Proc. Natl. Acad. Sci. USAVol. 80, pp. 4554-4558, July 1983Psychology

Attenuation of Mach bands by adjacent stimuli(lateral inhibition/spatial contrast/retina/neural networks/vision)

FLOYD RATLIFF, NORMAN MILKMAN, AND NANCY RENNERT

The Rockefeller University, 1230 York Avenue, New York, New York 10021

Contributed by Floyd Ratliff, April 20, 1983

ABSTRACT Pronounced bright and dark bands are seen atthe bright and dark edges of half-shadows and similar distribu-tions of illumination. These are the so-called Mach bands. A pairof vertical Mach bands was generated with a ramp pattern in thecentral strip of a horizontal tripartite oscilloscope display. Thispattern consisted of two uniform fields (one of low luminance, oneof high luminance) joined by a gradient of uniform slope. The up-per and lower strips were uniform throughout. A coupled pair ofpointers could be displayed in these two strips and adjusted by theobserver to match the apparent location and width of either of theMach bands in the central strip. Insertion of a vertical bar in thecentral strip nearby and on either side of the ramp attenuates thecorresponding Mach band. The closer the bar is to the Mach band,the stronger the attenuation. The attenuation is nearly indepen-dent of the sign of the contrast of the bar, but it does depend uponthe magnitude and sharpness of the contrast. Also, the attenuationis independent of the width of the bar; a narrow line is as effectiveas a broad bar of the same contrast. No net luminance change isrequired; a bipolar stimulus with equal parts above and below themean is as effective as a monopolar stimulus. These results pointto two competing physiological mechanisms with different spatialsensitivities-one that generates Mach bands and one that atten-uates them.

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Mach bands are the bright and dark stria that appear at the edgesof a penumbra or half-shadow. As was shown by Ernst Mach,who first reported these bands in 1865, the phenomenal max-ima and minima do not have direct physical counterparts-thatis, there are no corresponding maxima or minima in the phys-ical distribution of the light itself (Fig. 1A).Many factors influence the appearance of the Mach bands:

slope of the gradient, contrast, mean luminance, state of ad-aptation, color, and the like. Most of these have been inves-tigated in some detail-first, by Mach himself, as reported ina series of six papers published during the period 1865-1906.For translations of Mach's original papers and a review of sub-sequent work by others (up to 1965) see Ratliff (2). Later re-search on the Mach bands (up to 1972) was reviewed by Fior-entini (3) and that on colored Mach bands (up to 1978), by Pease(4).

In Mach's first report, over a century ago, he suggested thatthese phenomena result from reciprocal inhibitory interactionswithin the visual system. Such an interpretation has been widelyaccepted ever since that time, but the actual physiologicalmechanisms have not yet been fully elucidated. If the bandsare indeed caused by inhibitory interactions that extend overany significant distance, then it should be possible to alter theappearance of the bands by altering the spatial pattern of il-lumination adjacent to them. In the very last paragraph of hisfinal (1906) paper on this subject, Mach reported one prelim-inary experiment along these lines. First he projected a ramp

FIG. 1. Mach bands. (A) Schema of apparent brightness (curved line)of a ramp pattern (straight lines) of luminance (Z), as a function of dis-tance (X). See Fiorentini and Radici (1). (B) Schema of Mach's exper-iment (2) on attenuation of dark Mach band (curved line) by a mask(diagonal lines) occluding bright half of ramp pattern (dashed straightlines).

pattern of light on a screen. With a straight-edged mask, par-allel to and on the ramp side of the dark band, he then graduallyoccluded light from the projected pattern (Fig. 1B). The firsteffect on the dark Mach band was a slight weakening when thelight was occluded by the mask at a distance from the band equalto a visual angle of 16'. The dark band became very thin andindistinct as the mask approached an angle of 11' and disap-peared altogether when the mask reached an angle of 7'.

Mach interpreted his experiment as showing the passive ef-fect of diminishing, or removing altogether, the contrasting partof the pattern which (presumably) generates the bands. But adecrease or complete occlusion of light in the region adjacentto the Mach bands (as in Mach's original experiment) is not es-sential. Indeed, many years ago one of us (F.R.) noticed thatboth the dark and the bright bands in a photograph of a ramppattern may be attenuated by adjacent stimuli, which increasethe amount of light. Not only does a black card laid over thephotograph parallel to and nearby one of the bands produceattenuation similar to that described by Mach, so does a whitecard. Also, attenuation may be produced by a light thread orby a dark thread laid parallel to and nearby one of the Machbands. Evidently the attenuation of the bands in these dem-onstrations is an active process. The recent finding by Ratliffet al (5) that the Mach bands may be attenuated equally by theseand a wide variety of other different stimuli raises the question

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of what their common denominator, if any, may be. The ex-ploratory experiments reported here address this question.

APPARATUS AND METHODSThe ramp patterns and other stimuli were displayed on a Tek-tronix 608 monitor oscilloscope with P31 phosphor. The x axisof the display consisted of 256 raster lines and the y axis, 256points on each line. Frame rate was 270 Hz. The parameters,such as dimensions and positions of patterns, relative contrasts,slopes of gradients, etc., were all digitally controlled by mod-ulation of the z axis. A similar two-dimensional visual displayhas been described elsewhere (6). The display consisted of threehorizontal strips (Fig. 2). The central strip contained the basicramp pattern-two uniform fields (one of high luminance, oneof low luminance) joined by a gradient of uniform slope. Ad-ditional stimuli of any chosen profile, whose effects on the Machbands were to be studied, could be placed at any position in theramp pattern. The narrow horizontal strips at the top and bot-tom of the display provided uniform backgrounds for the com-parison stimulus. This stimulus consisted of a coupled pair ofpointers (vertical bars)-one superimposed on the top strip, theother on the bottom strip. By means of a keypad linked to theinstrument the observer could adjust the position and width ofthe coupled pointers to match the position and width of theMach bands. (The contrast of the pointers could also be ad-justed, but because width and contrast of the Mach bands arerelated, we chose to measure only location and width in theseinitial experiments.)

At a viewing distance of 1 m the ramp pattern used in all ofthe experiments reported in this paper subtended an angle of5.220 in height and 5.390 in width. The gradient between theuniform bright and dark fields subtended 0.42° of visual angle.The mean luminance of the pattern, measured with a Spectrabrightness spotmeter, was 13.5 footlamberts (1 footlambert =3.426 candelas/m2). With respect to the mean luminance, thecontrast of the bright field was +20%; the contrast of the dark

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FIG. 2. (Upper) Map ofx-y oscilloscope display of typical ramp pat-tern and adjacent bar stimulus (center horizontal strip) and of the ad-justable coupled pointers (upper and lower horizontal strips). (Lower)Relative luminance (Z)across ramp pattern and adjacent bar at levelr in center strip (solid line, r) and across upper and lower strips withcoupled pointers at levels p (dashed line, p).

field was -20%. Each of the strips at the top and the bottomof the pattern subtended 5.390 of visual angle in width and 0.650in height. Their luminance was set equal to the mean lumi-nance of the Mach pattern. The coupled pair of pointers wasset at a positive contrast (+20%) with respect to the mean whenstudying the bright Mach band and at a negative contrast (-20%)when studying the dark Mach band. The adjacent stimuli, whoseeffects were to be studied, could be set at any location on theramp pattern and at any contrast (with respect to the luminanceof the basic pattern at that location) in the range of ± 20%.

For the study of the effect of location of adjacent stimuli eachexperimental session consisted of 130 observations. Of these,120 were with an adjacent stimulus-10 observations each for12 different locations of the stimulus. The remaining 10 ob-servations of the ramp pattern were made without any adjacentstimuli present. The 130 patterns were presented in a quasi-random fashion.

For the study of the effect of the contrast of the adjacentstimuli, the stimulus was set at a fixed location on the ramppattern and its contrast (with respect to the pattern at that lo-cation) was varied in a quasi-random fashion from one obser-vation to the next. Each session consisted of 110 observations-10 each at ±2 +4, +6, . ±20% contrast and 10 at 0% con-trast (no adjacent stimulus).The observer's task, in all cases, was to adjust the position

and width of the coupled pair of pointers so that they matchedthe apparent width of one or the other of the two Mach bands.The patterns were not presented for a fixed interval. They re-mained on for whatever time the observer required to adjustthe pointers-usually only a few seconds. When satisfied withthe adjustment of the pointers, the observer pressed a key thatsignaled the end of the observation. Information about the ramppattern, any adjacent stimuli, and the width and location of thepointers was automatically stored in the computer.The next pattern from the series was then displayed. With

the presentation of each new pattern the pointers were auto-matically relocated and changed in width in a quasi-randommanner and over a considerable range so that their initial po-sition would not bias the setting by the observer. It is knownthat paired stimulus patterns in the outer regions of a tripartitefield can, when moving or flickering, induce the appearance ofa similar pattern connecting the two through the unpatternedcentral field (7, 8). No such "phantom" effects were observedin these experiments; the pointers always were stationary andof fixed contrast when finally compared with the Mach bands.However, if disturbed in any way by the presence of the point-ers, the observer could remove and restore them at will bypressing the proper key.Mach bands may fade considerably under steady fixation and

will disappear altogether if completely stabilized on the retina(9). Therefore, the observer was instructed to scan the patternso that the Mach bands would be "refreshed" continuously andthus always have essentially the same appearance while thepointers were being adjusted.

The display oscilloscope provided the only light in the roomin which the experiments were carried out. The observer wasadapted to that level of illumination before the experiment. Sixobservers participated in the experiments. As is well known,there are very large interobserver differences in the appear-ance of Mach bands (10). Our data from different observers allshow the same general trends, but absolute values mavdiffergreatly. Furthermore, the relation between widths (and con-trasts) of the dark and the bright bands varies significantly fromone observer to another. It is uncertain whether these differ-ences result from different criteria or whether they representdifferences in underlying mechanisms.

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RESULTSLocation of Adjacent Bar Stimuli. A vertical bar stimulus

that subtended 25' of visual angle in width and was of 20% pos-itive contrast with respect to its local background was placed atvarious locations, parallel to the Mach bands, in the uniformfields of the ramp pattern. When the bar was on the same sideas, and far removed from, the light Mach band, it had no effect.The band appeared to be of the same width as when there wasno bar stimulus present in the field. However, with patternsin which the bar was closer to the light Mach band, there camea point at which the width (and contrast) of the band began todecrease (Fig. 3 Left). Finally, a position was reached at whichthe light band disappeared altogether. When the bar stimuluswas placed in the uniform field adjacent to the dark Mach band(Fig. 3 Right) the effect was similar to that exerted on the lightband. For this particular observer, the two bands appeared tobe about the same width when there was no adjacent stimulus,and the attenuation of each, when the bar stimulus was adjacentto it, was about the same in both cases. In neither case, withthis particular pattern, did the adjacent bar stimulus have ameasurable effect on the more distant Mach band on the op-posite side of the gradient.

Width of Adjacent Bar Stimuli. The attenuation of the Machbands is independent of the width of the adjacent bar stimulus(Fig. 4). The effect of a very wide bar that extended almost tothe outer edge of the pattern (subtending a visual angle of 100')differed little from the effect of the narrowest bar stimulus thatcould be generated on the display used in these experiments(one subtending only 2.5' of visual angle).

Sign of Contrast. The effects of adjacent bar stimuli are prac-tically independent of the sign of the contrast. For example, abar of 20% negative contrast had essentially the same effect asa bar of 20% positive contrast placed in the same location (Fig.5). However, there were some observable differences, which

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FIG. 4. Attenuation of bright Mach band by adjacent bar stimuliof various widths. Contrasts of all stimuli were -20% with respect tothe uniform bright field. Widths of adjacent stimuli: 2.5' (o), 12.5' (A),25.0' (0), 50.0' (m), 75.0' (A), and 100.0' (A) of visual angle. *, Control(no adjacent stimulus).

are not revealed in these data. These differences resulted mainlyfrom the slight border contrast at the edge of the bar that isopposite in sign to the contrast of the bar itself. Most observersreported that the dark (negative) border contrast of a bar withpositive contrast sometimes tended slightly to "attract" and tofuse with the dark Mach band when the bar was nearby. On theother hand, the bright (positive) border contrast of a bar of neg-ative contrast sometimes tended slightly to "repel" and to stayseparate from the dark Mach band when the bar was nearby.Similar, but inverse, relations held for bar stimuli on the uni-form bright side of the ramp pattern adjacent to the bright Machband. Although the border contrast of a closely adjacent barstimulus might slightly distort the appearance of a Mach band,the attenuation was the much stronger influence. As the barapproached the Mach band, or increased in contrast, the at-

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FIG. 3. Attenuation of Mach bands by adjacent bar stimuli. Lo-cation and width of bands are plotted as a function of proximity of ad-jacent bar stimulus. Inset are diagrams of the stimulus patterns usedfor bright band (above) and dark band (below). The data points for thetwo edges of the bright band (pairs ofopen circles) and for the two edgesof the dark band (pairs of filled circles) are shown in lateral positionsrelative to the corresponding flections (indicatedby two vertical dashedlines) in the ramp patterns. The Mach bands disappeared completelyat the locations indicated by the uppermost points.

Proximity of adjacent stimulus, arc min

FIG. 5. Attenuation of dark band by adjacent bar stimuli of thesame width (25' of visual angle) but of opposite contrast, either +20%(A) or -20% (), with respect to the uniform dark field. Compare withattenuation by a biphasic bar stimulus (a) in which the bar was verynarrow (2.5' of visual angle), and there was no net luminance change:the contrast of the leading bar was +20% and that of the trailing barwas -20%. *, Control (no adjacent stimulus). The configurations of thethree patterns are shown in the three insets.

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tenuation dominated all other effects and the band disappearedaltogether.

Biphasic Bar Stimuli. Most interesting of all our findings withthese bar stimuli is that not only are width and polarity of con-trast all relatively insignificant-no net luminance change at allis required in the interfering stimulus for it to be effective. Forexample, as shown also in Fig. 5, biphasic bar stimuli that haveequal parts above the mean and below the mean were just aseffective as were monophasic stimuli of equal area and of eitherpolarity. (Whether the leading edge of the biphasic stimuluswas positive or negative made some difference in the resultsbecause the sign of the slight border contrast at the edge of thebiphasic stimulus reversed accordingly.)

Triangular and Gaussian Stimuli. All bar or line stimuli wehave tested attenuate the Mach bands. However, other typesof stimuli may either enhance the Mach bands or have no effecton them at all. Thus far, in exploratory experiments, we havefound one example of each (Fig. 6).A triangular stimulus of positive contrast enhanced the dark

Mach band when placed in the field nearby it; a triangular stim-ulus of negative contrast enhanced the bright Mach band whenplaced in the field nearby it. The triangle was constructed sothat it had the same area as one of the bar stimuli that had pre-viously been shown (in Fig. 3) to be an effective attenuator ofthe Mach band. A positive triangular increment on a uniformfield produced dark Mach bands itself. As this triangle changedposition, the bands it produced moved along with it. When theMach band produced by the standard ramp pattern and thenearest Mach band produced by the triangular stimulus werewidely separated they were seen as distinct and nearly equal.As the triangle and its attendant Mach band approached thestationary Mach band in the ramp pattern one band fused withthe other and produced an enlarged Mach band. Finally, as thetriangle reached the ramp, the two bands merged into one band,approximately the same size as the original (Fig. 6B). Similarresults were obtained with a triangle of negative contrast ad-jacent to the bright Mach band.

Triangular stimuli with contrast opposite to those describedabove produced a bright Mach band adjacent to the dark bandin the ramp pattern and a dark Mach band adjacent to the bright

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FIG. 6. Triangular and Gaussian stimuli. (A) Effect of an adjacentGaussian stimulus of positive contrast and with same area as the tri-angular stimulus in B. The width and location of the dark Mach bandproduced by the standard ramp pattern in the presence of the Gaussianstimulus is represented by the filled circles. *, Control (no Gaussianstimulus). (B) Effect of an adjacent triangular stimulus of positive con-trast and same area as Gaussian stimulus in A and as bar stimulus inFig. 3. The width and location of the dark Mach band produced by theleading edge of the triangle is represented by the open triangles. Thewidth and location of the dark Mach band produced by the standardramp pattern in the presence of the triangular stimulus is representedby the filled circles. *, Control (no ramp pattern).

band. When the stimuli were brought close to the ends of thegradient in the ramp pattern, the contrasting bands attenuateone another. If the gradient and the side of the triangle wereof the same slope, both the band in the ramp pattern and theband produced by the triangle disappeared when the triangletouched the end of the gradient. If they were of different slopes,there would be a flection in luminance at the junction of thetwo and, depending on whether the flection was convex or con-cave, there would be a light or dark band.A truncated Gaussian stimulus was designed (Fig. 6A) so that

it had the same area as that of the bar stimulus (in Fig. 3) thatattenuated Mach bands and as that of the triangular stimulus(in Fig. 6B) that enhanced Mach bands. The truncation was im-perceptible, and the stimulus produced no border contrast atits edges. This truncated Gaussian stimulus had little or no ef-fect on the Mach bands in the standard ramp pattern, whethernear or far (Fig. 6A).

DISCUSSIONThese experiments confirm, and greatly extend, Mach's orig-inal observations in 1906 that the Mach bands may be signif-icantly altered by an adjacent stimulus. Mach's interpretationwas that his mask over part of the ramp pattern simply removeda portion of the stimulus that generates the bands. However,our results indicate that this and similar alterations of the pat-tern actually introduce new features into the stimulus that mayactively attenuate the bands.

Controlling Features of the Stimuli. Observations thus farpoint to three significant features of the most effective stimuli:proximity, contrast, and sharpness. The effect does not extendbeyond about 15' from the leading edge of the adjacent stim-ulus, and it increases with decreasing distance between thestimulus and the Mach band. With the stimulus positioned atan effective location, the attenuation increases with increasingcontrast. Finally, it appears that the leading edge of the stim-ulus must be sharp: a line, a bar, or a step stimulus is effective;a truncated Gaussian is not (but see the discussion of this pointbelow).

Features of the stimuli that are not effective are as inter-esting and informative as those that are. These include the find-ing that beyond a certain small minimum, the width of a baror step stimulus is not significant. Also important is the findingthat although contrast itself is significant, polarity of contrastis not. Most important, perhaps, is the finding that no net lu-minance change at all is required; a biphasic stimulus with equalparts above and below the mean can be just as effective as amonophasic stimulus that is either greater or less than the mean.

All of these findings point to, but do not yet fully substan-tiate, one major conclusion: the most important single featureof an effective stimulus is a sharp edge-an abrupt change inluminance. The location and the contrast of the change is sig-nificant, but the polarity of the change in luminance, whetherthe change is sustained over some distance-indeed, whetherthere is a net change in luminance at all-does not appear tobe significant. This conclusion is supported by the finding thata truncated Gaussian stimulus of the same area as an effectivebar stimulus is ineffective. But the experiments on this latterpoint are not definitive. The truncated Gaussian is much broaderthan the bar stimulus of equal area. The sharp leading edge ofthe bar stimulus (supposedly the effective part) can, in the limit,be immediately adjacent to the flection that produces the Machband. But the peak of the Gaussian (possibly the effective part)is always rather far removed from the flection in the Mach pat-tern-even when the truncated end of the Gaussian is im-mediately adjacent to it. Further experiments will have to bedone before the conclusion can be firmly drawn that it is the

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high spatial frequency content of sharp-edged stimuli that pro-duces attenuation of the Mach bands.

Theory and Physiological Mechanisms. For over 100 yearsMach bands have been attributed to reciprocal inhibitory in-teractions, but, although such interactions have been observedin the visual pathways of numerous and diverse species, no clear-cut relation to the Mach bands has yet been established. Thereare three major problems: (i) Mach bands are maximal when thewidth of the gradient is finite-about 0.20, or so. Indeed, Machbands are ephemeral or absent when the illumination profile isan abrupt step. However, all existing theories predict maximalMach bands at an abrupt step. (ii) Simple linear network theorypredicts symmetrical Mach bands, but in fact the bands areasymmetrical. Proposed nonlinear networks that might accountfor this asymmetry have no physiological bases; they are largelyad hoc. (iii) Current theoretical work treats the Mach bands inthe steady state, but the Mach bands are dynamic phenomenaand so are all likely underlying physiological mechanisms. Letus consider each of these problems in turn.

Frequently, the term Mach bands is used to refer to the bor-der contrast at a square edge or step (10) and to the familiarfluted appearance of a series of steps (the Chevreul illusion).These contrast phenomena are not Mach bands (1, 11, 12). In-deed, it has long been puzzling why maximal Mach bands areseen at the two ends of a gradient and not at all (or very faintly,if at all) at the edges of a step. Tolhurst (13) has proposed anexplanation in terms of the interactions of two competingmechanisms. The results of an experiment that directly su-perimposes the stimuli for these two mechanisms (unpublisheddata) support this general notion. The point we wish to em-phasize here is that if the border contrast at a step were itselfa Mach band then a step should enhance rather than attenuatethe Mach band at the end of a nearby gradient (as does the tri-angular stimulus in Fig. 6B). Evidently, border contrast andMach bands are different phenomena dependent upon differ-ent mechanisms (14-16).

The asymmetry of the Mach bands was noted when they werefirst discovered. It has been evident from the very beginningthat simple linear models are inadequate (2, 17). There is noshortage of nonlinear models, but they are largely ad hoc andhave no direct physiological bases. The recent quantitative ob-servations of essential nonlinearities in the vertebrate visualsystem (18, 19) provide a remedy for this lack of concretenessand may offer physiological explanation for the asymmetries.Another concrete possibility is offered by the known differ-ences in structure and location of the on-center and the off-cen-ter cells (20-22) that may mediate the bright and the dark lines,respectively. The function of those cells, which may be criticalin the Mach band phenomena, may not be mirror images of oneanother and thus may account for the asymmetry of the Machbands.

There is little to say about the largely neglected field of dy-namics of Mach bands. Only a few exploratory studies have beencarried out thus far (23-25). However, the subject has becomemore accessible in recent times mainly because of the devel-opment of extremely versatile electronically controlled visualdisplays.

Whatever the mechanisms underlying the Mach bands maybe, the one thing that is certain about them now is that we haveno clear understanding of them.

This research was supported in part by The National Eye Instituteand the Esther A. and Joseph Klingenstein Fund. Some special facil-ities were provided by Grant S07 RR07065 awarded by the BiomedicalResearch Support Program, Division of Research Resources, NationalInstitutes of Health. N. R. was a Guest Investigator at The RockefellerUniversity under the auspices of the College Work Study Program, DukeUniversity, Durham, NC.

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