Effect of Binocular Cortical Misalignment on Ocular Dominance and Orientation Selectivity

E�ect of Binocular Cortical Misalignment on Ocular Dominanceand Orientation Selectivity�Harel Shouval, Nathan Intratory, C. Charles Law, and Leon N Cooper.Departments of Physics and Neuroscience andThe Institute for Brain and Neural SystemsBox 1843, Brown UniversityProvidence, R. I., 02912Revised Oct, 1995AbstractWe model a two-eye visual environment composed of natural images and study its e�ect onsingle cell synaptic modi�cation. In particular, we study the e�ect of binocular cortical mis-alignment on receptive �eld formation after eye opening. We show that binocular misalignmenta�ects PCA and BCM learning in di�erent ways. For the BCM learning rule this misalignmentis su�cient to produce varying degrees of ocular dominance, whereas for PCA learning binocularneurons emerge in every case.1 IntroductionIt is now generally accepted that receptive �elds in the visual cortex of cats are dramatically in u-enced by the visual environment (for a comprehensive review see, Fr�egnac and Imbert, 1984). Innormally reared animals, the population of sharply tuned neurons increases monotonically, whereasfor dark reared animals it initially increases, but then almost disappears (see, for example, Imbertand Buisseret, 1975). Ocular dominance is dramatically in uenced by such manipulations as monoc-ular deprivation (Wiesel and Hubel, 1963) or reverse suture (Blakemore and van Sluyters, 1974;Mioche and Singer, 1989). It has even been shown that preferred orientations can be directly alteredby pairing the preferred orientation with a negative current, and the non-preferred orientation witha positive current (Fr�egnac et al., 1992).The issue of cortical input misalignment and its relation to receptive �eld development, hasbeen studied by Pettigrew (1974). He has found that in area 17 of young kittens, there is a largemisalignment between the receptive �elds of the two eyes, of as much as 30 deg. The misalignmentin normally reared kittens decreases to an average of 5 deg1 within 5 weeks from birth, and remainsaround that level through adulthood (Nikara et al., 1968; Joshua and Bishop, 1970). Binocularly�To Appear in Neural Computation 8.5yalso at Faculty of Exact Sciences Tel-Aviv University1Some examples near the fovea, e.g. Figure 3 of (Pettigrew, 1974) show a much smaller misalignment which issmaller than the receptive �eld size. 1

deprived kittens retain a high degree of misalignment. Furthermore, the reduction in misalign-ment, occurs concurrently with the development of orientation selectivity, ocular dominance anddisparity tuning (Movshon and van Sluyters, 1981). Thus, it follows that modeling of receptive�eld development should take into account this misalignment.Blakemore and Van Sluyters (1975) have suggested that plasticity may be needed in animalswith binocular vision, in order to overcome a developmental misalignment between cortical inputs.Van Sluyters (1977) and Van Sluyters and Levitt (1980) have tested the e�ect of prismatic deviationbetween the eyes. They have found that with a small deviation, kittens showed normal binocularity,while larger deviations reduced binocularity.The relationship between ocular dominance orientation selectivity and disparity was studied incats by Levay and Voigt (1988). They found that no relationship could be established betweencell's best orientation and ocular dominance or any aspect of its disparity tuning. There was alsono relation between ocular dominance and the sensitivity to disparity, while ocular dominance andbest disparity were related; Binocular cells were mostly zero disparity tuned, while more monocularcells were tuned to a broader distribution of best disparity.Di�erent models attempting to explain how cortical receptive �elds evolve have been proposedover the years (von der Malsburg, 1973; Nass and Cooper, 1975; Perez et al., 1975; Sejnowski,1977; Bienenstock et al., 1982; Linsker, 1986; Miller, 1994). Such models are composed of severalcomponents: the exact nature of the learning rule, the representation of the visual environment,and the architecture of the network. Most of these models assume a simpli�ed representation of thevisual environment (e.g. von der Malsburg, 1973), or replace the visual environment by a secondorder correlation function (Miller, 1994).A variant of Hebbian learning rule with subtractive decay and a visual environment representedby a second order correlation matrix, has been shown to achieve monocular receptive �elds (Milleret al., 1989). With a di�erent set of parameters, this learning rule develops orientation selectivecells as well (Miller, 1994). Dayan and Goodhill (1992) have shown that with a uniform positivecorrelation between corresponding pixels in the left and right eye, only binocular cells emerge. Bernset al. (1993) have performed simulations using correlations between a simpli�ed one dimensionalinput to both eyes. They were able to get deviations from totally binocular cells using two phases oflearning and sticky saturation bounds. In the �rst learning phase (prenatal) there was no correlationbetween the eyes, in the second phase (postnatal) the two eyes were positively correlated. Duringthe �rst phase monocular receptive �elds developed, and did not become totally binocular, in thesecond phase, due to the saturation bounds.A recent paper by Erwin, Obermayer and Schulten (1995) has compared the predictions ofseveral cortical plasticity models to experimental results. This comparison applied a di�erent visualenvironment for each model, mostly of a simpli�ed low dimensional type, or a type described only bya second order correlation function, but did not use an environment composed of natural images.They have found that some of the symbolic input models simultaneously developed orientationselectivity and varying degrees of ocular dominance. No misalignment between two eyes, or betweencortical inputs, was considered directly.Realistic representations of the visual environment have only very recently been considered(Hancock et al., 1992; Law and Cooper, 1994; Liu and Shouval, 1994; Shouval and Liu, 1996),and only in recent years have the statistics of natural images been studied and used for predictingreceptive �eld properties (Field, 1987; Field, 1989; Baddeley and Hancock, 1991; Atick and Redlich,1992; Ruderman and Bialek, 1994; Liu and Shouval, 1994; Shouval and Liu, 1996). Baddeley and2

Hancock (1991) have performed simulations of the Principal Component (PC) learning rule in avisual environment composed of natural images. They have found that the �rst PC is radiallysymmetric and that the second one is orientation selective and horizontal. This is due to a slightbias to the horizontal direction in the correlation function of natural images. Liu and Shouval(1994) have analyzed this situation and have shown that it depends on the scale invariant natureof the power spectrum of the natural images (Field, 1987). They have also analyzed the situationwith retinal preprocessing (Shouval and Liu, 1996). Due to the preprocessing, the �rst principalcomponent may become oriented, however, it was always found to be horizontal. Recently the sametype of realistic environment was used for simulations of the BCM rule (Law and Cooper, 1994).They have found receptive �elds selective to all orientations.Once actual visual scenes are used, it is possible to realistically represent two-eye input, andaccount for the fact that the two eyes are not looking at exactly the same visual scene. For example,Li and Atick (1994) have used natural images to extract detailed two-eye power spectra from stereoimages. They have used these results to predict properties of cortical receptive �elds.In this paper, we study the e�ect of a �xed synaptic density (arbor function) misalignmentbetween the cortical inputs coming from both eyes, on the development of ocular dominance andorientation selectivity. This type of misalignment may be caused by an imprecise developmentalalignment of the arbor functions from both eyes.We compare the outcomes that result from two learning rules: PCA (Oja, 1982) and BCM(Bienenstock et al., 1982; Intrator and Cooper, 1992). We have chosen to examine these twobecause they are well de�ned, and have stable �xed points. Many other proposed learning rules(Sejnowski, 1977; Linsker, 1986; Miller et al., 1989, for example), are closely related to the PCArule. Their outcome depends only on �rst and second order statistics. The BCM rule, in contrast,also depends on third order statistics.We show that binocular misalignment a�ects these two learning rules in a di�erent manner. Forthe BCM learning rule, misalignment is su�cient to produce varying degrees of ocular dominance,whereas for the PCA learning rule, binocular neurons emerge independent of the misalignment.2 A Binocular Visual Environment Composed of Natural ImagesWe have used a set of 24 natural scenes. These pictures were taken at Lincoln Woods State Park,scanned into a 256 X 256 pixel image. No corrections have been used for the optical distortions ofthe instruments. We have avoided man-made objects, because they have many sharp edges, andstraight lines, which make it easier to achieve oriented receptive �elds.In this paper, we limit ourselves to study receptive �eld formation near the fovea, and thus donot model the change in resolution which corresponds to the complex log retinotopic mapping.The anatomy of the visual pathway is such that light which falls on the retina is encoded bythe receptors. The signal is then processed by the retinal circuitry and is projected by the ganglioncells onto the optic nerve. Signals from both eyes cross at the Optic Chiasm and continue to theLateral Geniculate Nucleus (LGN). In the LGN inputs from the two eyes are segregated in di�erentlayers. From the LGN, signals project up to the visual cortex.The receptive �elds of both ganglion cells and LGN projections have a center surround shape(Figure 1). Some are \on center", which means that they are excited by spots of light falling ontheir centers, and inhibited by light on the surround; Others are \o� center" and are excited bylight falling on their surround, and inhibited by light in their center. In the cat, the most abundant3

type near the fovea are the X cells. They are linear, i.e., their response to an image that is composedof several components is roughly the sum of the response to the independent components (Orban.,1984).The e�ect of the retinal preprocessing is modeled by convolving the images with a di�erence ofGaussians (DOG) �lter, with a center radius of one pixel (�1 = 1:0) and a surround radius of three(�2 = 3)2. The e�ect of this preprocessing is shown in �gure 1.−

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Figure 1: Three of the natural images used (top) processed by a Di�erence of Gaussians �lter(middle) are shown (bottom.)As illustrated in Figure 2, the input vectors from both eyes are chosen as small, partiallyoverlapping, circular regions of the preprocessed natural images; these converge on the same corticalcell.The input from the right and left eye respectively are denoted by dl and dr, and the output ofthe cortical neuron then becomes: c = �(dl �ml + dr �mr); (1)2This ratio between the center and surround is biologically plausible (e.g., Enroth-Cugell and Robson, 1966).4

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aFigure 2: Schematic diagram of the two eye model, including the visual input preprocessing. Theoverlap parameter O is de�ned as O = s=2a. When O = 1 the receptive �elds are completelyoverlapping, when O = 0 they are non overlapping but touching O < 0 means that they are nonoverlapping and not touching.where � is the non linear activation function of each neuron and ml,mr are the left and rightsynaptic weight vectors respectively. We have used a nonsymmetric activation function to accountfor the fact that neuronal activity as measured from spontaneous activity has a longer way to goup than to go down to zero activity.In order to examine the e�ect of varying the overlap between the receptive �elds we de�nean overlap parameter O = s=2a, where a is the receptive �eld radius in pixels, and s is the linearoverlap in pixels, as shown in Figure 2. O = 1 when the left and right receptive �elds are completelyoverlapping, and O � 0 when the receptive �elds are completely separated. We are interested inthe dependency between ocular dominance and the degree of misalignment between the left andright eye. First, we measure the maximal response of the left and right eye termed L and R. This isdone by �nding the optimal spatial grating frequency, and then �nding the response at the optimalorientation with this grating for each eye. We consider the following ocular dominance measure Bwhich is based on the left and right eye responses:B = L� RL+ R: (2)This measure has been motivated by that used by Albus (1975) in de�ning the bin boundariesfor a seven bin ocular dominance histogram3. Our bin boundaries are given by:�0:85;�0:5;�0:15; 0:15; 0:5; 0:853Since there is always some activity from both eyes, we have extended bin 1 and 7 slightly with respect to thoseused by Albus. 5

3 Cortical plasticity learning rulesWe have employed these realistic visual inputs to test two of the leading visual cortical plasticityrules: Principal Components Analysis (PCA) and the Bienenstock Cooper and Munro (BCM)model. The two algorithms di�er by their information extraction properties as discussed in Intratorand Cooper (1994); PCA extracts second order statistics from the visual environment, while BCMextracts information contained in third order statistics as well. There are other Hebbian learningrules that are related to PCA. These models may produce somewhat di�erent results, but are notstudied here.3.1 Principal Components AnalysisPrincipal components analysis (PCA) is one of the most widely used feature extraction methodfor pattern recognition tasks. PCA features are those orthogonal directions which maximize thevariance of the projected distribution of the data. They also minimize the mean squared errorbetween the data and a linearly reconstructed version of it based on these projections. Principalcomponents are optimal when the goal is to accurately reconstruct the inputs. They are notnecessarily optimal when the goal is classi�cation and the data are not normally distributed (seefor example, p. 212, Duda and Hart, 1973).A simple interpretation of the Hebbian learning rule is that, with appropriate stabilizing con-straints, it leads to the extraction or approximation of principal components. This has often beenmodeled (see for example; von der Malsburg, 1973; Sejnowski, 1977; Oja, 1982; Linsker, 1986;Miller et al., 1989) . The learning rule that we use was proposed by Oja (1982), and has the form:�mi = �[dic� c2mi] (1)where di is the presynaptic activity at synapse i, c is the postsynaptic activity, and mi is thestrength of the synaptic e�cacy of junction i. �, is a small learning rate. This learning rule hasbeen shown to converge to the principal component of the data.3.2 BCM learning ruleThe BCM theory (Bienenstock et al., 1982) was introduced to account for the striking dependenceof the sharpness of orientation selectivity on the visual environment. We shall be using a variation,due to Intrator and Cooper (1992), for a nonlinear neuron with a nonsymmetric sigmoidal transferfunction. Using the above notation, synaptic modi�cation is given by:_mj = ��(c;�M)dj; (2)where the neuronal activity is given by c = �(m �d), �(c;�M) = c(c��M), and �M is a nonlinearfunction of some time averaged measure of cell activity, which in its simplest form is:�M = E[c2]; (3)where E denotes the expectation over the visual environment. The transfer function � is nonsymmetric around 0 to account for the fact that cortical neurons show a low spontaneous activity,and can thus �re at a much higher rate relative to the spontaneous rate, but can go only slightly6

below it4. We have tested several sigmoidal functions to verify the important features needed forrobust results.It has been shown (Intrator and Cooper, 1992) that this version of the BCM learning rule leads toa neuron which seeks multi-modality in the projected distribution (rather than simply maximizingthe variance) and is thus suitable for �nding clusters in high dimensional space. Simulations usingthis learning rule were found to be in agreement with the many experimental results on visualcortical plasticity (Clothiaux et al., 1991; Law and Cooper, 1994). A network implementationof this neuron, which can �nd several projections in parallel while retaining its computationale�ciency, was found applicable for extracting features from very high dimensional vector spaces(Intrator et al., 1991; Intrator, 1992).We have used the modi�cation rule used by Law and Cooper (1994) with a variable learningrate: _mj = � 1�M �(c;�M)dj ; (4)produced faster convergence with qualitatively similar results.4 ResultsFor the results reported here we used �xed circular receptive �elds with diameters of 20 pixels.We tested the robustness of the results to receptive �elds of sizes 10 to 30 pixels and found noqualitative di�erence.4.1 Completely overlapping receptive �eldsWith completely overlapping receptive �elds, BCM neurons develop various orientation preferences,all highly selective. A typical example of such receptive �elds and orientation selectivity is presentedin Figure 3. A less typical result would be a slight ocular preference with high orientation selectivity.It should be noted that high orientation selectivity is obtained for a single neuron with no needto introduce lateral inhibition between neurons; cells produce receptive �elds, of all orientations5,similar to simple cell receptive �elds observed in visual cortex (Jones and Palmer, 1987; see Kandeland Schwarts, 1991, for review). These results are in sharp contrast to those of a PCA neuron.PCA neurons developed receptive �elds with horizontal orientation selectivity only (Figure 3).Orientation selectivity in PCA neurons depends on the retinal preprocessing. When PCA neuronsare trained on raw natural images, the dominant solution is radially symmetric (Liu and Shouval,1994). However, when retinal preprocessing is included, oriented solutions can be attained (Shouvaland Liu, 1996). The strong preference to the horizontal direction is due to a slight bias in thecorrelation function of natural images to this direction (Baddeley and Hancock, 1991; Shouval andLiu, 1996). If the images are rotated by an angle � then so are the preferred orientations of thePCA neurons. The results hold for a linear and nonlinear (sigmoidal) neuron. The results are alsoinvariant to a sign change (weight vectors m and �m are eigen-vectors).4The actual sigma used in the simulations is (ex � e�x)=(0:1ex + 3e�x).5Although all orientations are represented the probability of attaining di�erent orientations may di�er (Law andCooper, 1994). 7

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LeftrightFigure 3: Top: Receptive �elds for a BCM neuron with completely overlapping inputs. Tuningcurves are selective and similar in both eyes. Bottom: Receptive �elds for completely overlappinginputs using the PCA rule.4.2 Partially and non-overlapping receptive �eldsBCM neurons acquire selectivity to various orientations in both the partial and the non-overlappingcases. When cortical inputs are misaligned, various ocular dominance preferences may occur (Fig-ure 4) even for the same overlap. This result stands in sharp contrast to the one obtained by PCAneurons; only binocular neurons emerge for the PCA rule.The BCM receptive �eld formation results are summarized in Figures 5 and 6. Cortical inputsmisalignment does not a�ect orientation selectivity of the dominant eye, but does produce varyingdegrees of ocular dominance; depending upon the degree of overlap between the receptive �elds.The main result is that ocular dominance (even for single cell simulations) depends strongly on thedegree of overlap between visual input to the two eyes. Figure 6 presents a box-plot summary of700 runs showing the dependence of ocular dominance on visual input overlap. It is evident thatbinocularity as well as the spread of ocular dominance increase as the degree of overlap increases.The PCA results for partially overlapping receptive �elds are presented in Figure 7. As men-tioned above, it can be seen that the degree of overlap between receptive �elds does not alter theoptimal orientation, so that whenever a cell is selective its orientation is in the horizontal direction.The degree of overlap does a�ect the shape of the receptive �elds and the degree of orientationselectivity that emerges under PCA; orientation selectivity decreases as the amount of overlap de-creases. However, when there is no overlap at all, one again obtains higher selectivity. For PCA,there is a symmetry between the receptive �elds of both eyes that imposes binocularity. This arisesfrom the invariance of the two-eye correlation function to a parity transformation (see Appendix).We also studied the possibility that under the PCA rule di�erent orientation selective cellswould emerge if the misalignment between the cortical inputs was in the vertical direction. Thistests the e�ect of a shift orthogonal to the preferred orientation. The results show that there is nochange in the orientation preference; even in this case only horizontal receptive �elds emerge.The PCA results described above were quite robust to introduction of nonlinearity in a cell's8

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LeftrightFigure 4: Receptive �elds with a small overlap (O = :2) using the BCM rule. Results vary from fullybinocular cells with a moderate orientation tuning (top) to less binocular cells with well de�nedreceptive �elds as well as high orientation selectivity in one eye (middle), and �nally, monocularhighly selective cells (bottom).activity. There was no qualitative di�erence in the results when a non symmetric sigmoidal transferfunction was used.5 ConclusionsIn this paper we study whether unsupervised learning can produce both orientation selectivity andvarying degrees of ocular dominance with the same set of assumptions about the visual environment.We use a visual environment composed of preprocessed natural images. The two eyes view portionsof the same image and we test how varying the degree of overlap a�ects ocular dominance.The PCA and BCM learning rules were chosen since they are representative of rules based onsecond and third order statistics; they have been used before with natural images environment, andhave stable �xed points. Using the same visual environment, we have shown that misalignmentin the synaptic density function generates various degrees of ocular dominance under a BCM rule,but fails to do so under a PCA rule, where only binocular cells emerge. This makes the BCM moreconsistent with biological �ndings (Hubel and Wiesel, 1962). We have also shown that while thismisalignment does not prevent cells from becoming selective to all directions under a BCM rule, itis unable to overcome the slight horizontal bias in natural images so that only horizontal selectivecells emerge under a PCA rule. This is especially surprising when the misalignment is not in the9

horizontal direction.The result of the BCMmodel are in agreement with biological �ndings; Oriented receptive �eldsof various degrees of binocularity emerge (Hubel and Wiesel, 1962; van Sluyters and Levitt, 1980;Orban., 1984) and in general, the degree of misalignment in mature receptive �elds is smaller thanthat between the arbor functions (the original cortical input misalignment); it has been observedthat the degree of receptive �eld misalignment is reduced by normal rearing (Pettigrew, 1974).The more binocular BCM neurons are, the smaller the misalignment between their RF centers. Incontrast, the displacement of cortical inputs, with single cell PCA learning always leads to binocularand horizontal cells. The result about the �rst PC being horizontal is in agreement with previoustheoretical results (Hancock et al., 1992; Shouval and Liu, 1996). The misalignment between RFcenters is in this case, close to zero for overlapping RF's and large for non overlapping RF's.Experiments with vertical prismatic deviation between the eyes as well as experiments withhorizontal surgical and optical strabismic kittens (van Sluyters and Levitt, 1980) correspond directlyto our simulations with a shift between receptive �elds in the vertical and horizontal direction(Figures 4-7). Experiments with a vertical displacement are most similar to our simulations, sincenaturally the two eyes are not displaced on the vertical axis and one expects a that the animalwould have a lesser ability correct for displacement in the vertical direction. The �ndings of theseexperiments, that with a small deviation kittens show normal binocularity while larger deviationsreduce binocularity, give support to the idea that displacement may indeed be the origin of thevarying degrees of binocularity. These results are in agreement with the BCM result, and not withthe PCA result.A cell which is tuned to a non zero disparity, responds optimally to stimuli which falls either infront or behind the focal plane. This corresponds to a shift of the optical axis of one eye with respectto the other eye. Thus, one might expect that by displacing the cortical receptive �elds of the twoeyes, disparity tuned neurons would emerge, and the disparity would correspond to the degree ofdisplacement. Surprisingly, this is not the case, for both BCM and PCA neurons; Our simulationsshow, that when cells develop binocular receptive �elds, they maximally respond to stimuli in theoverlapping region of the arbor functions. In that region, their maximal response turns out to beon the focal plane (zero disparity) (Figures 5,7). The more monocular BCM neurons also tend todevelop a small non-zero disparity. The results for BCM neurons are in agreement with biological�ndings (LeVay and Voigt, 1988) and with a, recently introduced, disparity tuning model (Bernset al., 1993).Despite this agreement with some disparity tuning results, it is important to remember that oursimulations do not attempt to model disparity tuning. A model for the development of disparitytuning has to account for the fact that in a realistic 3D visual environment, objects that lie in frontof the focal point would be shifted in the opposite direction to objects which lie behind the focalpoint. Thus, the statistics of 3D stereo images is far more complex than what one would use in a2D model. Such a representation of stereo images has recently been used by Li and Atick (1994).We believe that the fact that cells typically emerge with zero disparity tuning in our simulationsis due to the lack of stereo information in our images. We expect that for PCA neurons, stereoinformation will not alter the fundamental result that PCA neurons are binocular (see the analysisin the appendix). However, it remains to be seen how the distribution of best disparity will bealtered.In an analytic study of the symmetry properties of the eigenstates of a misaligned two-eyeproblem, we show that the binocularity of PCA neurons results from the invariance of the two eye10

correlation function to a parity transformation 6.6 AcknowledgmentsThe authors thank the members of Institute for Brain and Neural Systems for many fruitful con-versations and the referees for helpful comments. This research was supported by the Charles A.Dana Foundation, the O�ce of Naval Research and the National Science Foundation.A Appendix: Symmetry properties of the eigenstates of the twoeye problemThe evolution of neurons in a binocular environment under the PCA learning rule, according toequation 1, reaches a �xed point when Qm = �m; (1)where mT = (ml;mr), the left and right eye synaptic strengths; the two-eye correlation functionQ has the form: Q = 0B@ Qll QlrQrl Qrr 1CA ; (2)where Qll and Qrr are the correlation functions within the left and right eyes, Qlr and Qrl arethe correlation functions between the left-right and right-left eyes. We denote by upper case R0sthe coordinates in each receptive �eld with respect to a common origin, and by lower case r0s thecoordinates from the centers of each of the receptive �elds. Thus, R0l and R0r are the coordinatesof the centers of the left and right eyes, Rl and Rr are the coordinates of points in both receptive�elds, and rl and rr are the coordinates of the same points with respect to the centers of the left andright receptive �eld centers. For a misalignment s between receptive �eld centers, R0l + s = R0r.Therefore, Rr � Rl = R0r � R0l + rr � rl = s+ rr � rl (see Figure 8).Using translational invariance, it is easy to see thatQll = E �d(rl)d(r0l)� = Q(r� r0)Qrr = E �d(rr)d(r0r)� = E �d(rl + s)d(r0l + s)� = Q(r� r0)Qlr = E �d(rl)d(r0r)� = E �d(rl)d(r0l + s)� = Q(r� r0 + s)Qrl = E �d(rr)d(r0l)� = E �d(rl + s)d(r0l)� = Q(r� r0 � s)where E denotes an average with respect to the environment and where, occasionally, for simplicity,we replace rl by r. Since Q(r� r0) = E (d(rl)d(r0l)) then Q(r� r0) = Q(r0 � r).We now introduce a two-eye parity operator P which inverts the coordinates as well as the twoeyes:6It is important to note that this symmetry applies only to single principal components, but not to combinationsof several PC's. In the model proposed recently by Erwin and Miller (1995) the �nal states are combinations ofseveral PC's. 11

P : 8><>: rl ) (�rl)rr ) (�rr)s) (�s) (3)It follows that under P; Rl �Rr = rr � rl + s) �rr + rl � s.The two-eye parity operator can also be written in matrix form in terms of the one eye parityoperator P , thus: P = � 0 PP 0 � : (4)The e�ect P has on the two-eye receptive �elds m is:P�ml(rl)mr(rr)� = �mr(�rl)ml(�rr)�Any correlation function that is invariant to a two-eye parity transformation P has eigen-functions mT (r) = �ml(rr); mr(rl)�, which are also eigen-functions of P. This imposes symmetryconstraints on the resulting receptive �elds that force them to be binocular.Any correlation function7 of the form:Q = 0B@ Q(r� r0) Q0(r� r0 + s)Q0(r� r0 � s) Q(r� r0) 1CA (5)is invariant to the two-eye parity transform P ( that is PQP = Q), as long as Q(x) = Q(�x) andQ0(x) = Q0(�x). Thus the eigen-functions of Q are also eigen-functions of P. The eigen-value is�1, since P 2 = 1. Therefore we deduce that:� ml(rl)mr(rr)� = ��mr(�rl)ml(�rr)� : (6)Thus m(r) = � ml( rl)�ml(�rr)� : (7)This means that the receptive �elds for the two eyes are inverted versions of each other up to asign. Therefore for this learning rule the receptive �elds are always perfectly binocular.7This class includes the type of correlation function described in equation 2, as well as the type postulated byMiller et al., (1989), in which s = 0 and Q0 = �Q. There, monocularity for non-oriented receptive �elds is attainedby choosing � < 0 and restricting weights to be positive.12

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Figure 5: BCM neurons with di�erent overlap values; O = 1; 0:6; 0:2;�0:2 from top to bottom. Theocular dominance histograms summarize the ocular dominance of 100 cells at each overlap value.The dependence of ocular dominance on visual overlap is evident.16

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Figure 7: Receptive �elds for partially overlapping inputs using the PCA rule. Receptive �eld foran overlap value of O = :6 (top left). Receptive �eld for a small overlap, O = :2 (middle left).Receptive �eld for no overlap , O = �:2 (bottom left). Receptive �eld for shift in the verticaldirection between the visual inputs when O = :5 (top right). Receptive �eld for shift at 36 deg,O = :5 (middle right). Receptive �eld for images that were rotated by 45 deg, O = :5(bottomright). In all cases the cell is binocular and horizontal. The symmetry property evident in thesereceptive �elds is analyzed in the appendix.18

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Figure 8: Coordinates for the two eyes. For a shift s between the two eyes, R0l+s = R0r. ThereforeRr �Rl = R0r �R0l + rr � rl = s+ rr � rl19