Research Note Cortical Magnification, Scale Invariance and Visual Ecology VEIJO VIRSU,* RIITTA HAR17 Received 14April 1995; in Revisedform 29 December 1995

VisionRes., Vol. 36, No. 18, pp.2971–2977,1996 Pergamon Couvright(0 1996Elsevier ScienceLtd. All riehts reserved 0042-6989(95)00344-4 ‘- Printed in breat Britain 0042-6989/96$15.00+ 0.00

Research Note Cortical Magnification, Scale Invariance and Visual Ecology VEIJO VIRSU,* RIITTA HAR17 Received 14April 1995; in revisedform 29 December 1995

The visual world of an organism can be idealized as a sphere. Locomotion towards the pole causes translation of retinal images that is proportional to the sine of eccentricity of each object. In order to estimate the human striate cortical magnification factor M, we assumed that the cortical translations, caused by retinal translations due to the locomotion, were independent of eccentricity. This estimate of M agrees with previous data on magnifications, visual thresholds and acuities across the visual field. It also results in scate invariance in which the resolution of objects anywhere in the visual field outside the fixated point is about the same for any viewing distance. Locomotion seems to be a possible determinant in the evolution of the visual system and the brain. Copyright 0 1996 Elsevier Science Ltd.

Cortical magnification Acuity Scale invariance Primates Ecology

INTRODUCTION itself must not disturb the processing of visual information sincevision is importantfor guidingthe movements. Several factors have determined the phylogeneticdevel- The locomotionof man and other primates can thus be a opmentof the brain and the visualsystem.For example,it phylogeneticdeterminantof the topographyof the visual has been desirable to maximize neuronal sampling cortex if its organizationdecreasesthe largevariationthat density because it limits visual acuity. Since the density the locomotion causes in the displacement, movement itself is limited by the sizes of the eyes, neurons, optic and shape of retinal images of objects as a function of nerve, and brain, a good evolutionary solution was that eccentricity.Our study indicatesthat observationson M, the samplingis dense and acuity high in the center of the scale invariance and resolution can be explained visual field, whereas the sampling could be gradually correlativelyif it is assumed that the wiring of the visual sparserin the periphery.The corticalmagnificationfactor system compensates for some effects of locomotion in M (Daniel & Whitteridge, 1961) may be seen as a the brain. solution to the sampling problem, but this evolutionary The scale invariance means that changes of viewing notion is not sufficientto explainwhy the valuesofM are distance do not alter the resolution of extrafoveal object quantitatively what they are or how the sampling is details significantlywhen the point of fixation does not spatially distributed throughout the visual field. There- change (Van Essen et al., 1992).The scale invariance is fore, we found it interesting to study whether the only approximate,like anythingin biology,because it is a empirically observed values of M and acuity would compromiseof many demands.It indicates,however,that correlate with the visual effects of locomotion. The changesof viewing distancedo not significantlyalter the qualitative requirements of evolution can be satisfied amount of informationthat the brain receives of a detail quantitatively in many different ways, but adding to be processed. Locomotion of an organism is the constraintscan lead to quantitativesolutions. primary cause of distance changes, and M can be one Varying lifestyles of different animal species have computational device that contributes to the scale affected the structure of their visual systems, and the invariance. crude features of topographic maps in the brain have The linear M indicates how many millimeters on the developed according to such phylogenetic constraints surface of the primary visual cortex correspond to 1 deg (Hughes, 1977; Whitteridge, 1973; Kaas, 1988). One of visual angle at different eccentricities. Its inverse, general constraint is that the movement of the animal M-l, is almost directly proportional to eccentricity beyond 1 deg (Van Essen et al., 1984). The scale ., invariance follows from this because M–l is approxi- *Towhom all correspondenceshouldbe addressed at the Department mately directly proportional to the inverse of visual of Psychology, P.O. Box 4 (Fabianinkatu 28), FIN-00014 acuity and resolution at different loci of the visual field University of Helsinki, Finland [Fax: 358-0-1912-3493;hail (Daniel & Whitteridge, 1961; Cowey & Rolls, 1974; [email protected]]. _/LowTemperature Laboratory, HeIsinki University of Technology, Virsu & Rovamo, 1979). Since the striate cortex can be FIN-02150Espoo, Finland. assumed to be a uniform structure (Hubel & Wiesel, 2971 2972 RESEARCHNOTE

deformed visual information in the periphery would deserve less weight. In fact, the visual systems of mammals emphasize the central parts of the visual field much more than the periphery (Whitteridge, 1973; Hughes, 1977):retinal samplingis more dense, receptive fields are smaller, and cortical representation is more extended for the central than the peripheral retina. Different developmental pressures and needs of maximum informationvary according to animal species and affect the specifics of the sampling, however. For example, animalsfor which the horizon and other lateral information is especially important have a horizontal visual streak in the retina (Whitteridge, 1973; Hughes, 1977),but the visual system of primates and many other mammals emphasizesdifferentmeridiansalmost equally (Daniel & Whitteridge, 1961). A spherewas selectedto representthe visualworld as a mathematical idealization because it was difficult to imaginehow more complex three-dimensionalflow-field FIGURE 1. The optical flow field of an organism in locomotion generating configurations would be better justified. A towardspole P. The deformationdiscussedby Gibson(1950)refers to sphere agrees with the forest ancestry of humans, it is the fact that only the front of an object is seen in P while only the side of the same object would be visible at E = 90 deg. suitablefor many other mammalshaving useful information above their head, and it fits the approximate meridional symmetry of M. Spherical flow fields have been commonly used for insect vision also (Wehner, 1974),a constant amount of cortical machinery seems to 1981;Horridge, 1992).Severalother configurationswere correspond to similar performance measures in many examined, however, but they did not change our results visual tasks across the whole visual field (Virsu et al., significantlyfor small eccentricities.For example, when 1987). For example, human thresholds for perceiving the visual objects approachedwere on a pIane perpendi- various aspects of movement are predicted across the cular to the direction of movement and the activations whole visual field quite accurately by the cortical caused by the objects moved the same constant cortical magnification factor (Virsu et al., 1982; Johnston & distances independentlyof eccentricity (E), our predic- Wright, 1983, 1985;Wright & Johnston,1985;Wright& tionswere not affected for eccentricitiesup to 20-30 deg. Gurney, 1992). Otherconfigurations,such as a cone, cylinderor a higher- degree(e.g. parabolic)surface,causedtoo steepgradients with E at large eccentricities to agree with empirical METHODS results. In addition to a parameter for flow, we had to The displacement and deformation of retinal images assume a sampling interval parameter based on the can be illustrated by the optical flow field of Fig. 1 anatomy of the foveal center in the visual system. The (Gibson, 1950;Wehner, 1981;Warren& Hannon, 1988; retinal projections were also assumed to be spherical Horridge, 1992;te Pas et al., 1996).For an animal, with because the relation between the environment and the forward facing eyes, moving linearly among stationary striate cortex, not the retina, was relevant here. objects, the pole in front of the animal (the focus of outflow)is stationary,whereas for each constantvelocity and object distance, the retinal images of visual objects RESULTS outside the pole move the faster the more peripherally The spherical flow field can be reduced to an arc of they are located in the visual field. It is assumed that the circle with radius r as in Fig. 2. Locomotion of observer objects are at such an intermediatedistancethat the self- O for a distanceAr in the directionof P causes the image movement causes image displacementon the retina, but of an object at eccentricityE (Oc E c 90 deg) to translate not so close that the displacementof the organism itself on the retina. Point A moves to point C and the visual becomes significant.For rotatorymovementsof the eyes, angleof the translationis M. Let BC be perpendicularto head and body, the pole is not accurately in the direction OA and distanceOB = r – M. The length of BC is then of heading (Regan & Beverley, 1982; van den Berg & Ar sin E and Brenner, 1994), but since the pole is crucial for the AE = arc tan[(Ar sin E)/(r – M)]. guidanceof locomotion,it is usuallykept in the center of the visual field. Since AR is small, r – AR % r. Then: In addition to decreasing the variation caused by M = arc tan[(Ar sin E)/r]. locomotion,it would be advantageousevolutionarilyalso to obtain maximum information about the target in the A reasonable approximationis that arc tan (x)x x (in pole and process it with a large machinery, whereas the radians). The error is negligible for small angles and RESEARCHNOTE 2973

P points (deg/mm) as a function of eccentricity across the whole visual field. The magnification factor (mm/deg) would then be the inverse M = (F+ k sinE)-l. (2) At smaII eccentricities, the additive constant F will violatethe geometricalinvarianceobtainedby pure flow- field calculations,but F is necessary because the size of neurons and tissues in the brain depends on biological factors other than visual ecology. Since term k sinE is much bigger than F for large eccentricities, M-l H k sin E at eccentricities larger than about 1 deg, and M–l z F when E approachesOdeg.

o ● M-1Levi OM

FIGURE 2. The displacement of image points for objects seen at 12 M ❑w M-J~ Grusser different eccentricities E when the pole P is viewed and the organism IIA has advanced a certain distance. The intermediate stage of retinal M-1 ~ M Van Essen images in the relation between environment and the cortex was 8 bypassed in the present calculations. ~~-’ Present A M-1 11 A 1 increases with the size of the angle, being 870 at 30 deg and 23Y0at 60 deg. The simplificationallowed: AE = Ar sin E/r. ‘7 w 0 10 20 30 40 50 60 If displacement Ar takes At seconds, the average EccentricityE (deg) velocity q of change of the visual angle is: ~ = ~/At= [(Ar sinE)/r]/At = (Ar sinE)/(rAt). The velocity of the observer is w = Ar/At, and then M q = (w/r) sin E. v Thus, apartfromE, the visual angularvelocitydepends on the velocityof the observerand the distanceof objects, A but the displacement of image points at any constant distance and velocity is approximately proportional to sin E. As far as the evolution of cortical topography is 0.1 w concerned,we can omit the effects of changesin velocity and distancebecause the organismshave experiencedall M-1 B II I kinds of velocitiesand distancesrelevant to their habitats 0.01J! I independentlyof eccentricity. I I I 0.1 1 10 Assuming that the retinal displacements caused by 0.01 locomotionresult in constantimage displacementsin the EccentricityE (deg) visual cortex for all visual-fieldlocations,k sinE would FIGURE 3. The inverse striate cortical magnification M-l (solid then reveal how many degreesof visual anglecorrespond symbols) and M (open symbols) as functions of eccentricity [linear to a unit length along the cortex, a relationshipcalled the plot in (A) and logarithmic plot in (B)]. Our estimates of inverse cortical magnificationfactor M–l (e.g. displace- M-l = 0.0685+ 4.32 sin E and its inverse, with constants derived ments in mm on the surface of the cortex; k as a scale from the averaged data of Levi et al. (1985) and Griisser (1995) at E = Oand E = 40 deg, are shown as solid lines without symbols. The constant).We also have to add a constant(F) to represent best-fitting empirical functions suggested by different authors are the spacing of neural elements for the central fovea: the indicated by circles for Levi et al. (1985); M–’ = 0.08(E + 0.8), by value of F is not determinedby the visual flowbut by the squaresfor Griisser(1995);M–l = 0.073+ 0.059E,andby trianglesfor limits of anatomy, and therefore it must be estimated VanEssenetal.(1984);&fz= 140(E+ 0.78)-2”2.The results of Gattass separately.Thus: et al. (1987) and Tootell et al. (1988) for monkeys have not been plotted: they are similar to other data in the figure.As the authorsstate M-] = F + k sin E (1) themselves,the preliminaryfunctionscalculatedfor humansby Sereno et al. (1995)yieldunrealisticallyhighfovealvalues that donot fit those describes the cortical movement of projected image of the others. 2974 RESEARCHNOTE

COMPARISONWITH OTHER ESTIMATIONS organism, agrees with these results: it assumes that the The accuracy of the approximationscan be evaluated mostcentralmagnificationincreasesalmostby a factor of by comparisons with empirical data for different 2 as comparedwith thevaluesderivedfrom the densityof eccentricities. Our estimates are plotted in Fig. 3 with retinal ganglion cells by Rovamo and Virsu (1979). The previously published experimental data for the striate retino-cortical mappings proposed by Schwartz (1980) and Johnston (1989) assume a stationary organism, and cortex. The values of constantsF = 0.0685 and k = 4.32 the mappingsconsideredin artificialintelligenceempha- in these graphs were chosen so as to agree with human size local properties (e.g. Mallet et al., 1990; Mallet et anatomical and psychophysical data (Cowey & Rolls, al., 1991).The importanceof global visual flow patterns 1974; Levi et al., 1985; Drasdo, 1991: Griisser, 1995). for local processing is pointed out by the preferred The valuesF and k dependon the size of the brain and the sensitivity of some cortical areas of animals for move- density of neurons, and they also vary somewhat ment directions away from the central visual field meridionally (Van Essen et al., 1984; Gattass et al., (Rauschecker et al., 1987; Albright, 1989). Specific 1987;Tootellet al., 1988).With the chosenF, the highest organizationsfor perceivingcentrifugalmovementsseem foveal magnification is 14.6mm/deg. The values of M to exist in the human visual system (Regan & Beverley, remain nearly constant for E <1 deg [Fig. 3(B)], 1979; Morrone et al., 1995). Different cortical neurons approximating the size of the rod-free area (Curcio & (e.g. Graziano et al., 1994; Duffy & Wurtz, 1995) and Sloan, 1992). areas (e.g. de Jong et al., 1994; Uusitalo et al., 1995) A variety of magnificationfunctionshave been used to respond differentially to flow stimuli. The shapes of describe particular experimental data. Usually the func- magnificationfunctionsfor differentvisual cortical areas tions have been radially symmetric and locally isotropic are quite similar (Sereno et al., 1995; Tootell etd., approximationslike the present one. Levi et al. (1985) 1995). used a linear estimateik-l = k(E +X) to describemany After developing the simple derivation above, we human psychophysical results. An essentially linear searched for its antecedents.Visual flow field similar to relationshipbetweenM–l andE has often been suggested ours was used by van de Grind (1990) as an explanation for other reasons also (Cowey & Rolls, 1974;Schwartz, of M based on the retinal ganglioncell density functions 1980; Drasdo, 1991; Strasburger et al., 1994: Grusser, derived by Drasdo (1977). The fit was not good because 1995), including direct measurements from the human M cannot be predicted by the ganglion cell densities and monkey striate cortex, density measures of cells (Azzopardi & Cowey, 1993) and because van de Grind along the retino-cortical pathway and estimations from (1990) did not apply the necessary additive constant. migraine phosphenes.At eccentricities >60 deg the sine Because of the poor fit he did not develop the theory functionbeginsto deviatefrom linearfunctions,but there further. Johnston (1986, 1989) has considered how the the limits of binocular vision are surpassed, and for topography of the striate cortex depends on the visual smallereccentricitiesthe sine functionresemblesa linear world idealizedas a conical surface. He (Johnston,1989, fimction so closely that the predictions of these two p. 1495) mentions the locomotion principle in one modelswould agree within experimentalnoise:it may be sentence, but did not develop the idea further because very difficult to distinguish between these two alter- he did not find a teleological justification for the natives experimentally with the present means. Our approximate radial symmetry found in primate topo- estimate is close to that of Levi et al. (1985) and thus it graphic maps. We consider the radial symmetry as a fitsthe same humanpsychophysicaldata that they used.It concomitantof the forest ancestry. also agreeswith the most recent direct observationsfrom monkeys, as illustrated in Fig. 3. Assuming a linear estimateofM–l, Rovamoand Virsu (1984)calculatedthe COMPARISONWITH ACUITY DATA topography of the three-dimensionalconvex surface of Figure 4 showsthe acuity data of Wertheim (1894)for the striate cortex and found a good agreement between the principal half-meridians.The solid line presents our their model and empirical macaque data. Their three- prediction of the data based on M in equation (2) with dimensional model of retinotopic mapping is a suffi- F = 0.0685 and k = 4.32. The function is obtained by ciently accurate approximation of the retinotopic map- multiplyingthe acuity data by the inverseM in order to ping that would follow from the present estimateof M-l. find out how many cycles each acuity is in terms of The biological background of M-l considered here can cortical distances, averaging the c/mm, and using the be understood as a theoretical justification of the linear average as a constantmultiplierof M in order to obtain a approximation. function to describe acuity in c/deg for different In order to obtain an estimatefor the humanM, a direct eccentricities. The cortical acuity suggested by these proportionalityof areal magnificationwith the densityof data, obtained in the orientation discriminationof small retinal ganglion cells is often assumed (Drasdo, 1977; square-wave gratings, is 3.45 c/mm. The value depends Rovamo & Virsu, 1979; Wassle et al., 1990), but the on the observer and method, and agrees with values 3.8 results of Azzopardi and Cowey (1993) show that the for movement direction discrimination and 5.3 for foveal cortical magnificationis larger than the densityof detection reported by Rovamo and Virsu (1979), and extrafoveal retinal ganglion cells would indicate. Our 6.2 for detection by Virsu and Rovamo (1979), all these estimate of M, derived from the ecology of the moving calculated from the cut-off frequencies of contrast RESEARCHNOTE 2975

substantially,but their average is near the acuity value ❑ Temporal A given by Wertheim. The upper data symbols show the e Nasal A A o Superior Nyquist frequencies derived from the determinationsof A Inferior A cone spacing in the human retina by Hirsch and Curcio —3.45M 0 (1989). f. ❑ ‘, A ~. The Nyquist frequencies represent the calculated ~. ~. A \ ~. 6 maximum spatial frequency for each eccentricity at which the retinal samplingmosaic allows vision without aliasing under optically unlimited conditions (the Ny- quist frequency is half the sampling frequency, and samplingartifacts such as aliasing occur when a discrete array samples stimuli that contain spatial frequencies higher than the Nyquist frequency). Detection with aliasing is anatomically possible at much higher spatial 1 10 frequencies (Williams, 1985) but the optical limitations Eccentricityf (deg) of the eye and noisekeep visual acuitybelow the Nyquist limit in the center of the visual field.In the periphery,the FIGURE4. Visual acuity at different half-meridiansoutside the fovea opticallimitationsdo not restrictthe resolution,but if the according to the measurements and tabulated values of Wertheim method is such as in the orientationdiscriminationof the (1894).Data points are plottedonlyfor principalmeridiansbecause all Wertheim (1894) study, in which correct responses do data for oblique meridians fall between the extremes shown in the not tolerate aliasing, the peripheral and foveal acuity figure and plotting them would clutter the figure. At eccentricities values are comparable because they both remain below <2.5deg the visual acuityvalues have been linearly interpolatedto the centralmost value 30 c/deg (acuity = 1.0) as reported by Wertheim. the Nyquist limit. The dashed line indicates the slope of –1 producinga complete scale The course of our function 3.45A4in the fovea is the invariance in which visual acuity is independentof viewing distance same as that of the Nyquist criterion, but the function because a change of distance is then completely counteracted by a remains below the criterion like the acuity data. The change of eccentricity. For example, if viewing distance is halved, value ofM forE = Odeg can vary individuallyby a factor visual acuity is halved because eccentricity doubles,but object size is then also doubled, and thus the amount of detail resolved remains of 2 or so (see Griisser,1995),but as far as a generalvalue constant. is suggested, a value near the 14.6mm/deg used in the figure seems reasonablefor the humans. The values 20- 25 mm/deg at E = Odeg estimated by Tolhurst and Ling sensitivityfunctions.Acuity valuesin detectionare better (1988) and Horton and Hoyt (1991) seem too high for than in orientationor directiondiscriminationbecausethe good predictions of acuity from the performance of detectionof correct time intervalsof grating presentation peripheralvision. can be done on fewer cues than the discrimination concerning the orientationof the grating or the direction of its movement. Although some meridional asymmetry does exist, our 70,, symmetric function 3.45M fits the data very well (about 97% of the total variance of the data is explained). A ❑ Nyquist criterion 60 o SIMreymouth complete scale invariance would require a straight line o S2 with slope –1 (see the dashedline) in the logarithmicplot A s3 of Fig. 4, but the slopeof the data (the exponentof a least- —3.45M L squarespower function)is –0.85. The scale invarianceis ( thus not perfect, as shown by the figure, but at eccentricities >1 deg the data come close to it. Due to the additive foveal constant,our function cannot express < the perfect scale invarianceeither,but the functionfitsthe data and deviates from the complete scale invariance in 20 the same way as the data. The scale invarianceis not needed in the central fovea because the images of foveal visual objects do not move significantlytowards the periphery when the objects get o 1 2 closer—the images of fixated objects are then only EccentricityE (deg) enlarged.The linear plot of Fig. 5 showswhat happensat small eccentricities. The solid line represents function FIGURE5. Visualresolutionat small eccentricities.The lower rowsof data symbolsrepresent three different subjects studied by Weymouth 3.45i14again, but it was not fitted to these results but to et al. (1928). The upper data symbols show the Nyquist frequencies the Wertheim data. The lower rows of data symbols derivedfromthe determinationsof conespacingin the humanretina by representthree differentsubjectsstudiedby Weymouthet Hirsch and Curcio (1989). The solid line represents function 3.45M al. (1928). The subjects differ from each other quite with its constant derived from the Wertheim data in Fig. 4.

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CONCLUSION tionalto sin E, and an inversetransformationcan be made An implementation of visual magnification in the to counteract the deformation in wide-angle imaging. brain, of course, cannot be in strict accordancewith any Separate movements of visual objects are easy to detect singleecologicalfeaturebecausethere are other selection in the system because self-generated linear movement pressures as well, but it is interesting that a model with produces a similar constant background throughout the two rational parameters agrees with empirical data so visual field. well: correlations between our estimates and empirical magnificationor acuitydata are typicallybetter than 0.95. REFERENCES The good fit seems to justify a sphere as the representa- Albright,T. (1989).Centrifugaldirectionalbias in the middletemporal tion of general featuresof the humanvisualworld. We do visual area (MT) of the macaque. 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