Vision Res. Vol. 34. No.8. pp. 1089-1104. 1994

CoPYlright ~ 1994 Elsevier Science LtdPrinted in Great Britain. All rights reserved

0042-6989/94 $6.00 + 0.00

An analysis of the reset of visual cortical circuits responsible for the binding or segmeilltation of visualfeatures into coherent visual forms yields a model that explains properties of visuall~rsistence. Thereset mechanisms prevent massive smearing of visual percepts in response to rapidly moving images.The model simulates relationships among psychophysical data showing inverse relatiollS of persistenceto flash luminance and duration, greater persistence of illusory contours than real cont()iurs, a U-shapedtemporal function for persistence of illusory contours, a reduction of persistence due to adaptation witha stimulus of like orientation, an increase of persistence due to adaptation witlll a stimulus ofperpendicular orientation, and an increase of persistence with spatial separation of a m:asking stimulus.The model suggests that a combination of habituative, opponent, and endstopping meclflanisms preventsmearing and limit persistence. Earlier work with the model has analyzed data about boundaryformation, texture segregation, shape-from-shading, and figure.-ground separation. ThIIJS, several typesof data support each model mechanism and new predictions are made.

Vision Neural networks Visual cortex Visual persistence Feature binding Illusory co~ Off-cells

INTRODUcrJON test stimulus decreases; but when subjects adaptHumans and other animals form useful visual represen- to a stim~lus of a pe~ndicular orienta~ion totations of rapidly changing scenes, The visual system ~he test stimulus, persistence of the test stimulusrapidly resets the segmentations of changing parts of a Increases, , .scene to prevent image smearing, This article explains .The subsequ.ent on~et of a maskIng s~lmulushow a neural network theory of early visual processes greatly curtails persistence of a target stimulus.

proposed by Grossberg and Mingolla (1985a, b, 1987) Before presenting the details of model mechanisms, weaccounts for many of the data on visual persistence, The briefly describe how the model addresses each of thesetheory suggests that a key process governing these data data sets,is the time taken to reset a segmentation, We simulatereset dynamics that help to force a rapid return of the Inverse relation of persistence to luminance and tonetwork to a state unbiased by prior segmentation in stimulus durationorder to better process incoming data, We explain how Figure I (a), taken from Bo,wen, Pola and Matinhysteresis in the segmentation network is a rate-limiting (1974), shows that, for each luminance curve, persistencefactor in visual persistence, and show that properties of is inversely related to stimulus dll1ration. Except for verythe hysteresis match key psychophysical data. Psycho- short stimulus durations, persi:>tence is also inverselyphysical studies of visual persistence have revealed four related to stimulus luminance, Similar results have beenkey sets of data, which are all explainable by the model, found by many ~uthors (see Coltheart, 1980; Breitmeyer,

P ' '0 I I d . I 1984 for reviews)..erslstence IS Inverse y re ate to stlmu us I h d f B ' (1974) b'd t' d t t . I I ' n t e stu y 0 owen et aj. , su ~ects were

ura Ion an 0 s Imu us ummance, "III t . t h I th I asked to match the perceived offset of a target stimulus

.usory con ours persIs muc onger an rea, 0 0t d ' II t d t b th with the perceived onset of ~i probe stimulus, The

con ours an I usory con ours 0 no 0 eye, , .,, I t ' h' b t ' t d physical mterstlmulus Interval Ibetween the target andInverse re a Ions IP e ween persIs ence an , ...t o I d t. h t ' t ' f I .mask stimuli provided a measure of the target's persist-S Imu us ura Ion c arac ens IC 0 ummance- L d G'

ld (1981) d h .

db d t ence, ong an I ea argue t at perceive

ase con ours. ffi ' d f .bWh b . t d t t t . I f th 0 set IS not a goo measure 0 persistence ecause some

.en su ~ec s a ap 0 a s Imu us 0 e same ..," " parts of the stimulus may continue to persist beyond the

onentatlon as the test stimulus, persistence of the 0 d t' I ffi t S k' d L (1979) dperceive S Imu us 0 se, a III an ong an~-~ Long and McCarthy (1982) sho,wed that when subjects

were told to attend to any resid"lal trace of the stimulus,and not just perceived offset, the duration of totalpersistence was directly related to stimulus luminance,

1089

.Center for Adaptive Systems and Department of Cognitive andNeural Systems, Boston University, III Cummington Street.Boston, MA 02115, U.S.A.

tTo whom all correspondence should be addressed.

GREGORY FRANCIS,* STEPHEN GROSSBERG,*t ENNIO MINGOLLA*

Received 4 August 1992; in revised form 8 February 1993; in final form 15 June 1993

1090 GREGORY FRANCIS el 01.

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Measures of total persistence have, in turn, beencriticized as being the result of afterimages or iconicmemory (Coltheart, 1980; Breitmeyer, 1984; DiLollo,1984). Perceived offset and total persistence are thusdifferent features of the dynamic processing of achanging stimulus. In this paper we model persistencedata based upon perceived offset, and we do not considerthe properties of total persistence until the conclusion.When we refer to persistence we mean the time betweenphysical offset and perceived offset.

The inverse relationships between persistence andstimulus duration and luminance imply that persistencecannot be modeled as a simple decay of activity of someneural stimulus representation. The initial strength ofsuch a representation at the moment of stimulus offsetwould presumably increase with stimulus duration orluminance, yielding a higher starting point from whichdecay would begin, and thus longer persistence. FigureI(b) (solid lines) demonstrates that persistence of signalsin the model is inversely related to stimulus duration and

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FIGURE 2. (a) Illusory contours persist longer than real contours.Persistence of illusory contours is maximal at an intermediate durationof the stimulus. [Reproduced with permission from Meyer and Ming(1988).) (b) Computer simulation of real and illusory boundarycontour persistence as a function of flash duration. The boundariesproduced in response to the illusory contours persist longer than theboundaries produced in response to the real contours. Persistence ofillusory contours peaks at an intermediate stimulus duration. as in thedata. The solid lines connect the points sampled in the data of Meyer

and Ming (1988).

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luminance at all but the shortest durations. The modelachieves this close match with the psychophysical databy generating an active reset signal at stimulus offsetwhich inhibits the persisting signals~ of the stimulus(Grossberg, 1991). We later quantitatively analyze howthe strength of the reset signal increases with stimulusduration and luminance.

150 :==--6--~---6 ~'-'100

Parado.\'ical increase of persistence of illusor)' contours

Figure 2(a) (taken from Meyer & Ming, 1988) showsthat illusory contours have different persistence proper-ties than contours defined by luminance edges. Not onlydo illusory contours persist substantially longer than realcontours, but persistence of an illusory contour peaks atan intermediate stimulus duration. In contrast, thepersistence of a stimulus defined by luminance edgescontinually decreases as stimulus duration increases.These data place strong constraints on the source of

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FIGURE I. (a) Persistence is inversely related to flash luminance andflash duration. [Reproduced with pennission from Bowen 1'101. (1974).)(b) Computer simulation of boundary signal persistence as a functionof flash duration and flash iuminance. Dashed lines simulate modelperfonnance ,,'ithout habituative transmitter gates that fonn the basis

of the reset mechanism required to explain data on persistence.

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CORTICAL DYNAMICS OF BINDING AND RESET 1091

signals used to reset a changing visual segmentation. Inour model, only changes in luminance-derived edgesgenerate reset signals. Thus, figural boundaries that in-clude illusory contours persist longer than contours ofcorresponding length that are defined entirely by lumi-nance edges, because the former contain luminance edgesas a smaller proportion of the total contour. Figure 2(b)shows that the model's responses to illusory contourspersist longer than real contours. Persistence of illusorycontours in the model is not inversely related to stimulusduration at short durations because illusory contourstake some time to fully develop (Reynolds, 1981) and, ifnot fully developed, they can more quickly disappear.

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Effects of orientation-specific adaptationFigure 3 (hatched bars) shows that adaptation to

stimuli can also influence persistence duration (Meyer,Lawson & Cohen, 1975). This figure demonstrates thatwhen subjects adapted to a stimulus of the sameorientation as the test stimulus, persistence of the teststimulus decreased; but when subjects adapted to astimulus of a perpendicular orientation to the teststimulus, persistence of the test stimulus increased. Ineach case, persistence could be changed by as much as:i: 20 msec. These data provide two clues about thehypothesized reset signal. First, it suggests thatadaptation or habituation drives the reset signal. Second,it indicates that opponent interactions between pathwayssensitive to opposite orientations regulate the inhibitionthat forms the reset signal. Below we explain how aneural circuit consistent with these observations cangenerate a transient response at stimulus offset that actsas a reset signal. Figure 3 (shaded bars) shows thatadaptation influences persistence of signals in the modelin the same way revealed by psychophysical studies.

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FIGURE 4. (a) The persistence of thin lines moving in stroboscopicmotion depends on the spatial separation between successive images.[Reproduced with permission from Farrell el 01. (1990).) (b) Computersimulation of boundary signal persist'~nce as a function of the spatialseparation between contours of a target and a mask. The dashed linesimulates model performance without the spatial competition. Notethat the size of our simulation plan'~ did not permit testing spatial

separations larger than shown.

Same Orthogonal

Orientation of adaptation stimulus

FIGURE 3. Hatched bars: change in persistence depending onwhether the adaptation stimulus had the same or orthogonal orien-tation as the test grating. [plotted from data in Meyer el al. (1975).]Shaded bars: computer simulation of boundary signal persistencedepending on whether the adaptation stimulus had the same or

orthogonal orientation as the test grating.

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1092 GREGORY FRANCIS et at.

Shortening of persistence by a spatially proximal mask

Figure 4(a) from Farrell, Pavel and Sperling (1990)shows that the influence of a mask on the persistence ofa target depends on the spatial distance between thestimuli, with closer masks decreasing the persistence ofthe target. Other studies (Farrell, 1984; DiLollo &Hogben, 1987) have found similar results. Mostresearchers interpret this result as being due to spatialinhibition, which prevents smearing of moving stimuli.The model contains this type of inhibition and Fig. 4(b)(solid line) demonstrates that the persistence of signalsin the model correlates well with the psychophysicaldata.

The model that we use to simulate persistence datawas originally developed to explain many other types ofpsychophysical and neural data, such as data aboutboundary completion, illusory contour fonnation, tex-ture segregation, shape-from-shading, three-dimensionalfigure-ground pop-out, brightness perception, andfilling-in of three-dimensional surface percepts(Grossberg, 1987a,b, 1994; Grossberg & Mingolla,1985a, b, 1987; Grossberg & Todorovic, 1988). Modelmechanisms have also been derived from several basicprinciples about visual infonnation processing

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CORTICAL DYNAMICS OF BINDING AND RESET 1093

not implement the inhibition in a specific architecturecapable of explaining the properties of visual persistence.

Simulations of the model generated Figs l(b), 2(b), 3,and 4(b) with a single set of parameters (except where themodel is intentionally "dissected" for analysis). All theequations and parameters governing the model behaviorare described in the Appendix. These simulationsdemonstrate the model's competence to explain qualitat-ive relationships between persistence and variousstimulus qualities. The simulations do not in every caseprovide a quantitative match with the psychophysicaldata. Why this is so is discussed in the concludingremarks.

(a)

(b)

FEATURE BINDING AS A SOURCE OF VISUALPERSISTENCE

In this section we describe how the boundary contoursystem (BCS) (Grossberg & Mingolla, 1985a, b, 1987)addresses the psychophysical properties of visual persist-ence. Figure 5 schematizes the model, with each cell'sicon drawn to indicate its receptive field structure.

We base our first observation on the adaptationstudies of Meyer et al. (1975), which show that persist-ence can be specifically modified according to stimulusorientation. These data are explained using model cellsthat locate and represent oriented boundaries. Thus,although cells in the first level contain unorientedcenter-surround receptive fields, the remaining cells inthe network respond best to a boundary of a specificorientation. In agreement with neurophysiological dataon receptive field properties in visual cortex (Hubel &Wiesel, 1965), the first level of oriented cells in the BCScorrespond to cortical simple cells. These cells respondto oriented luminance edges of a specific polaritycentered at a particular point on the retina. Cells in thenext level, corresponding to complex cells in visualcortex, receive rectified inputs from like-oriented simple

with distance. Grossberg and Mingolla (1985a, b) callthis process theftrst competitive stage. This competitionoccurs across positions in the direction of preferredorientational tuning (endstopping) as well as acrosspositions lateral to the preferred direction. Activitiesfrom the first competitive stage feed into a process ofcross-orientation inhibition at each position called thesecond competitive stage, which partially accounts for theopponent processing implied by the results of Meyer etal. (1975) in Fig. 3.

Signals surviving the competitive stages input tocooperative bipole cells that are sensitive to more globalproperties of the configuration of image contrasts.

1094 GREGORY FRANCIS el al.

Figure 5, for example, shows a horizontally tuned bipolecell, which received excitatory inputs from a horizontalrow of horizontally tuned cells and inhibitory inputsfrom vertically tuned cells at the same locations. Thisorientation-specific inhibition helps to explain the orien-tation-specific data of Meyer et al. (1975) in Fig. 3.Grossberg and Mingolla (1985b) showed that this inhi-bition provides the network with a property of spatialimpenetrability, which prevents boundary linkings fromforming across intervening boundaries of roughlyperpendicular orientation. Every bipole cell has twoindependent lobes to its receptive field, and each lobemust receive a sufficient amount of excitatory input fromthe second competitive stage for the bipole cell togenerate a response. A bipole cell triggers a responseonly if its receptive fields are each stimulated by one ormore boundary components. For example, a bipole cellwhose receptive field center is at a corner of a boundarycannot generate a response because the contourstimulates only one side of its receptive field. On the

other hand, a bipole cell centered between two illusorycontour inducers as in, say, a Kanizsa square, willgenerate a bipole response if the inputs are sufficientlystrong. In this fashion, parallel arrays of bipole cells cangenerate an illusory contour. von der Heydt, Peterhansand Baumgartner (1984) have found evidence support-ing the existence of bipole cells in area V2 of monkeycortex.

Bipole-to-hypercomplex feedback carries out a spatialsharpening process similar to the first competitive stage.Here, each bipole cell feeds back on-center, off-surroundsignals to hypercomplex cells of the same orientationsensitivity. Spatial sharpening is another part of themodel that correlates with t.he spatial inhibitionindicated in Fig. 4(a).

The positive feedback from the bipole cells to lowerlevel hypercomplex cells provides the rate-limitingsource of persistence in the modc~l because the activitiesgenerated by the feedback from one bipole cell can exciteparallel arrays of other bipole cells, which in turn feed

(b)(a)

(c) (d)

FIGURE 7. (a) Stimulus input to the system, a bright square on a dark background. (b) Boundary response to the squareshortly after the input returns to the background level. (c) Boundary signals start to erode from the comers of the square toward

the middle or the contours. (d) Boundary erosion is almost complete.

CORTICAL DYNAMICS OF BINDING AND RESET 1095

back signals that can excite the original bipole cell, asindicated in Fig. 6(a). A self-sustaining feedback loop isgenerated by the cooperative interactions of thepathways marked by heavy lines. Thus, within themodel, persistence is due the positive feedbackinteractions that choose a coherent boundary segmenta-tion from among many possible groupings, and inhibitpotential groupings that are weaker. This positive feed-back loop causes hysteresis that is controlled by othermodel mechanisms.

For example, as indicated in Fig. 6(b), the end of asurface contour does not support these feedback inter-actions. In this figure the leftmost boundary signal ofFig. 6(a) has been deleted. Without this boundary signal,the left bipole cell does not fire because only one side ofits receptive field receives stimulation. Upon stimulusoffset, the effect of this organization is that, even withoutan active reset signal, the boundary signal at the end ofa contour passively decays away and exposes a newcontour end [as in Fig. 6(a,b)]. At this new contour end,the process repeats itself for another bipole cell and soon from the ends of the surface contour inward towardthe middle of the contour [as in Fig. 6(b,c)]. It is thisinward erosion of boundary signals that we correlatewith the persisting visual percept beyond stimulus offset.In the simulations described below, we show that, with-out an active reset signal, this passive erosion is insuffi-cient to model the psychophysical data on persistence.

Figure 7 summarizes a simulation of boundary signalerosion. Figure 7(a) shows the stimulus presented to

2.5

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FIGURE 8. Time plot of boundary signal activities at a cross-section of Fig. 7 (lower horizontal contour). The signals dropmarkedly upon stimulus offset as the bottom-up input is removed. The top-down signals maintain the boundaries for a

substantia! length of time. but the boundaries erode away from the ends toward the middle of the contour.

the system, a bright square on a dark background.Figure 7(b-

GREGORY FRANCIS el al.1096

FIGURE 9. At stimulus offset, a gated dipole circuit produces a transient rebound of activity in the non. stimulated opponent

pathway.

a transient rebound of activity in the previously non-stimulated pathway.

The time plot next to each cell or gate describes thedynamics of this circuit. In the case shown, the sharpincrease and then decrease of the time plot at the lowerright of Fig. 9 indicates that an external input stimulatesthe horizontal pathway. In response to the strongersignal being transmitted to the next level, the amount oftransmitter in the gate inactivates during stimulation andthen rises back toward the baseline level upon stimulusoffset. Notice that the inactivation and reactivation oftransmitter occur more slowly than changes in the activi-ties of the neural cells. Each slowly habituating transmit-ter multiplies, or gates, the more rapidly varying signal inits pathway, thereby yielding net overshoots and under-shoots at input onset and offset, respectively. Duringstimulation, the horizontal channel wins the opponentcompetition against the vertical channel as indicated inthe top right time plot. However, upon offset of thestimulation to the horizontal channel, the input signalreturns to the baseline level but the horizontal transmit-ter gate remains habituated below its baseline value. Asa result, shortly after stimulus offset, the gated tonicinput in the horizontal channel has a net signal below thebaseline level. Meanwhile, the vertical pathway main~

Tonicinput

PhasicOn-input

tains the baseline response at all cells and gates beforethe opponent competition. Thus, when the horizontalchannel is below the baseline activity, after stimulusoffset, the vertical channel win:s the opponent compe-tition and produces a rebound 01" activity as shown in thetop left time plot. As the horizontal transmitter gaterecovers from its habituated sta1:e, the rebound signal inthe vertical channel weakens and finally disappears.

Figure 10 shows how this rebound of activity acts as areset signal in the full BCS arlchitecture. Figure 10(a)schematizes how inputs in a "horizontal pathway excite ahorizontal bipole cell. As described above, thesehorizontal bipole cells can generate a hysteresis thatcorresponds to persistence. Due to the interactions of thegated dipole circuit, offset of the horizontal inputgenerates a rebound of activity in the vertical pathway,which, as Fig. 10(b) demonstral:es, inhibits the horizon-tal bipole cell. This inhibition greatly speeds up theerosion of boundary signals al1ld decreases persistence.Note that, apart from its crucial role in explainingpersistence data, the inhibition of bipole cells by offsetsignals from perpendicularly oriented pathways of thegated dipole circuit has been shown to play an equallycrucial role in preventing unwanted boundary groupingsacross intervening surfaces (spatial impenetrability).

CORTICAL DYNAMICS OF BINDING AND RESET 1097

(a)

(b)

FIGURE 10. (a) A horizontal input excites a horizontal bipole cell,which supports persistence. (b) Upon offset of the horizontal input, arebound of activity in the vertical pathway inhibits the horizontalbipole cell. This inhibition resets the hysteresis of the feedback loop

and reduces persistence.

The inverse relationships between persistence andstimulus duration and luminance, as shown in Fig. l(a),follow from the properties of the gated dipole. Thelonger the stimulus duration, or the stronger (moreluminous) the input to a gated dipole, the more habitu-ated becomes the transmitter gates and, thus, thestronger becomes the reset signals. The significance ofthe strength of the reset signals is evident in Fig. l(b),which shows that persistence of signals in the model isinversely related to stimulus duration and luminance,except at very short stimulus durations. At short stimu-lus durations, there is only a very weak reset signalgenerated by the gated dipoles. However, because thestimulus presentation is so brief, the BCS does notestablish a strong hysteresis in the feedback loop.Indeed, because stimuli of a greater duration or a higher

luminance create stronger boundary signals and canmore quickly establish strong activities in the feedbackloop, at the shortest stimulus durations persistence isdirectly related to stimulus duration and luminance.Haber and Standing (1970) reported that persistenceincreases with stimulus duration v{hen the stimuli arebriefer than 20 msec. At longer stimulus durations, thegated dipoles begin to generate stronger reset signals,which more quickly remove the persisting boundarysignals in the feedback loop. Thus, the inverse relation-ships between persistence and lumimance and durationoccur when a strong hysteresis has been established inthe BCS and when the gated dip-ole circuits producestrong reset signals.

To emphasize the role of the gated dipole circuit andits reset signal, we re-ran simulatiotilS for two luminancevalues and several different stimulus durations butmodified one parameter so that the transmitter gates didnot habituate. The dashed lines in Fig. l(b) showthat without the transmitter habituation, persistenceincreases, or does not change, with stimulus duration.Similarly, persistence increases with stimulus luminancewithout the gated dipole circuit; thus, it is the behaviorof habituative transmitters in the gated dipole circuitthat explains the data of Bowen et al. (1974) within theBCS model.

Because the habituative transmitters of the gateddipole circuit are located outside the feedback loop ofthe BCS, only the offsets of lumi:rlance-derived edgesgenerate reset signals. Therefore, only the reset signalsgenerated by illusory contour inducers inhibit thepersisting illusory contours. Inducers of illusorycontours have few luminance edges and so, at stimulusoffset, generate fewer reset signals than boundariesdefined entirely by real contours. With fewer reset signalsavailable to break the hysteresis of the feedback loop,illusory boundaries persist longer than real boundaries.We show the results of simulations of these properties inFig. 2(b). The luminous-based stimulus for these simu-lations was an outline square, while the illusory contourinducers were L-shaped stimuli at the corners of asquare. Our choice of inducer folmlS was limited bycomputational resources as explained in detail in theAppendix. This choice suffices to illulstrate the dynamicalproperties of contours that are formed at positionswithout luminance contrast. Other research concerningthe BCS has more fully analyzed the relationshipsbetween boundary signals and perceived illusorycontours through computer simulation (Gove,Grossberg & Mingolla, 1993) and psychophysicalexperimentation (Lesher & Mingollla, 1993). Grossbergand Mingolla (1985a) also analyzed! why some inducersproduced stronger illusory contours than others.

Because luminance edges define ,only a small part ofan illusory contour, the boundary representation takeslonger to become strongly established than the boundaryrepresentation of the outline square~. Therefore. illusorystimuli of short duration do not generate strong hys-teresis and can more quickly erode at stimulus offset.Stimuli of an intermediate duration generate strong

1098 GREGORY FRANCIS el at.

hysteresis, but do not produce strong reset signals atoffset. Thus, these stimuli persist longest. Stimuli of longduration generate a strong hysteresis, but they alsogenerate stronger reset signals, which shorten persist-ence. Thus, over the same range of stimulus durationsthat show an inverse relationship with persistence for theoutline square, persistence of boundary signals for anillusory contour is not inversely related to stimulusduration at short durations, but peaks at someintermediate value, as shown in Fig. 2(b). The solid linesin Fig. 2(b) connect simulation data points of thestimulus durations sampled in the psychophysical studiesof Meyer and Ming (1988). A comparison of these curveswith the data in Fig. 2(a) shows that they are very similarin shape, although the absolute persistence values aredifferent.

Finally, because the opponent processing andhabituative pathways are orientation-specific, the modelexplains the adaptation results of Meyer et al. (1975)shown in Fig. 3. Within the model, adaptation to, say,a horizontal stimulus habituates the transmitter inhorizontal pathways but leaves the vertical pathwaysunadapted. If one then tests persistence of a horizontalstimulus, the horizontal pathways further habituate thetransmitter during the test presentation. The strongerthan usual habituation of the horizonal pathways meansthat, at stimulus offset, the reset signals will be strongerthan usual and persistence will decrease. On the otherhand, if one tests persistence of a vertical stimulus, thenboth oriented pathways become habituated: the horizon-tal pathways from the prior adaptation and the verticalpathway from the habituation due to the target presen-tation. As a result of the opponent competition betweenthese habituated signals (Fig. 9), the reset signalsgenerated at offset of the vertical test stimulus will beweaker than usual and persistence will increase.

So far we have accomplished two major goals. First,we explained how interactions of a tonic input, habitua-tive transmitter gates, opponent processing, and rectifiedoutput signals in a gated dipole circuit generate a resetsignal upon offset of an oriented luminance edge.Second, we showed that the properties of this reset signalat the second competitive and bipole stages of the modelaccount for the inverse relationships between persistenceand stimulus luminance and duration (Bowen et al.,1974), the prolonged non-monotonic persistence proper-ties of illusory contours (Meyer & Ming, 1988), and theopposite influences on persistence of orientation-specificadaptation (Meyer et al., 1975). .

It remains to show that the spatial inhibition, orendstopping process, within the first competitive stage ofthe model can account for the decrease in persistenceshown in Fig. 4(a). The solid line in Fig. 4(b) demon-strates that the persistence of boundary signals doesdepend on the separation between target and maskstimuli. For comparison, the dashed line shows thepersistence of boundary signals when we removed theoriented spatial competition of the first competitive stagefrom the model (setting one parameter equal to zero).The second source of spatial inhibition, in the feedback

pathway of the bipole cells, remains intact. In the currentsimulations it is of a smaller range than the interactionsof the rrst competitive stage. The dasl1led line in Fig. 4(b)shows that without the spatial competition there are nochanges in persistence of the target stimulus until thetarget and mask boundaries are within the range of theinhibition in the feedback pathway. Thus, the spatialcompetition accounts for the abilit:f of the maskingstimulus to reduce persistence of the talrget (Farrell et al.,1990).

NEUROPHYSIOLOGICAL CORRELA TIS ANDPREDICTIONS

Grossberg (I 987a) reviews neural analogs of all stagesof the present model in visual cortex. In particular, vonder Heydt et al. (1984) found analogs of bipole cells inarea V2 of monkey visual cortex, 1Nhich Cohen andGrossberg (1984), Grossberg (1984), and Grossberg andMingolla (1985a) had modeled before these data werepublished. It may be possible to use additional neuro-physiological studies to verify the dynamic properties ofthese cells. For example, it should be possible to observethe inward erosion of boundary sign:als by observing asingle bipole cell in visual cortex. Aftelr finding the centerof the bipole cell's receptive field, one could run a seriesof experiments varying the position clf luminance edgesrelative to the center of the cell's receptive field. Becausethe model predicts that boundary sigI1Lals erode from thecontour ends, the cell should show its greatest persistingresponse when its receptive field is centered on thecontour and should show less persistence as the exper-imenter shifts the contour center to one side or the otherof the receptive field center. Note that this predictionfollows despite another prediction, derived from ananalysis of the BCS in response to :;tatic stimuli, thatbefore stimulus offset the amplitude of activity should benearly identical for bipole cells all allong the contour,regardless of the cell's distance from inducers beforereset occurs. Properties such as the inverse relationshipsbetween persistence and stimulus duration andluminance, greater persistence for illusory than realcontours, the effects of adaptation, alild the influence ofa masking stimulus should also be observable in aninvestigation of these cells.

Likewise, the ability of masking stimuli to reducepersistence of the target should be measurable at asuitable population of hypercomplex cells. Other hyper-complex cells should exhibit habitualtive gating of theirresponses, as well as opponent rebounds to offset ofstimuli that are oriented perpendicular to their receptivefields. The firing of these rebounding hypercomplex cellsshould correlate with diminished pe:rsistence in targetbipole cells that are tuned to a perpendicular orientation.

Lesher and Mingolla (1993) have tested the BCSmodel through psychophysical ex]periments on theinduction of illusory contours in a Kanizsa square usingvariable numbers of line ends that are perpendicular tothe illusory contour. They found that an inverted Ufunction relates illusory contour clarity to the number of

CORTICAL DYNAMICS OF BINDING AND RESET1099

line end inducers. The BCS explains the inverted U asfollows. More line end inducers at first produce more cellactivations at hypercomplex cells of the second competi-tive stage whose receptive fields are centered at andperpendicular to the line ends. These responses are calledend cuts (Grossberg & Mingolla, 1985b). They are dueto an interaction between the first and second competi-tive stages. The first competitive stage inhibits thosehypercomplex cells at a line end whose orientationalpreference is parallel to the line. These inhibited cellsdisinhibit (via the second competitive stage) hypercom-plex cells at the line end whose orientational preferenceis perpendicular to the line (see Fig. 5). These greaternumbers of end cuts, in turn, collinearly cooperate tomore strongly activate the bipole cells that generate theillusory contour.

As more line inducers are used, however, they getcloser together. They then inhibit each others' targethypercomplex cells via lateral inhibitory signals from thefirst competitive stage. Their net responses are herebyweakened by lateral inhibition, as are their end cuts andtheir illusory contours. Lesher and Mingolla (1993)noted that other models of illusory contour formationcannot explain this inverted U effect.

The BCS model suggests that an inverted U functionmay also relate illusory contour persistence to thenumber of line end inducers under these experimentalconditions, since weaker bipole cell activations shoulderode more quickly after stimulus offset and since agreafer number of reset signals would be generated. Thatis, as the spatial density of line inducers increases beyondthe point where illusory contour clarity reverses, thenillusory contour persistence should also reverse inresponse to line inducers that are on for a fixed amountof time. Stimuli in the rising portion of the Lesher andMingolla (1993) study provide both stronger bipole cellactivity and stronger reset signals, and so present a moreambiguous case. This type of experiment would probethe interactions between first and second competitivestages, as well as between the spatial and temporalproperties of emergent segmentation.

Grossberg (1987a, Section 30) linked properties of theBCS simple, complex, and hypercomplex cells to exper-imentally reported properties of spatial localization andhyperacuity. Badcock and Westheimer (1985a, b) usedflanking lines to influence the perceived location of a testline. They varied the position of the flank with respectto the test line as well as the direction-of-contrast offlank and test lines with respect to the background. Theyfound that two separate underlying mechanisms wereneeded to explain their data: a mechanism concernedwith the luminance distribution within a restrictedregion, and a mechanism reflecting interactions betweenfeatures. Within the central zone defined by the firstmechanism, sensitivity to direction-of-contrast wasfound, as would be expected within an individual recep-tive field. On the other hand, a flank within the surroundregion always caused a repulsion which is independentof direction-of-contrast. Thus "when flanks are close toa target line, it is pulled toward the flank for a positive

flank contrast but they push eacl1l other apart if the flankhas a negative contrast. A flank in the surround regionalways causes repulsion under th,e conditions presented"(p. 1263). To further test independence of direction-of-contrast due to the surround, they also found that "theeffect of a bright flank on one side can be cancelled bya dark flank on the other. Withiin the central zone thisprocedure produces a substantia:! shift of the mean of apositive contrast target line towards the positive contrastflank" (p. 1266).

Badcock and Westheimer (1985a) noted that theaverage of luminance within the 4~entral zone is sensitiveto amount-of-contrast and dir(:ction-of-contrast in away that is consistent with a difference-of-Gaussianmodel. Such a computation also occurs at the elongatedreceptive fields, or simple cells, of the BCS (Fig. 5). Pairsof simple cells with like positions and orientations butopposite directions-of-contrast t:hen add their rectifiedoutputs at complex cells which are, as a consequence,insensitive to direction-of-contra.st (Fig. 5). Such cellsprovide the inputs to the first (:ompetitive stage. Theoriented short-range lateral il1lhibition at the firstcompetitive stage is thus insensitive to direction-of-contrast, has a broader spatial Jrange than the centralzone, and, being inhibitory, would al\\rays causerepulsion-all properties of the Badcock and West-heimer (1985a) data. In summary, all the main effects inthese data mirror properties of the circuit in Fig. S.

In further tests of the existence and properties of thesedistinct mechanisms, Badcock and Westheimer (1985b,p. 3) noted that "in the surround zone the amount ofrepulsion obtained was not influenced by verticalseparation of the flank halves, even when they wereseveral minutes higher (or lower) than the target line. Inthe central zone attraction was only obtained when thevertical separation was small enough to provide someoverlap of lines in the horizontal direction". These datafurther support the idea that the central zone consists ofindividual receptive fields, whereas the surround zone isdue to interactions across receptive fields which are firstprocessed to be independent of direction-of-contrast, asin Fig. 5. In our computer simulations of boundarycompletion and segmentation (Grossberg & Mingolla,1985a, b), it was assumed that the lateral inhibitionwithin the first competitive stage is not restricted to anypreferred orientation, as is also true of the surroundrepulsion effect in the Badcock and Westheimer (1985b)data.

This theoretically predicted correlation betweenproperties of hyperacuity, persistence, and illusorycontour formation presents an opportunity to designnew types of psychophysical experiments with which tofurther test the model. Other new experimental opportu-nities are summarized in Grossberg (1994). Some ofthese are now being explored in our laboratory.

RELATED FINDINGS AND CONCLUDING REMARKS

Because the equations presented in the Appendix arefor a "single-scale" BCS, they cannot explain findings by

1100 GREGORY FRANCIS et a/.

Meyer and Maguire (1981) that the persistence of agrating percept increases with spatial frequency. However,the multiple scale BCS interactions described in Gross-berg (1994) can account for this finding without affectingthe explanations described in this paper. With multiplescales of oriented filters, a grating of a low spatialfrequency excites both large and small filters, whereas agrating of a high spatial frequency excites only smallscaled filters. The net result of this skewed excitationdistribution across scales is that the low spatial frequencygrating creates more reset signals than the high spatialfrequency grating. All activated filters of similar orienta-tional and disparity sensitivity at each position input tothe same set of bipole cells. As in the case of real vs il-lusory contours, more reset signals imply less persistence.

The model also suggests why different experimentalmethods find different properties of persistence (Sakitt &Long, 1979). The results in this paper have correlated thedisappearance of boundary signals with the perceivedoffset of the stimulus. However, due to the slow timeconstants of the habituation in the gated dipole circuit,reset signals generated at stimulus offset may persistbeyond the offset of the boundary signals. Perceptualawareness of these reset signals may be used by subjectswhen the experimental instuctions tell them to observeany residual trace of the stimulus, as in the studies ofSakitt and Long (1979). In particular, stronger luminanceimplies greater habituation, which implies a greaterlength of time for the rebounding channels of the gateddipole to return to baseline. Thus, the persistence of thereset signals may be directly related to stimulusluminance, in agreement with the psychophysical studiesof total persistence (Sakitt & Long, 1979; Nisly &Wasserman, 1989).

In reading the Appendix, note that a single set ofparameters was used to simulate all the properties ofvisual persistence. While modifying the parameters maychange the quantitative values given in Figs 1, 2, 3, and 4,the relevant functional properties expressed by the curvesremain the same. Specifically, regardless of the specificchoice of parameters (excluding cases where boundarysignals or reset signals are not created at all), persistenceof signals in the model is inversely related to stimulusluminance and duration, illusory contours persist longerthan real contours, orientation-specific adaptation hasopposite influences on persistence, and a spatial maskingstimulus inhibits target persistence. The relationshipsbetween persistence and stimulus properties are built intothe structure, or non-parametric design, of the model. Inthe present simulations, our goal has thus been to clarifyand illustrate the qualitative functional meaning ofpersistence data. In much the same way, quantitativepersistence values vary from subject to subject and withexperimental conditions. This being said, it needs also tobe noted that no alternative explanation of persistenceproperties has explained as wide a range of data,provided the level of detail implemented herein, orattributed these properties to fundamental designconstraints on the dynamic balance that regulatesperceptual resonance and reset.

Given that the entire structure and dynamics of themodel had previously been derived and tested on otherdata than persistence data, the model's ability to simulatethe important functional properties of persistence datalends even greater support to the neural reality of thesemodel mechanisms. The ability of this same small set ofmodel mechanisms to explain data from several percep-tual and neural paradigms also provides conceptuallinkages across paradigms whereby new types of exper-iments can be designed to further test these mechanisms,

In summary we have shown how an analysis of theBCS cortical model can offer new mechanistic andfunctional explanations of data on visual persistence,These explanations are consistent with the theory'sprevious explanations of boundary completion(Grossberg & Mingolla, 1985a), texture segregation(Grossberg & Mingolla, 1985b), shape-from-shading(Grossberg & Mingolla, 1987), and three-dimensionalvision (Grossberg, 1987b, 1994), among others, whileextending its explanatory range still further into thedifficult temporal phenomena of visual persistence, Thefunctional role of the feature binding and reset mechan-isms that the model predicts to be responsible for persist-ence suggests links between persistence and fundamentalissues in the formation and breakup of perceptual group-ings in a dynamically fluctuating environment. Thus, farfrom being esoteric laboratory phenomena, data onpersistence afford important clues about some of themost fundamental processes of preattentive vision,processes moreover that are being increasingly wellcharacterized by neural network architectures.

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DiLollo, V. & Hogben, J.. (1987). Suppression of visible persistence asa function of spatial separation between inducing stimuli. Perception& Psychoph}'sics, 4/, 345-354.

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Reynolds, R. (1981). Perception of illusory contours as a function ofprocessing time. Perception, 10, 107-115.

Sakitt, B. & Long, G. (1979). Cones determine subjective offset of astimulus but rods determine total persistence. Vision Research, 19,1439-1441.

Acknow/edgements- This material is based upon work during whichG.F. was supported under a National Science Foundation Graduate

Fellowship, the Air Force Office of Scientific Research (AFOSR90-0175) and the Office of Naval Research (ONR NOOOI4-91-J-4100).S.G.

was partially supported by the Air Force Office of ScientificResearch (AFOSR 90-0175), the Defence Research Projects Agency(AFOSR 90-0083), and the Office of Naval Research (ONR NOOOI4-91-J-4100).

E.M. was partially supported by the Air Force Office ofScientific Research (AFOSR 90-0175). Thanks to Diana J. Meyers forher valuable assistance in the preparation of the manuscript.

Farrell, J., Pavel, M. & Sperling, G. (1990). The visible persistence ofstimuli in stroboscopic motion. Vision Research, 30, 921-936.

Gove, A., Grossberg, S. & Mingolla, E. (1993). Brightness perception,illusory contours, and corticogeniculate feedback. Technical ReportCAS/CNS-TR-93-02I, Boston University, Mass. In Proceedings ofthe world congress on neural networks. Hillsdale, N.J.: Erlbaum.(Vol. I, pp. 25-28).

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Grossberg, S. (1987b). Cortical dynamics of three-dimensional form,color, and brightness perception: II. Binocular theory. Perception &Psychophysics, 41, 117-158.

Grossberg, S. (1991). Why do parallel cortical systems exist for theperception of static form and moving form? Perception & Psycho-physics, 49, 117-141.

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Grossberg, S. & Mingolla, E. (1985b). Neural dynamics of perceptualgrouping: Textures, boundaries, and emergent segmentations.Perception & Psychophysics, 38, 141-171.

Grossberg, S. & Mingolla, E. (1987). Neural dynamics of surfaceperception: Boundary webs, iIIuminants, and shape-from-shading.Computer Vision, Graphics, & Image Processing 37, 116-165.

Grossberg, S. & Mingolla, E. (1993). Neural dynamics of motionperception: Direction fields, apertures, and resonant grouping.Perception & Psychophysics, 53, 243-278.

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Grossberg, S., Mingolla, E. & Todorovic, D. (1989). A neural networkarchitecture for preattentive vision. IEEE Transactions on Biomedi-cal Engineering, 36, 65-84.

Grossberg, S., Mingolla, E. & Williamson, J. (1993) Processing ofsynthetic aperture radar images by a multiscale boundary contoursystem and feature contour system. Technical Report CAS/CNS-TR-93-024, Boston University, Mass. In Proceedings of the worldcongress on neural networks (Vol. III, pp. 785-788). Hillsdale, N.J.:Erlbaum.

Haber, R. & Standing. L. (1970). Direct measures of short-term visualstorage. Quarterly Journal of Experimental Psychology, 21, 43-54.

von der Heydt, R., Peterhans, E. & Baumgartner, G. (1984). Illusorycontours and cortical neuron responses. Science, 224, 1260-1262.

Hubel, D. & Wiesel, T. (1965). Receptive fields and functionalarchitecture in two nonstriate visual areas (18 and 19) of the cat.Journal of Neurophysiology, 27, 229-289.

Lesher, G. & Mingolla, E. (1993). The role of edges and line-ends inillusory contour formation. Vision Research, 33, 2253-2270.

Long, G. & Gildea, T. (1981). Latency for the perceived offset of brieftarget gratings. Vision Research, 21, 1395-1399.

Long, G. & McCarthy, P. (1982). Target energy effects on Type 1 andType II visual persistence. Bulletin of the Psychonomic Society, 19,219-221.

Meyer, G. & Maguire, C. (1981). Effects of spatial-frequency specificadaptation and target duration on visual persistence. Journal ofExperinlental Psychology: Human Perception and Performance, 7,151-156.

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APPENDIXNetwork Equations and Parameters

The simulations of the model are a simplification of the system'sinteractions described elsewhere (Grossberg & Mingolla. 1985a, b,1987). These simplifications were necessitated by the following fact.Simulations in the prior reports of the model concerned the spatialinteractions of the segmentation system and so had no need toimplement the temporal dynamics of resonance and reset, whosejuxtaposition of fast and slow time scales greatly increased thecomputational load in simulations. To manage this load. theexperimental stimuli were also simplified. as indicated in the text.without losing their essential characteristics. Thus. the quantitativematches of simulations to data are less significant in this paper thanthe analysis of how model mechanisms work together to explain thecomplex qualitative pattern of persistence data. These qualitativeproperties are, moreover, robust. In much the same way. differentpsychophysical studies of persistence do not produce identical quanti-tative values across observers, but do show consistent relationshipsbetween persistence and stimulus properties. The psychophysical stud-ies of persistence also do not give absolute measures. but provide onlyrelative values that can be ordered within an experimental paradigm.

The model contains a total of eight levels of model neurons ortransmitter gates. Each level, except for the first. consists of twoparallel pathways coding horizontal and vertical orientations at eachpixel location. Within the second level. each orientation-specificpathway contains two simple cells responsive to opposite polarities ofluminance gradients. Thus. associated with each pixel point of theimage are seventeen different cells and transmitter gates. Since wecarried out all simulations on a 40 x 40 pixel array, the simulatednetwork contained 27,200 cells and transmitter gates. A differentialequation describes the behavior of each cell and gate.

We eased the computational requirements of integrating 27.200differential equations with a number of simplifications. First, ratherthan integrate all the differential equations explicitly, we algebraicallycomputed some cell activities at the equilibrium value of the differentialequation. We made this simplification for the unoriented on-center,off-surround cells in Levell. the oriented polarity-specific simple cellsof Level 2, and the hypercomplex cells undergoing lateral inhibition inLevel 4. Computing the values of these cells at equilibrium makes theassumption that they operate on a faster time scale than the other cells.Since the rate-limiting time scales of the simulations involve dynamicsof habituative transmitter gates and feedback within the BCS, unfold-ing the dynamics of these cells would not adversely affect our

1102 GREGORY FRANCIS el 01.

decreasing for inputs further away from the oriental center-line of thein-field the parameter l' controls the rate of fall off. Then a simple cellthat is selectively responsive to a bright-to-dark luminance gradientobeys the differential equation

dx 2BD", 2BD +

d/= -X;" +[Fi,,-Gi,,) .(AS)

where [PJ+ = max(p. 0). A cell responsive to a dark-to-bright lumi-nance gradient obeys the equation

dx 2DBi" WB +-= -X". +[G,.-F".) .(A9)

dl I,. f,. ,.

To save computation. the activities of these cells were computed atequilibrium as:

X7JD = [Fijk -Gijk]+' (AIO)and

explanations of visual persistence but would greatly increase thecomputation required to simulate the model.

A second simplification took advantage of symmetries in the imageplane. If the images presented to the system on the left and right sidesof the plane are mirror images, then activities produced on the righthalf will have corresponding equal values on the left. Thus, to savecomputation, we could compute only the values on the right side ofthe image plane and extrapolate the results to account for the fullimage plane. Similarly, when the top and bottom quadrants of the rightside are mirror images, we only needed to compute the cell responseson the bottom quadrant. Finally, if the upper and lower diagonals inthe quadrant are mirror images, then the values of the horizontal cellsin the lower (upper) diagonal equal the values of the vertical cells inthe upper (lower) diagonal. Thus, by restricting the input stimuli to besquare-shaped, we only had to integrate the differential equations forthe horizontal cells in the bottom right of the image plane to accountfor the behavior of the entire system.

These simplifications reduced the system of 27,200 differentialequations to only 1600 differential equations. The LSODA integratorroutine (Petzold & Hindmarsh, 1987) performed the integration ofthese equations. We based all the network equations upon thosedescribed in Grossberg and Mingolla (1985b).

X7jB = (Gilt -F,It]+, (All)

Level 3: oriented complex cells

Each cell in Level 3 becomes insensitive to the polarity of constrastby summing the rectified activities of the cells in Level 2 of the samelocation and orientation. Each Level 3 cell obeys the differentialequation

Level 0: image plane

Each pixel has a value associated with retinal luminance. Wedescribe the pixel-luminance values of the different stimuli used in thesimulations below.

Level I: center-surround cellsThe activity Xlj of a Level I cell centered at position (i,j) obeys a

shunting on-center, off-surround equation

dX)j I I ~ I ~-d = -Xij+(A -XiJt;..Bi.b'f/,.-(X;j+ C)t;..Djj,.,I,., (AI)t pq pq

where Ipq is the retinal luminance at position (p, q), A is the maximumactivity of the cell, -C is the minimum activity of the cell, and

B;jpq = B exp( -a -2 log 2[(i -p)2 + (j -q)~ (A2)

D;jpq=D exp[-p-210g2[(i -pf+(j-qr']] (A3)

are excitatory and inhibitory Gaussian weighting functions, respect-ively. The term log 2 means the parameters a and p set the radius oftheir respective Gaussians at half strength. Parameters Band D areconstant scaling terms.

To save computation, the equilibrium response of the differentialequation is found by setting the left hand side of equation (I) equalto zero. The resulting algebraic equation can be solved to find

A L Bijpqlpq -C L DijpqlpqX)j= pq pq. (A4)

1.0 + L(B;jpq + Djjpq)/pqpq

The activities of cells at this level share some key propenies with thosefound in ganglion cells or LGN (Grossberg, 1987a). No off-centeron-surround cells were implemented in our simulations.

Level 2: oriented simple cellsThe following equations define oriented simple cells that are centered

at position (i,j) with preferred orientation k. To create a verticallyoriented input field, or in-field, that is specific to the polarity of theluminance gradient, divide an elongated region into a left half L;j. anda right half Rjj.' Add up the weighted sum of the Leve! I inputs withinthe range of the left side

Fi~ = L EiJpqX~.,eLi..

(AS)

and the right side

G/jk = L E;jpqX~...Hi..

(A6

of the region, with

orientation, the tenD NX'Jr is a feedback signal from the higher levelcell of the same position and orientation, and the termX~jk Lpq Pjjpq(X~ + J)X~ is the inhibitory input from the lower levelcells of the same orientation and nearby spatial positions. Theinhibitory weights fall off in strength as the spatial distance betweencells increases, as in

Pjjpq = P exp[ -cS-2Iog 2[(; -p)2 + U -q)2D, (AI5)

where P scales the strength of the inhibition, and cS controls the spread.E1jpq = exp( -y-21og 2(; -p)2j (A7)

dXl~ 3 2BD WBT=-X,~+H(Xi~ +Xi~ ). (AI2)

Parameter H scales the activities of the input signals to the complexcell.

Let'el 4: habituative transmitter gates

The signal in each oriented pathway is gated, or multiplied, by ahabituative transmitter which obeys the following equation(Grossberg, 1972)

dX.-if' = K[L(M -X~) -(X:~ +J)X1p.). (AI3)

This equation says that the amount of available transmitter X~accumulates to the level M, via term KL(M -X~), and is inactivatedby mass action at rate K(X~" + J)X~, where J is the tonic input ofa gated dipole and X:" is its phasic increment. We always set the rateK much smaller than 1.0 so that these equations operate on a slowertime scale than the equations describing cell activities. At the beginningof each simulation, each transmitter value is set to the non-stimulatedequilibrium value X:" = LM I(L + J).

Level 5: first competitive stage of h}percomple.\" cells

The gated signals of a fixed orientation compete via on-centeroff-surround spatial interactions. Along with the tonic signal comingup through the habituative transmitters, each cell also receives a tonicinput which supports disinhibitory activations at the next competitivestage (see Grossberg & Mingolla, 1985a, b). The activity of a Level 5cell obeys differential equation

dX~~ 5 , .IT = -X1~+J +(X;~ +J)Xi~ + NXi.A

-X~~LPijpq(X~k+J)X~, (AI4)H

where -X~~ models the passive decay, the parameter J establishes anon-zero baseline of activity for the cell, the term (X:~ + J)X1jk is thegated excitatory input from the lower level at the same position and

.

CORTICAL DYNAMICS OF BINDING AND RESET 1103

.... For the simulations in this paper, the differential equation was

solved at equilibrium as

X s J + (Xf" + J)X~ + NX~J;jk= .(AI6)/ 1.0 + L P;jpq(X~+J)X~k

pq

Level 6: second competitive stage of hypercomple.1: cells

The output signals from the first competitive stage compete acrossorientation at each position. The activity of a cell receiving thiscompetition obeys the differential equation

dXfjk 6 5 5d/ = -Xijk + (X;;.- XijK) (A17)

where X~J; and X~K represent orthogonal orientations.

Level 7: cooperative bipole cells and spatial impenetrability

The next level uses a simplified version of bipole cells. As in LevelI, we divide the in-field of each horizontal bipole cell into a left sideL;;. and a right side R;}: (top and bottom for vertically oriented bipolecells). Each bipole cell then sums up excitatory like-oriented signalsand inhibitory orthogonally-oriented signals within each side. Aslower-than-linear bounded function squashes the net signal of eachside. We then set the output threshold of the bipole cells so thatboundaries must stimulate both sides of the receptive field for the cellto generate an output signal. The differential equation describing eachbipole cell activity is

dxl;. 7 fl ~ I 6 )+ I 6 )+)-d = -Xi;, + ,:.. X,.,k -X,.,Kt oR...

+flE. IX~)+ -IX':",,)+) (AI8)

where

(c)300

100

so

0

Level 8: spalial sharpening

Output signals from the bipole cells are thresholded to preventfeedback unless inputs activate both sides. These output signals thenundergo a spatial sharpening much as in the first competitive stage ofLevel 5. The activities of cells in Level 8 obey the differential equation

~ = -x~~ + [Xl~ -R]+ -X~-" L T(X;'k -R]+ (A20)..oS"

where parameter R is the output threshold for bipole cells, parameterTscales the strength of the spatial inhibition, and Sijis the eight nearestneighbors to pixel (i,j). These signals are scaled by parameter N beforefeeding back to the cells in Level 5 to close the feedback loop.

FIGURE AI. Changes in persistence as parameters are modified. Ineach case only one parameter is varied. The large solid diamonds markpersistence with the default parameter value. (a) Parameter N inequation (AI4) scales the strength of the bipole pathway feedbacksignals. Increasing parameter N strengthens the feedback signals togenerate a stronger hysteresis in the network without affecting thestrength of the reset signals. Persistence increases with N. (b) ParameterK in equation (AI3) controls the rate of habituation of the transmittergates. Greater habituation makes stronger reset signals, so persistencedecreases as K increases. (c) Parameter H scales the inputs to thehabituative gates in equation (AI2). Increasing H creates a strongerresonance and stronger reset signals. Starting with small values,increasing H has a larger effect on the hysteresis than on the strengthof the reset signals, thus persistence increases. At larger values,increasing H has a larger effect on the strength of the reset signals than

on the hysteresis, thus persistence decreases.

Computation of images and persistence

We operationally defined the boundaries of an image to be persistingwhenever, after target offset, a cell in Level 6 at the location andorientation of the target image edge (real or illusory) had an activityvalue> 0.5. The computer checked the values every 0.5 time steps afterstimulus offset (I time unit in the simulation is equivalent to 10 msec).For all simulated images a value at each pixel in simulated ft-Lindicated luminance intensity. The background luminance was always10-6 simulated ft-L.

The simulated luminances and durations of the target !lashes forFig. l(b) are indicated in the figure and were all bright squares (26 x 26

pixels) on a dark background.

..

1104 GREGORY FRANCIS tl of.

....

The inducers for the illusory stimuli in Fig. 2(b) were luminanceincrements (pixel values of 0.15 simulated ft-L) in the shape of Lsoriented appropriately in each quadrant to line up the inducer edges.The real stimulus was a bright outline (3 pixels wide) square of thesame luminance and size (32 x 32 pixels) as the illusory square.

We did not simulate the remaining stimuli shown in Fig. 2(a) becausethe simplified BCS used in our simulations creates boundary signalsbetween the inducing stimuli for all of the stimulus sets. It is not thefocus of this paper to show that our simulations accurately createillusory contours. Rather, we investigated the persistence of boundariesgenerated without a luminance edge. Illusory contours are one exampleof these types of boundary signals. A full simulation of the BCS withmore orientations, and whose bipole cell weights are modulated in twodimensions of spatial position and one of orientation, can accuratelypredict the generation of illusory contours (Gove el al., 1993). Such asimulation is beyond the scope of this paper.

The test stimulus for Fig. 3(b) was a pair of horizontally orientedluminance bars (10 x I pixels, pixel values of 0.15 simulated ft-L)separated by 3 pixel spaces. The adaptation stimulus was eitheridentical to the test stimulus or six small vertical lines (4 x I pixelseach) evenly spaced and placed to intersect with the horizontal bars ofthe test stimulus. The adaptation and test stimuli were both presentedfor 100.0 simulated msec.

The target for Fig. 4(b) was a bright square of 20 x 20 pixels witha pixel luminance value of 0.323 simulated ft-L. The mask for Fig. 4(b)consisted of a bar (16 x 2 pixels) along each edge of the target withequal pixel luminance and an edge-to-edge separation from the targetof 3-9 pixel spaces, each pixel space corresponding to 0.05 deg of visualangle. Larger spatial separations could not be used due to the limitedsize of the simulation plane. We always presented the target flash fora duration of 50.0 simulated msec and matched its offset with the onsetof the mask. We kept the masking flash on until the boundaries of thetarget flash fell below threshold.

Parameter' selection

Because integration of nonlinear differential equations is computa-tionally expensive, we simplified the BCS equations as much aspossible. As a result, we could not use the same parameters as othersimulations of the BCS, which calculated the equilibrium response ofthe system (Cruthirds, Gove, Grossberg, Mingolla, Nowak &Williamson, 1992; Gove et al., 1993; Grossberg & Mingolla, I 985a, b,1987; Grossberg, Mingolla, & Williamson, 1993). In particular,whereas the present simulations used only vertically and horizontallyoriented cells, other BCS simulations have used oblique orientations aswell.

To remain consistent with earlier simulations and to explain theproperties of persistence, the parameters used in our simulations wererequired to meet several properties. First, the parameter set had toallow the BCS to locate oriented boundaries. For example, if P is settoo large, then spatial inhibition between cells of like orientation andnearby positions can mutually inhibit activities at the next layer somuch that no signal survives the competition. Similarly, the thresholdfor the bipole cell activities, R, cannot be set too large (relative to theparameter Q and the strength of the inputs to the bipole cells) or thebipole cells will never fire.

A second requirement of the parameter set was that the activities inthe feedback loop, once activated, needed to be strong enough topersist once the external inputs were turned off. The parameters N, Q,R, and T in the bipole feedback pathway control this property of thenetwork. These parameters were set to insure a persisting activity inthe network and proper creation of boundary signals.

Figure Al(a) shows how the strength of activities in the feedbackloop influences persistence. Parameter N scales the strength of thebipole feedback pathway to the lower stages. Increasing N strengthens

the boundaries in the BCS without changing the strength of the resetsignals. Figure AI shows the persistence of a 100 simulated msec,0.323 ft.L stimulus. Figure AI(a) shows that as N increases, persistencealso increases because the hysteresis in the feedback loop is stronger.

A third requirement of the parameter set was that it had to allowgeneration of reset signals upon stimulus offset. This was realized byproperly choosing parameters, J, K, L, and M, of the habituativetransmitter gates. Grossberg (1980, Appendix E) showed that thestrength of the reset signal increases with parameter M and decreasesas J or L are increased. These parameters also establish the lower limit(equilibrium) of the gate strength. If the gate was allowed to habituatetoo much, it would no longer pass on sufficient input to the higherlevels. In such a case, boundaries could disappear before the offset ofthe stimulus. It is easy to find parameters which avoid this problem.Parameter K controls the relative rate of habituation, and increasingK allows for faster habituation and stronger reset signals. Figure AI(b)shows that increasing K decreases persistence.

The final task of parameter setting was to control the value of theinputs to the habituative gates so that they created a strong feedbackloop and generated strong reset signals. The six parameters of uvelI act to compress the cell response to luminous inputs. Equation (I)could have been replaced with a function like 10g(/,J to get similarresults, but we choose equation (I) to remain consistent with othersimulations of the BCS (Gove et al., 1993; Cruthirds et al., 1992). Thisstage of compression explains why the two lower curves of Fig. I havesimilar persistence despite the fact that the stimulus of the lower curveis significantly more luminous. The parameters of uvels 2 and 3 simplyscale the activities of the oriented filters. Increasing the activities ofthese cells has two effects. First, stronger signals create strongeractivities in the feedback loop, which act to increase persistence. At thesame time, these stronger signals increase habituation to generatestronger reset signals upon stimulus offset. The balance of these factorsdetermines persistence. Figure AI(c) shows that as parameter Hincreases from 0.005 to 0.025 persistence increases. This increase inpersistence indicates that the influence of the additional strength givento the activities in the feedback loop is greater than the additionalhabituation caused by the stronger inputs. As H increases still further,the additional habituation, and stronger reset signals, tend to dominatethe increases in boundary signal strength. This same analysis was usedto explain the inverted-U shapes of the curves in Fig. I(b) andFig. 2(b), as a function of stimulus duration.

For our explanations of persistence properties, the only"disallowed" values of parameters are ones that would generate absurdconsequences even outside the domain of visual persistence. Forexample, parameters could be set to prevent bipole cells from perform-ing boundary completion. However, once they are set so as to permitcompletion, illusory contours persist longer than real contours. Thatthe data curves can be explained through an analysis of the modelnetwork architecture shows that the persistence properties of the modelare robust.

The network parameters remained unchanged across all simulations,only the image luminances, durations, or spatiotemporally adjacentstimuli were varied. The following parameters were used: A = 67.5,B =2.5, C=60.0, D =0.05, H=O.I, J =20.0, K=0.0003, L = 3.0,M = 5.0, N = 13.0, P = 0.0005, Q = 0.5, R = 0.61, T = 0.3, « = 0.5,P = 3.0, y = 1.5, IS = 3.0. Each side of the oriented masks in uvel 2,Lip., Rip., were rectangles of 4 x I pixels in size. Each side of a bipolecell was restricted to a single column (vertical) or row (horizontal)extending 18 pixels from the position of the bipolecell. For comparisonpurposes, the dashed lines in Fig. l(b) we computed with K = 0.0; andthe dashed line in Fig. 4(b) was computed with P = 0.0.

All simulations were carried out on an Iris 4/280 or an Iris 8/280Silicon Graphics Superminicomputer. The computation of eachsimulated data point in Figs l(b), 2(b), 3, and 4(b), required approxi-mately half-an-hour on a multi-user machine.