{"title": "A Computationally Robust Anatomical Model for Retinal Directional Selectivity", "book": "Advances in Neural Information Processing Systems", "page_first": 477, "page_last": 484, "abstract": null, "full_text": "477 \n\nA COMPUTATIONA.LLY ROBUST \n\nANATOlVIICAL MODEL FOR RETIN.AL \n\nDIRECTIONAL SELECTI\\lITY \n\nNorberto M. Grzywacz \nCenter BioI. Inf. Processing \nMIT, E25-201 \nCambridge, MA 02139 \n\nFranklin R. Amthor \nDept. Psychol. \nUniv. Alabama Birmingham \nBirmingham, AL 35294 \n\nABSTRACT \n\nWe analyze a mathematical model for retinal directionally selective \ncells based on recent electrophysiological data, and show that its \ncomputation of motion direction is robust against noise and speed. \n\nINTROduCTION \n\nDirectionally selective retinal ganglion cells discriminate direction of visual motion \nrelatively independently of speed (Amthor and Grzywacz, 1988a) and with high \ncontrast sensitivity (Grzywacz, Amthor, and Mistler, 1989). These cells respond \nwell to motion in the \"preferred\" direction, but respond poorly to motion in the \nopposite, null, direction. \n\nThere is an increasing amount of experimental work devoted to these cells. \nThree findings are particularly relevant for this paper: 1- An inhibitory process \nasymmetric to every point of the receptive field underlies the directional selectivity \nof ON-OFF ganglion cells of the rabbit retina (Barlow and Levick, 1965). This \ndistributed inhibition allows small motions anywhere in the receptive field center \nto elicit directionally selective responses. 2- The dendritic tree of directionally \nselective ganglion cells is highly branched and most of its dendrites are very fine \n(Amthor, Oyster and Takahashi, 1984; Amthor, Takahashi, and Oyster, 1988). 3-\nThe distributions of excitatory and inhibitory synapses along these cells' dendritic \ntree appear to overlap. (Famiglietti, 1985). \n\nOur own recent experiments elucidated some ofthe spatiotemporal properties of \nthese cells' receptive field. In contrast to excitation, which is transient with stimulus, \nthe inhibition is sustained and might arise from sustained amacrine cells (Amthor \nand Grzywacz, 1988a). Its spatial distribution is wide, extending to the borders \nof the receptive field center (Grzywacz and Amthor, 1988). Finally, the inhibition \nseems to be mediated by a high-gain shunting, not hyperpolarizing, synapse, that \nis, a synapse whose reversal potential is close to cell's resting potential (Amthor \nand Grzywacz, 1989). \n\n\f478 \n\nGrzywacz and Amthor \n\nIn spite of this large amount of experimental work, theoretical efforts to put \n\nthese pieces of evidence into a single framework have been virtually inexistent. \n\nWe propose a directional selectivity model based on our recent data on the \n\ninhibition's spatiotemporal and nonlinear properties. This model, which is an elab(cid:173)\noration of the model of Torre and Poggio (1978), seems to account for several \nphenomena related to retinal directional <;eledivity. \n\nTHE ]\\IIODEL \n\nFigure 1 illustrates the new model for retinal directional selectivity. In this modd, \na stimulus moving in the null direction progressively activates receptive field regions \nlinked to synapses feeding progressively more distal dendrites Of the ganglion cells . \nEvery point in the receptive field center activates adjacent excitatory a~d inhibitory \nsynapses. The inhibitory synapses are assumed to cause shunting inhibition. (\"'\"e \nalso formulated a pre-ganglionic version of this model, which however, is outside \nthe scope of this paper) . \n\nNULL \n\n.--\n\nFIGURE 1. The new model for retinal directional selectivity. \n\nThis model is different than that of Poggio and Koch (1987), where the motion axis is \nrepresented as a sequence of activation of different dendrites. Furthermore, in their \nmodel, the inhibitory synapses must be closer to the soma than the excitatory ones . \n(However, our model is similar a model proposed, and argued against, elsewhere \n(Koc,h, Poggio, and Torre, 1982). \n\nAn advantage of our model is that it accounts for the large inhibitory areas t.o \nmost points of the receptive field (Grzywacz and Amthor, 1988). Also, in the new \nmodel, the distributions of the excitatory and inhibitory synapses overlap along the \ndendritic tree, as suggested (Famiglietti, 1985). Finally, the dendritic tree of ON(cid:173)\nOFF directionally selective ganglion cells (inset- Amthor, Oyster, and Takahashi, \n\n\fA Computationally Robust Anatomical Model \n\n479 \n\n1984) is consistent with our model. The tree's fine dendrites favor the multiplicity \nof directional selectivity and help to deal with noise (see below). \n\nIn this paper, we make calculations with an unidimensional version of the model \ndealing with motions in the preferred and null directions. Its receptive field maps \ninto one dendrite of the cell. Set the origin of coordinates of the receptive field to be \nthe point where a dot moving in the null direction enters the receptive field. Let S \nbe the size of the receptive field. Next, set the origin of coordinates in the dendri te \nto be the soma and let L be the length of the dendrite. The model postulates that \na point z in the receptive field activates excitatory and inhibitory synapses in pain t \nz = zL/ S of the dendrite. \n\nThe voltages in the presynaptic sites are assumed to be linearly related to the \nstimulus, [(z,t), that is, there are functions fe{t) and li(t) such that the excitatory, \n{3e(t), and inhibitory, {3i(t), presynaptic voltages of the synapses to position ;r in \nthe dendrite are \n\n. \n. \nJ = e, l, \n\nwhere \"*\" stands for convolution. We assume that the integral of Ie is zero, (the \nexcitation is transient), and that the integral of Ii is positive. (In practice, gamma \ndistribution functions for Ii and the derivatives of such functions for Ie were used \nin this paper's simulations.) \n\nThe model postulates that the excitatory, ge, and inhibitory, gi, postsynaptic \nconductances are rectified functions of the presynaptic voltages. In some examples, \nwe use the hyperbolic tangent as the rectification function: \n\nwhere Ij and T; are constants. In other examples, we use the rectification functions \ndescribed in Grzywacz and Koch (1987), and their model of ON-OFF rectifications. \nFor the postsynaptic site, we assume, without loss of generality, zero reversal \npotential and neglect the effect of capacitors (justified by the slow time-courses of \nexcitation and inhibition). \n\nAlso, we assume that the inhibitory synapse leads to shunting inhibition, that \nis, its conductance is not in series with a battery. Let Ee be the voltage of the \nexcitatory battery. In general, the voltage, V, in different positions of the dendrite \nis described by the cable equation: \n\nd2~~:, t) = Ra (-Ee!}e (z, t) + V (z, t) (ge (z, t) + 9j (z, t) + 9,.)), \n\nwhere Ra is the axoplasm resistance per unit length, g,. is the resting membrane \nconductance, and the tilde indicates that in this equation the conductances are \ngiven per unit length. \n\n\f480 \n\nGrzywacz and Amthor \n\nFor the calculations illustrated in this paper, the stimuli are always delivered to \nthe receptive field through two narrow slits. Thus, these stimuli activate synapses \nin two discrete positions of a dendrite. In this paper, we only show results for square \nwave and sinusoidal modulations of light, however, we also performed calculations \nfor more general motions. \n\nThe synaptic sites are small so that their resting conductances are negligible, \nand we assume that outside these sites the excitatory and inhibitory conductances \nare zero. In this case, the equation for outside the synapses is: \n\nd2U \ndy2 = U, \n\nwhere we defined A = 1/(Ra y,,)1/2 (the length constant), U = V/Ee, and y = Z/A. \nThe boundary conditions used are \n\nwhere L = L/ A, and where if R, is the soma's input resistance, then p = R, /(RaA) \n(the dendritic-to-soma cond uctance ratio). The first condition means that currents \ndo not flow through the tips of the dendrites. \n\nFinally, label by 1 the synapse proximal to the soma, and by 2 the distal one; \n\nthe boundary conditions at the synapses are \n\nlim U = lim U, \n\n1I~lIj \nII> 111 \n\n1I~lIj \n1Idivity in the {.\u00b7ells\u00b7 receptive fields; the \nrobustness of these computations again~t noise: I h\" robustness of these computa(cid:173)\ntions against speed. \n\nFigure 2 plots the degree of directional selectivity for apparent lIlotions acti(cid:173)\n\nvating two synapses as function of the synapses' distan,'e in a dendrite (computed \nfrom Equations 2 and 3). \n\n1. ~---------------------------\n\n'\" \n\n.. \n.. \n\" c:: ... \n>-.. -~ \n.. u \n.. -~ -\n'\" .. u .. \"-\n\n., \nc: a \n\n'\" \nc \n\n.5 \n\n.0 \n\n.00 \n\n32 \n\n16 \n\nB \n\n.. \n\n2 \n\n0 . 5 \n\n1.5 \n\n2.0 \n\n. 50 \n\n1.0 \n\nDendritic Distance (>.1 \n\nl!~IG URE 2. Locality of lnkraction betwt't\"'n synap\"'t'~ a(I1\\-'(1l,'\" hv apparent mo(cid:173)\ntions. \n\nIt can be shown that the critical parameter controlling whether a certain synap(cid:173)\ntic distance produces a criterion directional selectivity is rj (Equation 1). As the \nparameter rj increases, the criterion distance decreases , Thus , \"ince in retinal di(cid:173)\nrectionally selective cells the inhibition has high gain (Amthor and Grzywa<.\"z, 1989) \nand the dendrites are fine (Amthor, Oyster and Takahashi, 1984; Amthor, Taka(cid:173)\nhashi, and Oyster, 1988), then rj is high, and motions anywhere in receptive field \nshould elicit directionally selective responses (Barlow and Levick, 1965). In other \nwords, the model's receptive field computes motion direction multiple times. \n\nNext, we show that the high inhibitory gain and the cells' fine dendrites help to \ndeal with noise, and thus . may explain the high contrast sensitivity (0.5% contrast-\n\n\f482 \n\nGrzywacz and Amthor \n\nGrzywacz, Amthor, and Mistler, 1989) of the cells' directional selectivity. This \nconclusion is illustrated in Figure 3's multiple plots. \n\n~ 1.0 -\n! \n.. \nJ .50 \n\nOUTPUT NOISE \n\nVCCJ1ATDA'f 1IfI\\/1 NOI. \n\n.. \n\n.. \n\nN , , \n\n.. \n\n- -Looo ~1 \n-NlgII ~I \n\n, \".., ...... a \n.. -\"11 \n\nINIIIIT DAY 1IfI\\/1 NOIS\u00a3 \n\n, \n\n.. .. .. .. \n\noo-===--e:;..........::.-....;::.,---==--..::.. \n\n00 \n\n\u2022 . 0 \n\nResponse \n\n10 \n\n\u2022. 00 \n\n.0 \n\n10 \n\nResponse \n\n.00 \n\n4.0 \n\n1.0 \n\nAupan .. \n\nFIGURE 3. The model's computatifm of direction is robust against additive noise \nin the output, and in the excitatory and inhibitory inputs. \n\nTo generate this figure, we used Equation 2 assuming that a Gaussian noise is added \nto the cell's output, excitatory input, or inhibitory input. {In the latter case, we \nused an approximation that assumes small standard deviation for the inhibitory \ninput's noise.) \n\nOnce again the critical parameter is the ri defined in Equation 1. The larger \nthis parameter is, the better the model deals with noise. In the case of output noise, \nan increase of the parameter separatt's the preferred and null mean responses. For \nnoise in the excitatory input, a parameter increase not only separates the means, \nbut also reduces the standard deviation: Shunting inhibition SHUnTS down the \nnoise. Finally, the most dramatic improvement occurs when the noise is in the \ninhibitory input. (In all these plots, the parameter increase is always by a factor of \nthree.) \n\nSince for retinal direc tionally selective ganglion cells, ri is high (high inhibitory \ngain and fine dendrites), we conclude that the cells' mechanism are particularly well \nsuited to deal with noise. \n\nFor sinusoidal motions, the directional selectivity is robust for a large range \nof temporal frequencies provided that the frequencies are sufficiently low (Figure \n4). (Nevertheless, the cell's preferred direction responses may be sharply tuned to \neither temporal frequency or speed- Amthor and Grzywacz, 1988). \n\n\fI \n\nis \n\n! \ni \nI \ni \n\n:: \n~ \n\ni , \ni \ni is \n\nA Computationally Robust Anatomical Model \n\n483 \n\n-HI\", rl \n'llHl rl \n-\n\n\" , \n\n, \n\nI \n\n\\ \n\nI \n\nI \n\nI \n\nI \n\nI \n\nI \n\nI \n\n-- 20.0 \n\n2.00 \n\n200. \n\n---------------\n\n2.00 \n\n20 .0 \n\nFrtQlltncy ~I \n\nIE-roo \n\nFIGURE 4. Directional selectivity is robust against speed modulation. To gener(cid:173)\nate this curve, we subtracted the average respons~ to a isolated flickering slit from \nthe preferred and null average responses (from Equation 2). \n\nThis robustness is due to the invariance with speed for low speeds of the relative \ntemporal phase shift between inhibition and excitation. Since the excitation has \nband-pass characteristics, it leads the stimulus by a constant phase. On the other \nhand, the inhibition is delayed and advanced in the preferred and null directions \nrespectively, due to the asymmdric spatial integration. The phase shifts due to this \nintegration are also speed invariant. \n\nCONCLUSIONS \n\nWe propose a new model for retinal directional selectivity. The shunting inhibi(cid:173)\ntion of ganglion cells (Torre and Poggio, 1978), which is temporally sustained, is \nthe main biophysical mechanism of the model. It postulates that for null direc(cid:173)\ntion motion, the stimulus activates regions of the receptive field that are linked to \nexcitatory and inhibitory synapses, which are progressively farther away from the \nsoma. This models accounts for: 1- the distribution of inhibition around points \nof the receptive field (Grzywacz and Amthor, 1988); 2- the apparent full overlap \nof the distribution of excitatory and inhibitory synapses along the dendritic trees \nof directionally selective ganglion cells (Famiglietti, 1985); 3- the multiplicity of \ndirectionally selective regions (Barlow and Levick, 1965); 4- the high contrast sen(cid:173)\nsitivity of the cells' directional selectivity (Grzywacz, Amthor, and Mistler, 1989); \n5- the relative in variance of directional selectivity on stimulus speed (Amthor and \nGrzywacz, 1988). \n\nTwo lessons of our model to neural network modeling are: Threshold is not \nthe only neural mechanism, and the basic computational unit may not be a neuron \n\n\f484 \n\nGrzywacz and Amthor \n\nbut a piece of membrane (Grzywacz and Poggio, 1989). In our model, nonlinear \ninteractions are relatively confined to specific dendritic tree branches (Torre and \nPoggio, 1978). This allows local computations by which single cells might generate \nreceptive fields with multiple directionally selective regions, as observed by Barlow \nand Levick (1965). Such local computations could not occur if the inhibition only \nworked through a reduction in spike rate by somatic hyperpolarization. \n\nThus, most neural network models may be biologically irrelevant, since they \nare built upon a too simple model of the neuron. The properties of a network \ndepend strongly on its basic elements. Therefore, to understand the computations \nof biological networks. it may be essential to first understand the basic biophysical \nmechanisms of information processing before developing complex networks. \n\nACKNOWLEDGMENTS \n\nWe thank Lyle Borg-Graham and Tomaso Poggio for helpful discussions. Also, we \nthank Consuelita Correa for help with the figures. N .M.G. was supported by grant \nBNS-8809528 from the National Science Foundation, by the Sloan Foundation, and \nby a grant to Tomaso Poggio and Ellen Hildreth from the Office of Naval Research, \nCognitive and Neural Systems Division. F.R.A. was supported by grants from the \nNational Institute of Health (EY05070) and the Sloan Foundation. \n\nREFERENCES \n\nAmthor & Grzywacz (1988) Invest. Ophthalmol. Vi\". Sci. 29:225. \nAmthor & Grzywacz (1989) Retinal Directional Selectivity Is Accounted for by \n\nShunting Inhibition. Submitted for Publication. \n\nAmthor, Oyster & Takahashi (1984) Brain Res. 298:187. \nAmthor, Takahashi & Oyster (1989) J. Compo Neurol. In Press. \nBarlow & Levick (1965) J. Physiol. 178:477. \nFamiglietti (1985) Neuro\"ci. Abst. 11:337. \nGrzywacz & Amthor (1988) Neurosci. Ab\"t. 14:603. \nGrzywacz, Amthor & Mistler (1989) Applicability of Quadratic and Threshold Mod(cid:173)\n\nels to Motion Discrimination in the Rabbit Retina. Submitted for Publication. \n\nGrzywacz & Koch (1987) Synapse 1:417. \nGrzywacz & Poggio (1989) In An Introduction to Neural and Electronic Networks. \n\nZornetzer, Davis & Lau, Eds. Academic Press, Orlando, USA. In Press. \n\nKoch, Poggio & Torre (1982) Philos . Tran\". R. Soc. B 298:227. \nPoggio & Koch (1987) Sci. Am. 256:46. \nTorre & Poggio (1978) Proc. R. Soc. B 202:409. \n\n\f", "award": [], "sourceid": 93, "authors": [{"given_name": "Norberto", "family_name": "Grzywacz", "institution": null}, {"given_name": "Franklin", "family_name": "Amthor", "institution": null}]}