ganglion cells (inset- Amthor, Oyster, and Takahashi, 
A ComputationsJly Robust Anatomical Model 479 
1984) is consistent with our model. The tree's fine dendrites favor the multiplicity 
of directional selectivity and help to deal with noise (see below). 
In this paper, we make calculations with an unidimensional version of the model 
dealing with motions in the preferred and null directions. Its receptive field maps 
into one dendrite of the cell. Set the origin of coordinates of the receptive field to be 
the point where a dot moving in the null direction enters the receptive field. Let $ 
be the size of the receptive field. Next, set the origin of coordinates in the dendrite 
to be the soma and let /, be the length of the dendrite. The model postulates that 
a point z in the receptive field activates excitatory and inhibitory synapses in point 
:e = z,/$ of the dendrite. 
The voltages in the presynaptic sites are assumed to be linearly related to the 
stimulus, (z,t), that is, there are functions f(t) and fi(t) such that the excitatory, 
/3,(t), and inhibitory, /3i(t), presynaptic voltages of the synapses to position a in 
the dendrite are 
where "." stands for convolution. We assume that the integral of fe is zero, (the 
excitation is transient), and that the integral of fi is positive. (In practice, gamma 
distribution functions for fi and the derivatives of such functions for fe were used 
in this paper's simulations.) 
The model postulates that the excitatory, g, and inhibitory, gi, postsynaptic 
conductances are rectified functions of the presynaptic voltages. In some examples, 
we use the hyperbolic tangent as the rectification function: 
71 j =e,i, 
gi('t) = 1 
where 7i and / are constants. In other examples, we use the rectification functions 
described in Grzywacz and Koch (1987), and their model of ON-OFF rectifications. 
For the postsynaptic site, we assume, without loss of generality, zero reversal 
potential and neglect the effect of capacitors (justified by the slow time-courses of 
excitation and inhibition). 
Also, we assume that the inhibitory synapse leads to shunting inhibition, that 
is, its conductance is not in series with a battery. Let E be the voltage of the 
excitatory battery. In general, the voltage, V, in different positions of the dendrite 
is described by the cable equation: 
g2v(z,t) 
= + v + + 
d;r 2 
where Ra is the axoplasm resistance per unit length, g. is the resting menbrane 
conductance, and the tilde indicates that in this equation the conductances are 
given per unit length. 
480 Grzywacz and Arethor 
For the calculations illustrated in this paper, the stimuli are always delivered to 
the receptive field through two narrow slits. Thus, these stimuli activate synapses 
in two discrete positions ofa dendrite. In this paper, we only show results for square 
wave and sinusoidal modulations of light, however, we also performed calculations 
for more general motions, 
The synaptic sites are small so that their resting conductdices are negligible, 
and we assume that outside these sites the excitatory and inhibitory conductdices 
are zero. In this case, the equation for outside the synapses is: 
dU 
dy  
where we defined A = t/: (the length constant), U - V'/Ee, and 
The boundary conditions used a.re 
dU [y= = O, dU = U(O), 
dy 
where /, = ,/A, and where if R0 is the soma's input resistance, then p = 
(the dendritic-to-soma conductance ratio). The first condition means that currents 
do not flow through the tips of the dendrites. 
Finally, label by 1 the synapse proximM to the soma, and by 2 the distal one; 
the boundary conditions at the synapses are 
lim U= lira U, j=l,2, 
dU dU 
>J <i 
where r, = ,RA and r i 
It can be shown that the relative inhibitory strength for motions in the prefeed 
direction decreases with L and increases with p. Thus, to favor conditions for 
multiplicity of direction selectivity in the receptive field, we perform cMculations 
with L   and p = t. The strengths of the excitatoy synapses are set such that 
their contribution to somatic voltage in the absence of inhibition is invariant with 
position. Finally, we ensure that the excitatory synapses never saturate. 
Unde these conditions, one can show that the voltage in the soma is: 
((vi,t + 2) ri,2 + 2vi,1 + 4) e 26 - vi,tvi,2 
where y is the distance between the synapses. 
A final quantity, which is used in this paper is the directional selectivity index 
n. Let v and Un be the total responses to the second slit in the apparent motion 
in the preferred and null directions respectively. (Alternatively, for the sinusoidal 
A Computationally Robust Anatomical Model 481 
motion, these quantities are the respective average responses.) We follow Grzywacz 
and Koch (1987) and define 
RESULTS 
FIGURE 
tions. 
This section presents the results of calculations based on the model. Ve address: 
the multiple computations of directional selectivity in the cells' receptive fields; the 
robustness of these computations against noise: th,' robustness of these computa- 
tions against speed. 
Figure 2 plots the degree of directional selectivit for apparent motions acti- 
vating two synapses as function of the synapses' distan,-e in a deadrite (computed 
kern Equations 2 and 3). 
3E 
 05 
.00 .50 t.0 t.5 .0 
Dendpitic Distance ( X} 
2. Locality of toteraction betwu synapse, tcavat,', hv apparent 
It can be shown that the critical parameter controlling whether a certain synap- 
tic distance produces a criterion directional selectivity is 'i (Equation 1). As the 
parameter v i increases, the criterion distance decreases. Thus, since in retinal di- 
rectionaJly selective cells the inhibition has high gain (Arethor and Grzywacz, 1989) 
and the dendrites are fine (Anthor, Oyster and Takahashi, 1984; Arethor, Taka- 
hashi, and Oyster, 1988), then vi iS high, and motions anywhere in receptive field 
should elicit directionally selective responses (Barlow and Levick, 1965). In other 
words, the model's receptive field computes motion direction multiple times. 
Next, we show that the high inhibitory gain and the cells' fine dendrites help to 
deal with noise, and thus. may explain the high contrast sensitivity (0.5% contrast- 
482 Grzywacz and Amthor 
Grzywacz, Arethor, and blistler, 1989) of the cells' directional selectivity. This 
conclusion is illustrated in Figure 3's multiple plots. 
1.5 
00 
OUTPU'T 
N N P P 
00 .0 
lponse 
[xci'rA'rlY IldPU1  
N 
N_ p 
B 0 ..00 ,4 0 B 0 .0o 4.0 B.O 
lponse $Vonlt 
FIGUHE 3. The mode['s computation of direction is robust against additive noise 
in the output, and in the excitatory and inhibitory inputs. 
To generate this figure, we used Equation 2 assuming that a Gaussian noise is added 
to the cell's output, excitatory input, or inhibitory input. (In the latter case, we 
used an approximation that assumes small standard deviation for the inhibitory 
input's noise.) 
Once again the critical parameter is the ri defined in Equation 1. The larger 
this parameter is, the better the model deals with noise. In the case of output noise, 
an increase of the parameter separates the preferred and null mean responses. For 
noise in the excitatory input, a parameter increase not only separates the means, 
but also reduces the standard deviation: Shunting inhibition SHUNTS down the 
noise. Finally, the most dramatic improvement occurs when the noise is in the 
inhibitory input. (In all these plots, the parameter increase is always by a factor of 
three.) 
Since for retinal directionally selective ganglion cells, ri is high (high inhibitory 
gain and fine dendrites), we conclude that the cells' mechanism are particularly well 
suited to deal with noise. 
For sinusoidal motions, the directional selectivity is robust for a large range 
of temporal frequencies provided that the frequencies are sufficiently low (Figure 
4). (Nevertheless, the cell's preferred direction responses may be sharply tuned to 
either temporal frequency or speed- Arethor and Grzywacz, 1988). 
A Computationally Robust Anatomical Model 483 
/ 
/ 
// 
2.00 Q.O 0. 
FIGURE 4. Directional selectivity is robust against speed modulation. To gener- 
ate this curve, we subtracted the average response to a isolated flickering slit from 
the preferred and null average responses (from Equation 2). 
This robustness is due to the invariance with speed for low speeds of the relative 
temporal phase shift between inhibition and excitation. Since the excitation has 
band-pass characteristics, it leads the stimulus by a constant phase. On the other 
hand, the inhibition is delayed and advanced in the preferred and null directions 
respectively, due to the asymmetric spatial integration. The phase shifts due to this 
integration are also speed invariant. 
CONCLUSIONS 
We propose a new model for retina[ directional selectivity. The shunting inhibi- 
tion of ganglion cells (Torre and Poggio, 1978), which is temporally sustained, is 
the main biophysical mechanism of the model. It postulates that for null direc- 
tion motion, the stimulus activates regions of the receptive field that are linked to 
excitatory and inhibitory synapses, which are progressively farther away from the 
soma. This models accounts for: 1- the distribution of inhibition around points 
of the receptive field (Grzywacz and Amthor, 1988); 2- the apparent full overlap 
of the distribution of excitatory and inhibitory synapses along the dendritic trees 
of directionally selective ganglion cells (Famiglietti, 1985); $- the multiplicity of 
directionally selective regions (Barlow and Levick, 1965); 4- the high contrast sen- 
sitivity of the cells' directional selectivity (Grzywacz, Amthor, and Mistlet, 1989); 
5- the relative invariance of directional selectivity on stimulus speed (Arethor and 
Gr.ywac., 1988). 
Two lessons of our model to neural network modeling are: Threshold is not 
the only neural mechanism, and the basic computational unit may not be a neuron 
484 Grzywacz and Panthor 
but a piece of membrane (Grzywacz and Poggio, 1989). In our model, nonlinear 
interactions are relatively confined to specific dendritic tree branches (Torre and 
Poggio, 1978). This allows locaJ computations by which single cells might generate 
receptive fields with multiple directionally selective regions, as observed by Barlow 
and Levick (1965). Such local computations could not occur if the inhibition only 
worked through a reduction in spike rate by somatic hyperpolarization. 
Thus, most neuraJ network models may be biologically irrelevant, since they 
are built upon a too simple model of the neuron. The properties of a network 
depend strongly on its basic elements. Therefore, to understand the computations 
of biological networks, it may be essential to first understand the basic biophysical 
mechanisms of information processing before developing complex networks. 
ACKNOWLEDGMENTS 
We thank Lyle Borg-Graham and Tomaso Poggio for helpful discussions. Also, we 
thank Consuelita Correa for help with the figures. N.M.G. was supported by grant 
BNS-8809528 from the National Science Foundation, by the Sloan Foundation, and 
by a grant to Tomaso Poggio and Ellen Hildreth from the Office of Naval Research, 
Cognitive and Neural Systems Division. F.R.A. was supported by grants from the 
National Institute of Health (EY05070) and the Sloan Foundation. 
REFERENCES 
Amthor & Grzywacz (1988) Invest. Ophihalmol. Vis. Sci. 29:225. 
Amthor & Grzywacz (1989) Retinal Directional Selectivity Is Accounted for by 
Shunting Inhibition. Submitted for Publication. 
Arethor, Oyster & Takahashi (1984) Brain Res. 298:187. 
Arethor, Takahashi & Oyster (1989) J. Comp. Neurol. In Press. 
Barlow & Levick (1965) J. Physiol. 178:477. 
Famiglietti (1985)Neurosci. Abst. 11:337. 
Grzywacz & Arethor (1988) Neurosci. Abst. 14:603. 
Grzywacz, Arethor & Mistlet (1989) Applicability of Quadratic and Threshold Mod- 
els to Motion Discrimination in the Rabbit Retina. Submitted for Publication. 
Grzywacz & Koch (1987) Synapse 1:417. 
Grzywacz & Poggio (1989) In An Introduction to Neural and Electronic Networks. 
Zornetzer, Davis & Lau, Eds. Academic Press, Orlando, USA. In Press. 
Koch, Poggio & Torre (1982) Philos. Trans. R. Soc. B 298:227. 
Poggio & Koch (1987) Sci. Am. 256:46. 
Torre & Poggio (1978) Proc. R. Soc. B 202:409. 
