physiological optics in the hummingbird hawkmothsuperposition eye, compound eye, diffraction limit,...

$: Physiological optics in the hummingbird hawkmothsuperposition eye, compound eye, diffraction limit, optics, vision. Summary Introduction PHYSIOLOGICAL OPTICS IN THE HUMMINGBIRD HAWKMOTH:$
The hummingbird hawkmoth Macroglossum stellatarumisone of Europe’s most delightful insects. Active almostexclusively in bright daylight, this moth is not only a fast andaerobatic flier but also an excellent hoverer, sucking nectarfrom flowers in much the same way as its namesake thehummingbird. Its visual behaviour has been the subject ofstudy since the early part of this century (Knoll, 1922), morerecently with regard to visual stabilisation of flight (Kern,1994, 1998; Kern and Varju, 1994, 1998; Farina et al., 1995),distance perception (Pfaff and Varju, 1991; Farina et al., 1994),colour vision (Kelber, 1996; Kelber and Henique, 1999) andflower recognition (Kelber, 1997; Kelber and Pfaff, 1997).Whilst much information has been obtained regarding thevisual behaviour of M. stellatarum, very little is known aboutits eyes.

There are many paradoxes associated with the eyes of M.

stellatarum. First, despite being day-active, it has superpositioneyes with well-developed tapeta, a compound eye design moretypical of insects active at night. Second, the eyes areextremely aspherical and inhomogeneous, and thereby fail toadhere to the accepted optical construction necessary to forma superposition image on the retina. In the accepted model, firstdeveloped by Sigmund Exner over 100 years ago (Exner,1891), superposition eyes are spherical and maintain a constantfocal length and angular magnification throughout the eye.Most superposition eyes conform more or less to this classicalmodel.

In a future publication (E. Warrant, K. Bartsch and C.Günther, in preparation), we will show that the eyes of M.stellatarumnot only form crisp superposition images but alsopossess the sharpest angular sensitivities yet known for asuperposition eye. In the present study, we take a closer look

497The Journal of Experimental Biology 202, 497–511 (1999)Printed in Great Britain © The Company of Biologists Limited 1999JEB1713

The fast-flying day-active hawkmoth Macroglossumstellatarum (Lepidoptera: Sphingidae) has a remarkablerefracting superposition eye that departs radically from theclassical principles of Exnerian superposition optics.Unlike its classical counterparts, this superposition eye ishighly aspherical and contains extensive gradients ofresolution and sensitivity. While such features are wellknown in apposition eyes, they were thought to beimpossible in superposition eyes because of the imagingprinciple inherent in this design. We provide the firstaccount of a superposition eye where these gradients arenot only possible, but also produce superposition eyes ofunsurpassed quality. Using goniometry andophthalmoscopy, we find that superposition images formedin the eye are close to the diffraction limit. Moreover, thephotoreceptors of the superposition eyes of M. stellatarumare organised to form local acute zones, one of which isfrontal and slightly ventral, and another of which providesimproved resolution along the equator of the eye. Thisangular packing of rhabdoms bears no resemblance to the

angular packing of the overlying corneal facets. In fact, thiseye has many more rhabdoms than facets, with up to fourrhabdoms per facet in the frontal eye, a situation whichmeans that M. stellatarum does not possess ommatidia inthe accepted sense. The size of the facets and the area ofthe superposition aperture are both maximal at the frontalretinal acute zone. By having larger facets, a wideraperture and denser rhabdom packing, the frontal acutezone of M. stellatarumprovides the eye with its sharpestand brightest image and samples the image with the densestphotoreceptor matrix. It is this eye region that M.stellatarumuses to fixate flower entrances during hoveringand feeding. This radical departure from classicalExnerian principles has resulted in a superposition eyewhich has not only high sensitivity but also outstandingspatial resolution.

Key words: hummingbird hawkmoth, Macroglossum stellatarum,superposition eye, compound eye, diffraction limit, optics, vision.

Summary

Introduction

PHYSIOLOGICAL OPTICS IN THE HUMMINGBIRD HAWKMOTH: A COMPOUNDEYE WITHOUT OMMATIDIA

ERIC WARRANT1,2,*, KLAUS BARTSCH3 AND CLAUDIA GÜNTHER3,‡1Department of Zoology, University of Lund, Helgonavägen 3, S-22362 Lund, Sweden, 2Institute for Advanced Study,

Wallotstrasse 19, D-14193 Berlin, Germanyand 3Lehrstuhl für Biokybernetik, Universität Tübingen, Auf derMorgenstelle 28, D-72076 Tübingen, Germany

*e-mail: [email protected]‡Present address: Public Communication Networks Group, Siemens AG, Siemensdamm 50, D-13623 Berlin, Germany

Accepted 7 December 1998; published on WWW 3 February 1999

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at the optics of the eye in an attempt to understand how Exner’srules can be broken and still permit excellent superpositionvision. In attempting to unravel this paradox, we discoveredthat the eye had two remarkable properties. First, as in manyapposition eyes, there are gradients of resolution andsensitivity throughout the eye. Gradients in resolution werepreviously deemed impossible for superposition eyes. Second,there is not one corneal facet per rhabdom as in a conventionalcompound eye, but as many as four, the exact ratio dependingon the location in the eye. Because an ommatidium containsone rhabdom and one facet, this means that M. stellatarumdoes not have ommatidia in the functional sense, and theconcept of an interommatidial angle is meaningless. The eyesof M. stellatarumrepresent a remarkable optimisation of thesuperposition design for vision in bright light, and theirphysiological optics are described here for the first time.

An abstract of this work has been published previously(Bartsch and Warrant, 1994).

Materials and methodsAnimals

Macroglossum stellatarumL. (Lepidoptera: Sphingidae)was reared in a laboratory culture that was regularly restockedwith wild-caught moths from Southern Europe. Full details ofthe rearing and maintenance of hummingbird hawkmoths canbe found elsewhere (Farina et al., 1994).

Coordinates

All visual field coordinates used in this study comply to theconventions given in Fig. 1. Angles of latitude and longitudewere defined with the origin (0 °,0 °) located at the lateral (side)point of the eye (L). Longitudes anterior (A) to this point arepositive, those that are posterior (P) are negative. The sameconvention applies to latitude: latitudes dorsal (D) of the lateralpoint are positive, those that are ventral (V) are negative.

Goniometry

Corneal measurements of eye glow area, facet diameter andinter-facet angle were made using a large goniometer togetherwith a low-power microscope whose objective could be placedat any position on the surface of an imaginary hemispherecentred on the eye of a living hawkmoth. Attached to themicroscope was a camera. The hawkmoth was restrainedwithin an Eppendorf tube whose tip had been sliced off toallow the moth’s head to protrude. The head was lightly fixedto the tube with a small quantity of beeswax to prevent it frommoving. Excess scales and hairs were carefully plucked fromthe head to allow a full view of the eye. Chalk dust was thensprinkled lightly onto the eye to provide landmarks. A mountholding an oriented glass coverslip was then placed over theend of the objective. This coverslip, oriented at 45 ° to theobjective’s long axis, acted as a beam splitter which divertedorthodromic light from a small lamp, placed to the side of theobjective, down onto the eye. This light caused a bright eyeglow which could be seen through the microscope. The details

and operation of the goniometer are described by Dahmen(1991).

The experimental procedure was as follows. The goniometerwas used to move the microscope through steps of 10 °longitude. At each longitude, the microscope was movedthrough steps of 10 ° latitude, beginning as far ventrally aspossible and progressing dorsally until the microscope had nofurther travel. At each position, the eye glow wasphotographed, and the photographs were used to determine thesize, shape and facet content of the eye glow. Facet diameterwas also determined at each position. On the basis of thenumber of facets traversed by the eye glow during the 10 °transition from one photograph to the next, it was possible tomeasure the angle separating neighbouring facets at each eyeposition (using a method analogous to that used by Horridge,1978, and Dahmen, 1991, in apposition eyes).

Ophthalmoscopy

Retinal measurements of line-spread functions, point-spreadfunctions and the angular packing density of rhabdoms weremade using ophthalmoscopic techniques (for a fullexplanation, see Land, 1984a). These methods allow the retina,and any images formed on it, to be viewed through the eyes’own optics. A short description of the principal methods usedwill be given below. Specific details of some procedures – bestexplained in the context of the Results – are given at theappropriate places in the Results section.

For measurements of the packing density of rhabdoms, the

E. WARRANT, K. BARTSCH AND C. GÜNTHER

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Fig. 1. The coordinate system used in this study. The origin (0 °,0 °)was defined at the lateral point (L) of the eye, and angles of latitudeand longitude were defined in the manner shown. D, dorsal; V,ventral; A, anterior; P, posterior.

499Physiological optics in the hummingbird hawkmoth

ophthalmoscope was a converted Zeiss photoscope. Thiscontained a half-silvered mirror which diverted light from alaterally placed light path through the objective (Fig. 2). Thelight was produced by a xenon arc lamp and filtered by aninterference filter transmitting light in the wavelength range550–650 nm. Hawkmoths, restrained as described above, wereplaced on a small goniometer that could be used together withthe rotating microscope stage to position the moth at a chosenlatitude and longitude under the objective. A Bertrand lensinserted into the beam path (Fig. 2) allowed the retina to beviewed in angular space. Slits, pin-holes, letters and gratingsplaced at the position of the angular diaphragm could then beimaged on the retina. Images so produced could be intensifiedby an image intensifier, collected by a CCD camera, enhancedwith an electronic image-analysis system and printed on avideo printer. Removal of the Bertrand lens allowed thecorneal surface to be viewed.

For optical measurements of rhabdom acceptance functions,the ophthalmoscope was a bench-mounted arrangement ofSpindler and Hoyer optical components (for a full description,see Land, 1984a). Images were photographed with an OlympusOM-2 camera using Kodak TMAX400 black-and-white slidefilm and developed in TMAX developer. The resultingnegatives, which contained an intensity range of over 3logarithmic units (sufficient for quantitative micro-densitometry), were illuminated from below by a 0.075 mmwide circular pin-hole of light. Densitometry was performedby reading the voltage output of a photodiode placed above thenegative as the light source was moved beneath the negativein 0.05 mm steps. To convert voltage readings into intensitiesand to standardise readings between different films, a slidemade up of strips of known neutral density (strips made fromgelatin) was photographed on every film. This slide providedthe calibration curve for the film.

Electron microscopy

Whole eyes were placed for 2 h at 4 °C in standard fixative(2.5 % glutaraldehyde and 3 % paraformaldehyde in150 mmol l−1 sodium cacodylate buffer, pH 7.2). Following a

rinse in buffer, the eyes were added to 1 % OsO4 for 1 h.Dehydration was performed in an alcohol series, and the eyeswere embedded in Araldite. Ultrathin sections for electronmicroscopy were stained with lead citrate and uranyl acetate.

ResultsAn aspherical superposition eye

The refracting superposition eye of M. stellatarumis highlyaspherical (Fig. 3), and ommatidial anatomy and optics vary

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Fig. 2. The ophthalmoscopic apparatus. A converted Zeissphotoscope, containing a half-silvered mirror, diverted light (600 nm,∆λ=100 nm) from a xenon arc source to the eye of a hawkmothplaced on a goniometer beneath the objective. Placement of aBertrand lens in the beam path allowed the retina to be imaged. Slits,pin-holes, letters and gratings placed at the position of the angulardiaphragm could then be imaged on the retina. Images could beintensified, captured by a CCD camera and printed on a videoprinter. A regular camera could also be used in place of theintensifier and CCD camera.

Fig. 3. The eye of Macroglossum stellataruminhorizontal section. This longitudinal view, in the plane ofthe eye’s equator, very clearly shows the highly non-spherical nature of the eye. Because neither the retina (r)nor the cornea (c) is spherical or concentric, the clearzone of the eye (cz) varies in size throughout the eye. Thelamina (l) and medulla (m) of the optic lobe are alsovisible. a, anterior; p, posterior. Scale bar, 200µm.

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throughout the eye. The aspherical nature of the eye is clearlyseen in horizontal (equatorial) longitudinal sections (Fig. 3).Neither the corneal surface nor the retinal surface is concentricor spherical, and the size of the clear zone and the lengths ofthe rhabdoms vary throughout the eye. In this respect alone,the eyes depart radically from the classical Exnerian norm ofsphericity and homogeneity. As we shall see, this departurealso extends to other aspects of the eye. Despite this, angular-sensitivity functions with acceptance angles as small as 1.3 °can be recorded from the photoreceptors, indicating that imagequality on the retina is very good (the anatomy and physiologyof the eye will be expanded upon by E. Warrant, K. Bartschand C. Günther, in preparation).

Optical image quality

To assess image quality, we used the ophthalmoscopictechniques pioneered for insect eyes by Land (1984a). Thesemethods allow the retina, and any images formed on it, to beviewed through the eyes’ own optics. One way of assessing theoptical quality of the eye is to use the ophthalmoscope to lookat retinal structures, such as the array of photoreceptors. If thearray is clearly visible, the optical quality is good. Because theimage we see has passed only once through the optics of theeye (outwards from the retina), any measurements made fromthese images are referred to as ‘single-pass’. An alternativeway of assessing the optical quality is to place objects (suchas points, lines and gratings) outside the eye and then examinetheir images on the retina. Such measurements are ‘double-pass’ because light from an object is imaged twice by theoptical system: once on its way to the retina, and once againon its way back out. The image of relevance to the animal isthat formed on the inward journey to the retina: the single-passimage. Luckily, any double-pass images we capture in theophthalmoscope can be converted to single-pass images usingstandard procedures (which are very clearly explained byLand, 1984a).

As a qualitative indication of image quality in M.stellatarum, we projected letters and square-wave gratings

onto the retina using the ophthalmoscope (Figs 4, 5). Theretinal photoreceptor array of the lateral eye (0 °,0 °; see Fig. 1)was clearly visible (Fig. 4A), indicating that the eyes are wellfocused. It also indicates that the quality of the optical systemin this part of the eye is better than the retinal ‘grain’, i.e. thatthe angular separation of neighbouring rhabdoms (∆φ) isgreater than the angular half-width (in object space) of thereceptive field of each rhabdom (∆ρ). According to Land(1984a), the photoreceptor array will only be visible when ∆φexceeds 0.9∆ρ. In the lateral eye ∆φ≈2.0 ° and ∆ρ≈1.8 ° (seebelow), so this indeed seems to be the case. In this regard, M.stellatarum can be compared with two other diurnallepidopteran species (measurements made by Land, 1984a):the eyes are at least as well focused as those of the agaristidmoth Phalaenoides tristificaand somewhat better focusedthan those of the skipper butterfly Ocybadistes walkeri.Interestingly, in the frontal part of the retina of M. stellatarum,the packing of the rhabdoms becomes much denser and theophthalmoscope can only just resolve the photoreceptor array.It seems that in this part of the retina ∆ρ approaches or isslightly larger than ∆φ. This notion is supported byelectrophysiological measurements of ∆ρ(1.3–1.4 °: E.Warrant, K. Bartsch and C. Günther, in preparation) andoptical measurements of ∆φ(approximately 1.1 °; see below).

Double-pass images of objects placed outside the eye are


Fig. 4. Ophthalmoscopic images of the retina in Macroglossumstellatarum; lateral eye (0 °,0 °). (A) The clearly visible matrix ofrhabdoms. (B) A musical ‘natural’ sign (height 18.4 °) imaged on theretina. The original object was placed in the plane of theophthalmoscope’s angular diaphragm (see Fig. 2). Scale bar (forboth parts), 4 °.

Fig. 5. High-contrast square-wave gratings of various angularperiods imaged on the equatorial frontal retina (longitude 70 °). (A) 12 ° period (0.08 cycles degree−1); (B) 8 ° period(0.13 cycles degree−1); (C) 6 ° period (0.17 cycles degree−1); (D) 4 °period (0.25 cycles degree−1). All gratings are clearly resolvable onthe retina, except the 4 ° grating whose contrast has declinedmarkedly. Scale bar (for all parts), 5 °.


also clearly visible, with a musical ‘natural’ sign of 18.4 °height being easily seen (Fig. 4B). When high-contrast square-wave gratings are projected (Fig. 5), those of lowest spatialfrequency (0.08 cycles degree−1) are easily resolved (Fig. 5A),although the sharp edges of the grating object are lost in theimage (due to diffraction). Gratings are still very clear even at0.17 cycles degree−1 (Fig. 5C), but by 0.25 cycles degree−1 theyare difficult to discern (Fig. 5D).

Quantitative measurements of optical quality can be madeby optically measuring the acceptance functions of singlerhabdoms. This was performed by observing the image of a0.49 ° point source centred on a single rhabdom in the lateraleye (Fig. 6, inset) and measuring the distribution of light in thisimage using densitometry. Whilst almost all of the light in thephotographed image arises from the single central rhabdom,some light does arise from the neighbouring six rhabdoms.Because the light is emitted through the same aperture fromwhich it entered, the resulting light distribution should beidentical to the rhabdom’s acceptance function whose half-width is ∆ρ(the acceptance angle). Moreover, this distributionshould also be ‘single-pass’ because a single rhabdom acceptsand emits almost all of the incoming point source blur-circle.Note that the image shown in the inset of Fig. 6 is somewhatelliptical, a phenomenon that is often encountered. Acceptancefunctions were measured by making a densitometric scan alongthe long axis of the elliptical image. The scan appears roughlyGaussian (circular symbols in Fig. 6) and, in fact, can be

accurately fitted by a function R(θ) composed of the sum oftwo Gaussian functions, one having a higher amplitude andnarrower half-width, and the other having a lower amplitudeand broader half-width:

where θ is a variable (angle), A1, a1, A2 and a2 are constantsand A1+A2=1. The half-width hi of each Gaussian function isrelated to the constant ai (the ‘natural radius’) via the simplerelationship ai=0.6hi. The receptive field data shown in Fig. 6can be accurately fitted by equation 1 with the following best-fit constants: a1=0.89 °, A1=0.621, a2=1.65 ° and A2=0.379(correlation coefficient r2=0.98). The half-width of the totalfunction is 1.81 °. The data can almost be fitted by a singleGaussian function of the same half-width (thin inner profile inFig. 6), but the data points lie above the flanks of this profile.This is the ‘image flare’ we observed in the photographedimage (Fig. 6 inset). Whilst not extremely large in M.stellatarum, it does have the potential to reduce image contrastsignificantly and thus to make vision worse (Land, 1984a).Interestingly, exactly the same thing is seen inelectrophysiologically measured acceptance functions (E.Warrant, K. Bartsch and C. Günther, in preparation), implyingthat these flanks have a real effect on cellular receptive fields.

The acceptance functions of rhabdoms in the lateral eye thushave acceptance angles (∆ρ) of 1.81 °. Knowing this angle alsomakes it possible to estimate the half-width ∆ρl of the blurredpoint source image (or blur-circle) formed on the retina. Theacceptance function is the mathematical convolution of theblur-circle and the circular top-hat function representing thestop at the tip of the rhabdom. This stop is defined by thecylindrical sheath of tracheal tubes that surrounds eachrhabdom and internally reflects light from the blur-circle intothe rhabdom (Land, 1984a; Warrant and McIntyre, 1991; E.Warrant, K. Bartsch and C. Günther, in preparation). Using amethod described by Land (see Land, 1984a, Fig. 9), it ispossible to derive ∆ρl from ∆ρ and the angular width of thestop (∆ρr). In the lateral eye of M. stellatarum, ∆ρr≈1.73 °.With ∆ρ=1.81 °, this gives ∆ρl=1.27 °.

How does this value of ∆ρl compare with the value expectedif the blur-circle quality was limited only by the diffraction oflight at each corneal facet? Images formed by an optical systemcan never be sharper than this diffraction limit. In a diffraction-limited eye, ∆ρl=180λ/πD, where λ is the wavelength ofincoming light and D is the diameter of a single facet (Land,1981). In the lateral eye of M. stellatarum D=29µm (seeFig. 8). Taking λ=0.6µm (the median wavelength used duringthese experiments) gives ∆ρl=1.19 °. This is narrower than ourexperimental value of 1.27 ° by 0.08 °, implying that the lateraleye of M. stellatarum, whilst being close to the diffractionlimit, may suffer very slightly from image-degradingaberrations. Diffraction-limited superposition eyes have beenfound in the diurnal agaristid moth Phalaenoides tristifica, butin the diurnal skipper butterfly Ocybadistes walkerithe

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Fig. 6. The acceptance function of a single rhabdom in the lateral eye(0 °,0 °) of Macroglossum stellatarummeasured optically from theimage of a point source on the retina (inset). Point-source images,which were essentially single-pass (see text), were generated bycentring the image of a pin-hole of light (0.49 ° wide) on a singlerhabdom. Experimental data (circles) were obtained from severaldensitometric scans across the image shown in the inset. Atheoretical acceptance function (heavy continuous line) was thenderived by fitting these data to equation 1 (see text). This function, ofhalf-width 1.81 °, fits the data better than a single Gaussian functionof the same half-width (thin continuous line). The inset imagemeasures 4 °×4 °.

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superposition eyes are somewhat worse than the diffractionlimit (Land, 1984a). M. stellatarum lies somewhere inbetween, but closer to Phalaenoidesthan to Ocybadistes.

Corneal facet diameter

The corneal facets of M. stellatarummaintain neither aconstant diameter nor a perfectly hexagonal shape throughoutthe eye (Figs 7, 8). This is not uncommon in many appositioneyes, but for superposition eyes it is very unusual.

Facets are largest and most hexagonal in the anterior part ofthe eye, with diameters exceeding 30µm along the equator atlongitudes around 60 ° (Fig. 8B). For longitudes furtheranterior, the facets maintain this shape and maximum diameterventrally of the equator, notably in the region of the eye usedby M. stellatarumto fixate the entrance of a flower during

hovering (not shown in Fig. 8). In more posterior regions ofthe eye, the facets lose their perfect hexagonal shape andbecome compressed in the anterior–posterior direction(Figs 7A, 8B). In the dorso-ventral direction, facets are largestand most hexagonal at the equator, reaching diameters ofapproximately 29µm (Fig. 8A). Both above and below theequator, facets become smaller but remain fairly hexagonal.

Eye glow

The superposition eye of M. stellatarumpossesses areflective tapetum, which lies in the retina. Light rays that havenot been absorbed within the eye can be reflected by thetapetum and returned to a distant observer through the sameaperture of facets as they entered. If the eye is illuminated andviewed coaxially with the illumination (i.e. orthodromicillumination), a bright patch of blue-green light can be seen onthe eye surface (E. Warrant, K. Bartsch and C. Günther, inpreparation). This patch, called the eye glow, represents thepupil through which light reaches the retina: a larger eye glowrepresents a larger pupil and a greater light catch. In mostnocturnal superposition eyes, a bright eye glow is usuallyassociated with a dark-adapted and fully open pupil. Continuedillumination of such an eye normally causes screeningpigments within the eye to migrate and ‘close’ the pupil. Theeye glow then fades. Remarkably, in many diurnalsuperposition eyes including those of M. stellatarum, the pupilis always fully open, as if in the dark-adapted state, eventhough the animals themselves fly only in bright sunshine (E.Warrant, K. Bartsch and C. Günther, in preparation). The pupilcan be partially closed, but only using unnaturally bright light(Warrant and McIntyre, 1996; E. Warrant, K. Bartsch and C.Günther, in preparation). The significance of this is notunderstood at present.


Fig. 7. The facet matrix at two different equatorial locations:longitudes −15 ° (A) and +60 ° (B). The top, bottom, left and rightedges of each panel represent the dorsal, ventral, posterior andanterior directions respectively. Facets are larger and more regularlyhexagonal at the front of the eye (B). Scale bar (for both parts),30µm.

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In a classical Exnerian superposition eye, the size and shapeof the eye glow remain constant in different parts of the eye.Again, M. stellatarumbreaks this classical rule: both the sizeand shape of the eye glow vary significantly (Fig. 9) and in away that shows parallels to the facet variations. The eye glowbecomes largest and roundest in the anterior part of the eye,with regard to both its area and the number of facets whichconstitute it. Its largest size (350 facets, 0.25 mm2) occursapproximately 10 ° below the equator at longitudes between70 ° and 80 °. This region of the eye is used by M. stellatarumfor fixating the entrance of the flower during hovering. Awayfrom this region, the eye glow becomes smaller, decreasing insize in smooth gradients (Fig. 9). The eye glow becomessmallest and somewhat elliptical towards the edges of the eye.Over large regions of the lateral eye, the eye glow maintainsapproximately the same circular shape and an area of between0.14 and 0.16 mm2. Interestingly, the eye glow area increasesslightly towards the back of the eye.

Rhabdom packing in the retina

Using the ophthalmoscope, it is possible to scan a narrowslit across the retina and accumulate a bright image of rhabdomrows using an image intensifier. The angle between these rowsin each orientation was measured from video prints calibratedfor angle. Such images show that the angular separation of

rhabdom rows varies considerably in different parts of the eye.For instance, at different locations along the equator, there arerhabdom rows oriented dorso-ventrally, and the angularseparation between these rows ∆φ decreases towards the frontof the eye (Fig. 10). At an equatorial longitude of 20 °, the rowsare 1.73 ° apart (Fig. 10A), but at a longitude of 70 ° they areonly 1.17 ° apart (Fig. 10C). Similar reductions in the angularseparations of rhabdom rows in the other two row directionsare also seen towards the front of the eye (see Fig. 11).

To quantify these changes, measurements of row separationswere made at 10 ° intervals along the posterior–anterior equator(latitude 0 ° line) and along different dorso-ventral meridians,in two moths (Fig. 11). At each angular position, the animalwas rotated under the ophthalmoscope in order to pick out rowsin each of the three rhabdom row orientations. We thenmeasured the separation of rows in the three orientations (inset,Fig. 11A). Large variations in row separation were found inboth moths. Along meridians running dorso-ventrally(longitude lines 15 ° in Fig. 11A and 0 ° in Fig. 11B), theseparation of dorso-ventral rows (squares in Fig. 11) remainsroughly constant. In contrast, the separation of rows in theother two orientations (triangles and circles in Fig. 11) reachesa minimum at the equator and increases in both the dorsal andventral directions. For instance, along the 0 ° longitude line(Fig. 11B), the dorso-ventral rows remain separated by

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Fig. 10. Images of dorso-ventralrhabdom rows at three differentequatorial locations: longitudes 20 °(A), 45 ° (B) and 70 ° (C). The imageswere made by scanning a narrow slit oflight across the retina parallel to thedesired rhabdom row orientation.During scanning, the image wasaccumulated by the image intensifierand analysis apparatus (see Fig. 2).The angular separation of rows (∆φ)decreases in a smooth gradient towards the front of the eye. ∆φ is 1.73 ° in A, 1.44 ° in B and 1.17 ° in C. Scale bar (for all parts), 2 °.

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approximately 1.9 °, whereas rows in the other two orientationsreach a minimum separation of 1.3–1.4 ° at the equator,increasing to well over 2 ° dorsally. The separations of rows inthese latter two orientations appears to be greater and to occurmore rapidly in the dorsal direction than in the ventraldirection. Measurements along the equator (Fig. 11C,D) revealthat, even though the dorso-ventral rows are usually separatedby a much larger angle than rows in the other two orientations,the separation of all rows decreases towards the front of theeye. Frontally, the separations of rows in all three orientationsbecome very similar (approximately 1.1–1.2 °), indicating thatthe angular packing of rhabdoms becomes regularly hexagonalin this part of the eye.

The angular packing of rhabdoms in the eye indicates thepresence of retinal ‘acute zones’, a feature that is not reflectedin the packing of facets in the overlying cornea (see below).Acute zones are regions of the eye where spatial resolution isimproved (and also quite often sensitivity) and are a well-known adaptation in apposition eyes (Land, 1989). In M.stellatarum, there is an acute zone along the equator of the eye

because the rhabdoms are mostly densely packed here(Fig. 11A,B). This equatorial acute zone becomes even moreintense anteriorly, the region of the eye possessing the densestpacking of rhabdoms.

When we use an ophthalmoscope to look at the retina, wedo not see rhabdoms in their true physical arrangement (i.e.with their physical separations in micrometres). Instead, we seethe rhabdoms projected in angular space, and their angularpacking may bear little resemblance to their physical packing.The angular separation of rhabdoms ∆φ (degrees) is related totheir physical separation d (µm) via the local focal length f(µm): ∆φ=180d/πf. If changes in ∆φ from one part of the retinato the other are accompanied by appropriate changes in f, dcould remain constant. If this were so, then one would notexpect to see significant differences in the physical spacing ofrhabdoms within the eye. Conversely, changes in ∆φcould alsobe obtained by changes in d at a fixed f, in which case thephysical spacing of rhabdoms would vary significantly withinthe eye. To determine whether the changes in angular packingwe have observed can be explained by changes in f or d or


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Fig. 11. The angles separatingrhabdom rows in twoMacroglossum stellatarumalong two different dorso-ventral meridians (longitudes15 ° in A and 0 ° in B) andalong the anterior–posteriorequator (latitude 0 °; C,D). Therhabdom matrix contains rowsin three different orientations,and the angles separating rowsin each of these orientationsare shown by squares, circlesand triangles respectively(inset in A). The angularpacking of rhabdoms is notconstant within the eye butinstead forms local acutezones, one at the front of theeye and another along theequator, the equatorial onebecoming weaker posteriorly.D, dorsal; V, ventral; A,anterior; P, posterior.


both, we made electron microscope sections from twoequatorial locations within the same eye: laterally (at longitude0 °) and frontally (at longitude 70 °). These sections show thatthe physical packing of rhabdoms is indeed denser frontallythan laterally (Fig. 12 and see Fig. 14E,F), but not denseenough to explain the observed differences in angular packing.This means that simultaneous changes in both d and f are

responsible for the changes in angular packing we haveobserved.

Estimations of focal length

Knowledge of the angular (∆φ) and physical (d)separations of rhabdom rows (Fig. 11 and see Fig. 14E,F)allows us to calculate the focal length (f) at the lateral andfrontal equator of the eye. At both locations in the eye, thecalculated focal length is different for each of the threedifferent rhabdom row orientations. In the lateral eye,f=319µm for the dorso-ventral orientation and 350µm and380µm for the other two orientations. An ‘average’ focallength for these three values is 350µm. In the frontal eye, thethree focal lengths are 387µm, 424µm and 417µm, with anaverage of 409µm. Thus, not only do the focal lengths differin different parts of the eye (being longer towards the front),they also differ between different rhabdom orientations at anysingle location in the eye.

In all optical systems, including eyes, the focal length isdefined as the distance between the system’s optical ‘nodalpoint’ and its focal plane. In a classical superposition eye,there is a single nodal point located at the eye’s centre ofcurvature, and the focal plane is located at the distal tips ofthe rhabdoms. Because classical superposition eyes are alsospherical, the distal tips of the rhabdoms lie on a sphericalsurface whose centre of curvature is the nodal point. Thismeans that the focal length is the same in all parts of the eye.The fact that the focal length of the eye ofM. stellatarumisso variable suggests that the position of the nodal point mayvary with retinal location. The fact that the focal lengthdiffers with orientation even at single retinal locationsconfirms this suggestion.

Facet packing in the cornea

The angles between neighbouring facet rows in the corneawere determined from angular displacements of the eye glowmeasured using a goniometer. Angles between facet rows weremeasured at 10 ° intervals along the posterior–anterior equator(latitude 0 ° line) and along the dorso-ventral meridian(longitude 0 ° line). At each angular position, we measured theangular divergence of rows in each of the three facet roworientations (inset, Fig. 13A). As with rhabdom rows, one ofthe facet rows has a dorso-ventral orientation (squares,Fig. 13).

For a large range of latitudes above and below the equator,the angles between facet rows in all three orientations havevalues of approximately 1.8–2.0 ° (Fig. 13A). For latitudesmore dorsal than 60 ° and more ventral than −50 °, the anglebetween rows in all orientations increases markedly, withangles between some rows reaching nearly 3.0 °. Along theequator (Fig. 13B), the angles between facet rows remainsimilar for all three orientations, but decline steadily towardsthe front of the eye until a longitude of approximately 60 °. Forlongitudes more anterior than 60 °, the angle separating dorso-ventral rows suddenly increases dramatically, reaching a valueof nearly 3 ° at 90 ° longitude.

Fig. 12. Transverse electron microscope sections through the retinaat two different equatorial locations, approximately at longitudes 70 °(A) and 0 ° (B), in the same moth. The physical rhabdom packing isdenser frontally (A) than laterally (B). Sections show the rings oftracheal tubes which surround each rhabdom and act as a reflectivesheath for trapping light (Land, 1984a; Warrant and McIntyre, 1991).The section in B is slightly more distal than the section in A and is,unfortunately, not exactly perpendicular to the rhabdom axis (whichslightly exaggerates the more dilute packing present here).Calculations based on this section (see text) have been corrected forthis. Scale bar (for both parts), 5µm.

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The number of rhabdoms per facet

To some degree the angles between the facet rows do reflectthe angles between the rhabdom rows lying directly beneath.The facet row angles, like the rhabdom row angles, are smallnear the equator (Fig. 13A) and generally decline along theequator towards the front of the eye (Fig. 13B). However, aninspection of Figs 11 and 13 quickly reveals that, even thoughthis loose relationship does exist, the angles between facets aremuch larger in all parts of the eye than the angle betweenrhabdoms. This can be readily seen in a comparison of theangular packing matrices of rhabdoms and facets at twoequatorial locations on the eye (Fig. 14): laterally (at longitude0 °) and frontally (at longitude 70 °).

The angular packing of facets at the frontal equator(Fig. 14B) is slightly denser than at the lateral equator(Fig. 14A), as we have already seen in Fig. 13. The same trendis also seen in the angular packing of rhabdoms, but the densityis very much greater (and the packing more regularlyhexagonal) at the frontal equator (Fig. 14D) than at the lateralequator (Fig. 14C). Most important, though, is the fact that therhabdoms are much more densely packed in both parts of theeye than are the facets that lie above them. If this is the casein all parts of the eye, then this means that there must be manymore rhabdoms than facets. Is this actually the case? To answerthis question, we used the data shown in Figs 11 and 13 tocalculate the local densities of rhabdoms and facets, and


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Fig. 14. The physical and angular packing of rhabdoms and facets inthe eye at two different equatorial locations, laterally (longitude 0 °;A,C,E) and frontally (longitude 70 °; B,D,F). The small circlesrepresent the centres of rhabdoms and facets. Both the angular (C,D)and physical (E,F) packing of rhabdoms is denser at the front of theeye than at the side, but the relative change is greater for angularpacking. The angular packing of facets (A,B) remains rather similar,but much less dense, than the packing of the underlying rhabdoms.


thereby the number of rhabdoms per facet, at 10 ° intervalsalong the posterior–anterior equator (latitude 0 ° line) andalong the dorso-ventral meridian (longitude 0 ° line).

At all positions we tested along the equator, there are at leasttwo rhabdoms per facet, increasing to possibly as many as fourrhabdoms per facet in the extreme anterior part of the eye(Fig. 15B). Along the dorso-ventral meridian, the number ofrhabdoms per facet is greatest near the equator (two rhabdomsper facet) and falls systematically both dorsally and ventrally(Fig. 15A). In the extreme dorsal part of the eye (and possiblyalso in the extreme ventral part), the number of rhabdoms perfacet falls to one. As far as we can tell, these are the only placesin the eye where the number of rhabdoms and facets becomeequal. Spot checks in several other parts of the eye alwaysrevealed at least one rhabdom per facet, with most checksrevealing much higher ratios. The only conclusion that one candraw from these results is that the eyes of M. stellatarumpossess many more rhabdoms than facets, a most remarkablesituation. In a classical superposition eye, and indeed in allcompound eyes, there should be one rhabdom per facet in allparts of the eye. After all, an ommatidium is supposed tocontain one facet lens and one rhabdom. The fact that this isclearly not the case in M. stellatarumis yet another indicationof how significantly this eye departs from classical Exnerianprinciples.

DiscussionDeviations from classical superposition design

The refracting superposition eye of the hummingbirdhawkmoth is, in many ways, the most remarkablesuperposition eye ever described. Its deviation from theprinciples of classical Exnerian superposition optics is soprofound that at first glance it is hard to imagine how the eyecan form a decent image at all. Yet it does, and with arguablythe sharpest receptive fields yet recorded from a superpositioneye, with acceptance angles as small as 1.3 ° in the frontal eye(E. Warrant, K. Bartsch and C. Günther, in preparation). There

is no question that the departure of M. stellatarum fromclassical superposition optics has given it an eye of outstandingquality.

The eye is very non-spherical and has a retina withrhabdoms whose packing density varies throughout the eye.The size of the facets, the area of the superposition apertureand the focal length also vary markedly. All these propertiesare at extreme odds with the classical model of Exner (1891).In this model, superposition eyes are spherical and maintain aconstant focal length and angular magnification throughout theeye (i.e. the eye has a single nodal point). One would alsoexpect an Exnerian superposition eye to have rhabdoms ofequal length and separation and to have facets of equal size.

The variable focal length is of particular interest, because itadds an extra degree of freedom in the design of the eye. Asingle glass lens which is not circularly symmetrical, butelongated, will often be astigmatic: the focal length in oneorientation (say X) will be different from that in theperpendicular orientation (say Y). Because the nodal point in aglass lens is the same for all orientations, the image planes inthe X and Yorientations will not coincide and the image qualitywill be poor. Astigmatism is not a problem for M. stellatarum.Because the nodal point can differ for different orientations ata single retinal location, differences in focal length that occurat that location need not mean that the image planes lackcoincidence. On the contrary, the image planes coincideexactly on the retinal surface. Moreover, differences in focallength mean that the image can be differently magnified indifferent orientations (magnification is proportional to 1/f).This extra degree of design freedom would permitimprovements in local spatial resolution simply by having amagnified image (in one or more orientations). To takeadvantage of this magnification, the underlying rhabdoms mayeven be packed more densely.

Variations in the size of the superposition aperture have alsobeen noted in another diurnal moth, the agaristid Phalaenoidestristifica (Horridge et al., 1977). In this moth, the aperturegradually decreases from a maximum width of 15 facets in the

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centre of the eye to much smaller values near the edge. In thedorsal part of the eye, the aperture diameter falls toapproximately four facets. The eye of P. tristificais much morespherical than that of M. stellatarum, and it is not knownwhether it possesses retinal acute zones.

The superposition eyes of M. stellatarumapparently departfrom Exnerian rules to achieve a single major benefit: theproduction of local acute zones. The frontal but slightly ventralacute zone not only has a higher spatial resolution but alsocollects more light from each point in space (i.e. has largersuperposition apertures). In this region of the eye, therhabdoms have a rhabdom separation (∆φ) of just 1.1–1.2 °(Fig. 11C,D) compared with more than 2 ° in some other partsof the eye, which represents an approximately fourfoldincrease in packing density. In addition to this frontal acutezone, there is also an equatorial acute zone (Fig. 11A,B), a kindof horizontal ‘visual streak’ to borrow the vertebrate term.

In an apposition eye, increases in ∆φ are directly visiblefrom the appearance of the pupil and the angular packing offacets in the cornea (Stavenga, 1979; Land, 1989). In M.stellatarum, however, changes in ∆φoccur only within theretina. They are not measurable from either the pupil or thepacking of facets. According to classical Exnerian principles,acute zones should be impossible in a superposition eye. Toquote Land (1989): ‘the option of inserting small higherresolution regions (in superposition eyes) does not seem to beavailable, because the image-forming system will not work ifthere are local variations. There are superposition eyes withregions of different resolution – in euphausiid crustaceans forexample (Land et al., 1979) – but here there are really two eyesjoined together rather than a single one with an opticalgradient.’ The eyes of M. stellatarumare the first superpositioneyes known to possess pronounced resolution gradients.

Why does the hummingbird hawkmoth have acute zones?

The rhabdom packing that leads to acute zones in thesuperposition eye of M. stellatarumis actually remarkablysimilar to the ommatidial packing found in the apposition eyesof several other insects. Just as with the rhabdoms of M.stellatarum, the ommatidia of many bees, butterflies, waspsand locusts have their densest packing at the front of the eye,with a smooth decrease in density occurring posteriorly alongthe equator, especially between the dorso-ventral ommatidialrows (Baumgärtner, 1928; del Portillo, 1936; Autrum andWiedemann, 1962; Horridge, 1978; Land, 1989). And, just asin M. stellatarum, they also possess a horizontal visual streak.Remarkably, the same optical adaptations have evolved in twocompletely different eye designs via two completely differentmethods.

Equatorial gradients of spatial resolution are thought to bean adaptation for forward flight through a textured environment(Land, 1989). When an insect (or any animal) moves forward,it experiences the movement of its surroundings as it passesthem, a so-called ‘flow field’ of moving features (Gibson,1950; Wehner, 1981; Buchner, 1984). Features directly aheadappear to be almost stationary, while features to the side of this

forward ‘pole’ appear to move with a velocity that becomesmaximal when they are located at the side of the eye, 90 ° fromthe pole. If the photoreceptors have a fixed integration time ∆t(which is not necessarily the case), the motion of flow-fieldimages from front to back across the eye will cause blurring.An object moving past the side of the eye (with velocity v) willappear as a horizontal spatial smear whose angular size (indegrees) will be approximately v∆t. This effectively widens thelocal optical acceptance angle (∆ρ) to a new value of√[∆ρ2+(v∆t)2] (Srinivasan and Bernard, 1975; Snyder, 1977).The extent of this widening is worse at the side of the eye(higher v) than at the front (lower v). To maintain an optimumsampling ratio of ∆ρ/∆φ (Snyder, 1977, 1979), the equatorialincrease in ∆ρposteriorly should be matched by an increase in∆φ, as indeed seems to be the case.

A very fast flying insect such as M. stellatarumcan easilyexperience a velocity of 100 ° s−1 at the side of the eye. In thelateral eye, ∆ρ=1.81 ° (Fig. 6), and assuming a value of 15 msfor ∆t, we arrive at a new widened acceptance angle of 2.35 °,an increase of approximately 0.5 °. The separation (∆φ) ofdorso-ventral rhabdom rows in the lateral eye is approximately1.9 ° (Fig. 11). If we assume that the ratio of these ∆ρ and ∆φvalues is the optimum for sampling (in this case2.35 °/1.9 °≈1.2), then we can use this ratio to predict ∆ρfrontally (say at approximately 70 °) where widening due tomotion-blurring would be minimal. We know the value of ∆φis approximately 1.2 ° here (Fig. 11) and, using a samplingratio of 1.2, leads to a ∆ρ of 1.2×1.2 °≈1.4 °. This is exactly thesame acceptance angle as we have measuredelectrophysiologically in the same part of the eye (E. Warrant,K. Bartsch and C. Günther, in preparation).

In the case of M. stellatarum, the denser packing ofrhabdoms at the front of the eye is not just a response to lowerflow-field velocities. This acute zone also serves quite anotherpurpose: to fixate the entrances of flowers. When M.stellatarumhovers and sucks nectar, it uses the frontal, andslightly ventral, part of the eye to maintain its distance to theflower (Knoll, 1922). If the wind blows and the flower bobsaround, it is amazing just how rapidly and effortlessly M.stellatarumcan follow the movements. To a human observer,the moth seems almost ‘glued’ to the flower entrance by itsoutstretched proboscis. This stunning ability must in part bedue to the extensive binocular overlap (E. Warrant, K. Bartschand C. Günther, in preparation) and excellent resolution foundin the acute zone viewing the flower entrance. The ability ofM. stellatarumto follow flower-like movements and to hold itsdistance has recently been the subject of excellent behaviouralstudies (Pfaff and Varju, 1991; Farina et al., 1994; Kern, 1998;Kern and Varju, 1998). Moreover, movement-sensitive cellswith frontal receptive fields have recently been identified in theoptic lobe of M. stellatarumthat would be perfect inputs tobinocular circuits designed to detect the looming of a flowerblown towards the moth (Wicklein, 1994; O’Carroll et al.,1997).

The horizontal ‘visual streak’ possessed by M. stellatarum,and by other fast-flying insects which live in open terrain, is



possibly an adaptation for horizon detection. The horizon is amajor visual cue in natural scenes, a large contrasting borderbetween a blue-ultraviolet sky and a green-brown landscape.In many insects, the ocelli play a major role in detecting thishorizon (Wilson, 1978; Stange, 1981), but in M. stellatarum,which has no ocelli, the eyes are solely responsible. Asurprising number of visually interesting things are located atthe horizon, familiar landmarks for instance. Because of this,visual streaks have evolved which sample the horizon muchmore densely than other parts of the vertical visual field. Visualstreaks are common in vertebrates (for a review, see Hughes,1977) and in the compound eyes of arthropods (for reviews,see Wehner, 1987; Land, 1989). They have never before beenreported in a superposition eye. Hovering insects such as M.stellatarummay find the horizon very useful for preventingthemselves from ‘rolling’ around their long axes duringhovering. Interestingly, cells have recently been described inthe ventral nerve cord of M. stellatarumthat respond stronglyto upward and downward motion of laterally placed horizontaledges (Kern, 1994, 1998; Kern and Varju, 1998). Moreover,the response of these putative roll detectors is strongest whena horizontal edge moves over the equator of the eye, a resultthat fits very well with the presence of an equatorial visualstreak.

A compound eye without ommatidia

The superposition eye of the hummingbird hawkmoth is acompound eye without functional ommatidia. An ommatidiumcontains one set of rhabdomeres, which together constitute arhabdom, and one pair of lenses, the corneal facet lens and thecrystalline cone. In M. stellatarum, there are clearly manymore rhabdoms than facets (Figs 14, 15), and therefore theommatidial ratio of one rhabdom per facet has beenabandoned. Simply put, there are no ommatidia in thefunctional sense, and there is no interommatidial angle.Instead, there is a matrix of photoreceptors and an overlyingimaging system made out of a completely independent matrixof lens elements. In a sense, this is a compound eye that hasdeveloped into a kind of simple eye, like that in a spider: aretina with a fovea receiving a bright sharp image from anindependent lens.

What does M. stellatarumgain by lacking true ommatidia?First, because the rhabdoms are not constrained by anommatidial matrix, they are free to aggregate and to formacute zones in a way impossible for a normal superpositioneye. Second, because the facets are not constrained by therhabdom matrix, they are free to vary optically andmorphologically, and this they do spectacularly. Their opticalvariation has allowed them to develop a gradient ofsuperposition apertures, the largest of which are centredexactly over the frontal acute zone in the retina, the regionresponsible for flower fixation (Fig. 9). Not only this, thefacets here are also the largest in the eye (Fig. 8), and largerfacets are less affected by the image-degrading effects ofdiffraction. The flower-fixating part of the eye produces thebrightest, sharpest image on the acutest part of the retina. This

region has the highest sensitivity and the greatest resolutionof the entire eye, and entirely due to the eye’s radicaldeparture from the classical superposition design.

Superposition eyes without ommatidia, whilst rare, arenot unheard of. Cases are known from the dorsal eyes ofmale mayflies (Zimmer, 1898; Wolburg-Buchholz, 1976; P.Brännström and D.-E. Nilsson, in preparation), the dorsal acutezones of euphausiid shrimps (Chun, 1896; Land et al., 1979)and the larval eye of the euphausiid Thysanopoda tricuspidata,which has 90 rhabdoms but only seven facets (Land, 1981,1984b). The most extreme example is the mysid shrimpDioptromysis paucispinosa, which has a superposition eye thatin all respects is quite classical apart from the presence of asingle enormous facet supplying light to its own private acutezone of 120 rhabdoms (Nilsson and Modlin, 1994). But, as wementioned above, if any changes in resolution are seen in theseeyes, it is because of the presence of two separate regions, eachwith uniform resolution, that have been joined together to forma single eye. The superposition eye of M. stellatarumis thefirst documented example of a superposition eye with truegradients. Interestingly, new data from the nocturnalhawkmoth Deilephila elphenoralso show weak gradients, butin an eye that looks quite spherical and has the same numberof rhabdoms as facets (P. Brännström and Y. Arroyo Yanguas,in preparation).

It is not yet understood how superposition eyes can developwith an unequal number of facets and rhabdoms. It would seemthat during the manufacture of ommatidia some must developwithout lenses.

How canM. stellatarumbreak all the rules and get away withit?

Unfortunately, the answer to this question still remains amystery. The fact that the focal length varies at different pointsin the eye, and even at the same point, presents a majorconceptual difficulty in understanding this eye. Quite clearly,the optical nodal point has no single location and probably hasnothing at all to do with the local curvature of the retina, as itdoes in a classical superposition eye. Both the retina and theoverlying cornea are highly aspherical and are not concentric,which means that the depth of the clear zone varies all over theeye. The depth of the clear zone indicates the distance betweenthe optics and the image plane at the distal tips of the rhabdoms(McIntyre and Caveney, 1985). The fact that it varies meansthere must be a compensatory change in the optics to maintaina crisp image on the rhabdom tips. This must somehow beachieved by systematically altering the optics and morphologyof the lenses, something that is difficult to conceive becausethe same crystalline cone can be used to focus light from twoentirely different incident angles to two entirely differentlocations on the retina. How can a single crystalline conemanage two different clear zone depths, compensating for thedifference by bending light by different amounts for the twodifferent locations? This would require that the angularmagnifications of individual cones are not constant (as inExnerian eyes), but vary with the angle of incidence of

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incoming light. Interestingly, recent work suggests that thiscould indeed be the case in the unusual dorsal superpositioneyes of krill and mayflies, which have highly curved cornealsurfaces and flat retinas (D.-E. Nilsson, P. Brännström and L.Gislén, in preparation). The same problem arises whenthinking about the variable superposition aperture. A singlecrystalline cone could be optically functional in two differentsuperposition apertures. Again it is difficult to understand howthis can work unless the optical magnifications of individualcones can vary with the angle of incident light. Land (1989)was not being naive by claiming that gradients are impossiblein superposition eyes. They really do seem impossible.Obviously, however, they are not only possible, they alsoproduce superposition eyes of unsurpassed quality. Exactlyhow is still unclear.

This study is dedicated to Professor Dezsö Varju on theoccasion of his retirement from the Chair of BiologicalCybernetics at the University of Tübingen. The authors areextremely grateful to Dan-Eric Nilsson for critically reading themanuscript and for many fruitful discussions. Both he, Hans-Jürgen Dahmen and Mike Land graciously allowed us to usetheir equipment and willingly assisted us with its operation.Dezsö Varju, Hans-Jürgen Dahmen, Mike Land, MartinaWicklein, Almut Kelber and Michael Pfaff were also a sourceof rich insight. We are extremely indebted to Rita Wallén andLina Hansen for expert histological work. This study wouldhave been impossible without the gratefully acknowledgedassistance of a Twinning Grant provided by the EuropeanScience Foundation. K.B. is grateful for support from theDeutsche Forschungsgemeinschaft (SFB 307). E.J.W. is deeplygrateful for the ongoing support of the Swedish Natural ScienceResearch Council and for the generous hospitality extended tohim by Professor Varju and the other members of his Tübingendepartment during many memorable visits. This paper wascompleted during a Fellowship at the Institute of AdvancedStudy in Berlin, for whose support and marvellous workingenvironment E.J.W. is particularly grateful.

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