The time course of binocular rivalry revealsa fundamental role of noise

Functional Neurobiology and Helmholtz InstituteUtrecht University Utrecht The NetherlandsJan W Brascamp

Physics of Man and Helmholtz InstituteUtrecht University Utrecht The NetherlandsRaymond van Ee

Functional Neurobiology and Helmholtz InstituteUtrecht University Utrecht The NetherlandsAndre J Noest

Functional Neurobiology and Helmholtz InstituteUtrecht University Utrecht The NetherlandsRichard H A H Jacobs

Functional Neurobiology and Helmholtz InstituteUtrecht University Utrecht The NetherlandsAlbert V van den Berg

When our two eyes view incongruent images we experience binocular rivalry An ongoing cycle of dominance periods ofeither image and transition periods when both are visible Two key forces underlying this process are adaptation of andinhibition between the imagesrsquo neural representations Models based on these factors meet the constraints posed by data ondominance periods but these are not very stringent We extensively studied contrast dependence of dominance and transitiondurations and that of the occurrence of return transitions Occasions when an eye loses and regains dominance without in-tervening dominance of the other eye We found that dominance durations and the incidence of return transitions dependsimilarly on contrast transition durations show a different dependence Regarding dominance durations we show that thewidely accepted rule known as Leveltrsquos second proposition is only valid in a limited contrast range outside this range theopposite of the proposition is true Our data refute current models based solely on adaptation and inhibition as these cannotexplain the long and reversible transitions that we find These features indicate that noise is a crucial force in rivalry frequentlydominating the deterministic forces

Keywords binocular rivalry bistable perception noise visual awareness temporal dynamics Leveltrsquos second propositionneural models

Introduction

When we present two discordant images to both eyesbinocular fusion gives way to binocular rivalry (eg Figure 1)a phenomenon characterized by ongoing perceptual alter-nations between the two images (Alais amp Blake 2005Blake 2001) similar to those occurring when viewingambiguous figures such as the Necker cube Understand-ing such bistable phenomena is important for the study ofvisual awareness Arguably they are an evident demon-stration of fundamental processes underlying awarenessreflecting an ongoing effort of the visual system to selectand reorganize sensory input to form consistent interpre-tations (Andrews amp Purves 1997 Leopold amp Logothetis1999) In addition because of the partial decoupling ofstimulus and percept rivalry is a valuable tool in studiesinto neural correlates of awareness in isolation fromstimulationAccording to current ideas binocular rivalry revolves

around cross-inhibition and slow self-adaptation (eg

Wilson 2005) That is both interpretations of an ambiguousstimulus have a neural representation and these representa-tions inhibit each otherrsquos activity When one representationbecomes stronger than the other its inhibitory force on theother also increases and the situation develops to fullactivity (dominance) for one and to minimal activity(suppression) for the other representation Then slow self-adaptation gradually causes the dominant representationrsquosactivity to decline until a point is reached at which thebalance tips in the opposite direction and so forthModels based on these ideas can account for existing data

on dominance durations (Kalarickal amp Marshall 2000Lehky 1988 Mueller 1990 Stollenwerk amp Bode 2003Wilson 2005) Notably they agree with Leveltrsquos secondproposition In its original form this proposition states thatchanging the strength (contrast) of one eyersquos stimulus whilefixing the other one affects dominance durations only of theeye with the fixed contrast (Fox amp Rasche 1969 Levelt1966) More recent work (Bossink Stalmeier amp De Weert1993 Mueller amp Blake 1989 see also Shiraishi 1977)demonstrated small changes in the eye with the variable

Journal of Vision (2006) 6 1244ndash1256 httpjournalofvisionorg6118 1244

doi 10 1167 6 11 8 Received April 25 2006 published October 26 2006 ISSN 1534-7362 ARVO

contrast as well thus the currently accepted interpretationis that unilateral contrast changes primarily affect domi-nance durations in the fixed-contrast eyeThe good agreement between models and psychophy-

sics is encouraging but somewhat misleading because theconstraints posed by current psychophysical data are notvery rigorous as witnessed by substantial differencesbetween the models cited above It would seem thenthat current data do not allow all too specific inferenceson the rivalry mechanism Fortunately there are indica-tions that rivalryrsquos alternation cycle is more diverse in itsfeatures than is usually taken into consideration andtherefore carries more information on underlying mech-anisms than the handful of rules constraining currentmodelsSpecifically each alternation cycle comprises not only

periods with complete dominance of either percept but alsosubstantial transition periods in which a compound of bothis perceived (Blake OrsquoShea ampMueller 1992 Bossink et al1993 Hollins 1980 Mueller amp Blake 1989 Wilson Blakeamp Lee 2001 Yang Rose amp Blake 1992) A further in-formative complication lies in the fact that although mosttransitions mediate a dominance change from one eye tothe other some end up with dominance returning to thepreviously dominant eye (Mueller amp Blake 1989) Theselatter events which we call return transitions have notyet been studied systematically Finally what we do knowof rivalryrsquos dynamics all comes from studies that mea-sured at a limited set of specific contrast settings (eg fix-ing one eyersquos contrast at a chosen value and varying theother one) and not throughout the range of contrasts that wemay present the eyes withIn this study we aim to get a more complete view of the

dynamics of binocular rivalry including dominance dura-tions transition durations and the occurrence of returntransitions We study these for a matrix of left-eye and right-eye contrast combinations spanning the entire range fromnear the detection threshold to the theoretical maximumWesubsequently test whether current rivalry models can accountfor these dynamicsUsing a conventional orthogonal grating stimulus (Figure 1)

we demonstrate that dominance durations transition dura-tions and the frequency of occurrence of return transitions(fraction of return transitions [FRT]) all systematically

depend on stimulus contrast This dependence is similar fordominance durations and the FRT but different for transi-tion durations Regarding dominance durations we showthat Leveltrsquos second proposition (in its currently acceptedform) is accurate only in a limited contrast range and maybe replaced more generally by the statement that unilateralcontrast changes mainly affect dominance durations of thehigher contrast eye These findings show that the data un-derlying current ideas represent only a fraction of the richbehavior of this system The more complete characteri-zation presented here allows new inferences on the mecha-nism of rivalry and poses more stringent model constraintsIndeed we demonstrate using simulations that existingmodels cannot reproduce our findings Specifically sys-tems based exclusively on cross-inhibition and slow self-adaptation cannot account for the many return transitionsand long transition durations that we find These featuresof the alternation cycle point toward neural noise as anessential driving force underlying rivalry

Methods

Four subjects one author and three naive individualswith normal or corrected-to-normal vision participatedThe stimulus (Figure 1) consisted of sine-wave gratings(65 cyclesdeg) filling a circular patch (r = 031-) withconstant contrast plus a surrounding region in whichintensity fell off following a Gaussian profile (half-width006-) that is a Bsoft stimulus edge[ Average luminanceof both stimulus and background was 15 cdm2 For fusionwe used an alignment ring (r = 1- 50 cdm2) with fourlines extending 027- outward in the cardinal directions anda binocular pattern of open squares (side 034- 50 cdm2)sparsely scattered across the screen from 13- above andbelow the center onward Subjects reported percepts bypressing and holding either of two buttons corresponding toexclusive visibility of either eyersquos image or releasing allkeys in case of a transition Because releasing the keys isalso a natural response when one is uncertain of the percept(eg due to temporally unaligned eyes) subjects wereinstructed to press a third button to indicate such episodesWe sampled a 4 4 matrix spanning the full domain of left-eyeright-eye contrast combinations The four contrastvalues were customized per subject based on theirdetection threshold We therefore determined for eachsubject the contrast yielding 75 correct for gratingspresented monocularly (an otherwise identical stimulus) in atwo-interval two-alternative forced-choice QUEST (Watsonamp Pelli 1983) procedure We verified that this value wassimilar for both eyes and orientations and we used theaverage to determine the contrast range The lowest onewas 075 log10 units Michelson above 75 detectionwhich meant 15Michelson on average for these subjectsthe highest one was 100 Michelson (the theoreticalmaximum) and we interpolated the other two in equal log

Figure 1 Our binocular rivalry stimulus

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1245

steps Contrast conditions were distributed randomly overtrials Sessions consisted of four 5-min experimental trials(see the Appendix for remarks on this duration) and a 2-mincontrol trial each The first minute of each trial was dis-carded Control trials in which contrasts were always 100100 Michelson were compared over sessions to verifysubjectsrsquo constant performance leading a fifth subjectrsquos datato be excluded from analysis Each subject produced an av-erage of about 180 dominance durations per condition pereye and the accompanying transition durations

Results

Figure 2 shows the results for a typical subject (Panel A)and averaged over all four (Panel B) The top and middlecharts show dominance and transition durations the bottomones show the FRT that is the fraction of transition pe-riods after which dominance returned to the previously dom-inant eye instead of crossing over Return transitions werenot included in the calculation of transition durations achoice that did not notably affect any of our conclusionsNone of the subjects showed a significant eye preferencethus data could be pooled over eyes Consequently regard-ing dominance durations contrasts are given for the ipsi-lateral and contralateral eye that is for the eye of whichdurations are plotted and for the other eye respectivelyConcerning transitions we use the terms Bdeparture[ anddestination contrast that is in the eye that was dominantbefore the transition and in the other one respectivelyFigure 2 shows that both dominance and transition du-

rations are on the order of seconds with dominance phasestaking up the most time Furthermore return transitions arealmost absent in some conditions but in other conditionsthey make up as much as about half of all transitions start-ing from a given eye Dominance durations (top) and theFRT (bottom) show similar patterns of contrast depen-dence both being enhanced by a high ipsilateraldeparturecontrast and attenuated by a high contralateraldestinationcontrast In addition both values increase slightly whencontrast is lowered symmetrically in both eyes but this riseis small compared to the off-diagonal effects Transition du-rations (middle) show a different pattern of contrast de-pendence with the roles of departure and destination contrastbeing largely equivalent As a result transition durations arelargest when both contrasts are minimal (cf Hollins 1980)

A restriction to Leveltrsquos second proposition

A hallmark of current ideas on rivalry known as Leveltrsquossecond proposition is the notion that a change in one eyersquoscontrast has a strong effect on the other eyersquos dominancedurations and only a weak effect (or none in the originalformulation) on those of the eye itself This has been shownin experiments in which one eyersquos contrast was fixed while

the other one was varied To test if our data support thisnotion we reconstructed four such experimental regimesfrom our data as shown in Figure 3 The bottom-rightinset depicting a schematic top view of a chart such as inFigure 2 illustrates this Taking a section through thematrix at a given ipsilateral contrast (dashed line egpurple) and combining it with the section at the samecontralateral contrast (solid line same color) yields thedata for one such regime The four reconstructions differ

Figure 2 Dominance durations transition durations and the FRTas a function of the two eyesrsquo contrasts for one subject (A) andaveraged over all four (B) Ipsilateral (ips) and contralateral (cont)refer to the eye whose dominance durations are plotted and to theother eye respectively Departure (dep) and destination (dest)refer to the eye that was dominant before a transition started andthe other eye respectively Contrasts were customized for eachsubject with lsquolsquoMinrsquorsquo meaning near detection threshold and lsquolsquoMaxrsquorsquomeaning 100 Michelson Durations in Panel B were normalizedper subject relative to the dominance duration in the 100100contrast condition (12 09 16 and 09 s for these subjects) henceproportional relations between dominance and transition durationsare preserved The figure shows that dominance and transitiondurations are both on the order of seconds and that the FRT variesbetween about 0 and as much as 05 depending on condition Allthree quantities show a systematic dependence on contrast whichis similar for dominance durations and the FRT but different fortransition durations

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1246

in the level of the fixed contrast (dotted arrows inPanels AndashD) Panels AndashD depict the reconstructions fromour across-subject averages with colors corresponding tothose in the inset Panels A and B show the classic result Asthe contrast in one eye is altered dominance durationschange mainly in the other eye However Panels C andespecially D show that the pattern subsides and actuallyreverses for lower values of the fixed contrast Here themain effect is in the eye in which the contrast is changedThe validity of the proposition therefore depends entirelyon the subsection of the data one considers valid at highvalues of the fixed contrast but invalid at lower ones

Return transitions and dominance durations

Figure 2 showed similar contrast dependencies for domi-nance durations and the FRT Here we examine this simi-larity more closely Figure 4 displays across-subject averagesof the FRT as a function of mean dominance duration in thedeparture eye that is it shows how transitions starting fromdominance for a given eye are related to dominance du-rations of that eye The horizontal extent and the verticalextent of the 16 rhombi symbolize ipsilateral and contra-lateral contrast respectively with smaller values correspond-ing to lower contrast There is a clear positive correlationbetween dominance durations and the FRT as quantifiedby the regression line It turns out that there is no directcausal link between the occurrence of a return transition andthe occurrence of a long-lived percept (see the Appendix)that is one does not cause the other Instead the correla-tion stems from a common dependence on an underlyingvariable strongly influenced by contrast In addition irre-spective of contrast our data showed a tendency for sub-jects with long dominance durations to experience manyreturn transitions strengthening the notion that both quan-tities are connected

Comparison with existing models

We performed simulations (see the Appendix) with threeprevailing models based on adaptation and inhibition whichagree with experiments to date (Kalarickal amp Marshall 2000Stollenwerk amp Bode 2003 Wilson 2005) To verify if these

Figure 3 Leveltrsquos second proposition In its current form this prop-osition states that a change in one eyersquos contrast primarily affectsthe other eyersquos dominance durations rather than those of the eyeitself It is based on experiments where one eyersquos contrast wasfixed and the other one was systematically varied From our datawe took four subsets corresponding to such experimental re-gimes as shown in the bottom-right inset schematically depictingour 4 4 ipsilateralcontralateral contrast matrix Each color de-notes one subset with the solid lines indicating the data pointsfor the variable-contrast eye (ie contralateral contrast fixed) andthe dashed lines those for the fixed-contrast eye (ie ipsilateralcontrast fixed) Panels AndashD show across-subject averages usingthe same colors as in the inset (A and B) At high values of thefixed contrast (dotted arrows) we see the classic pattern (C and D)At lower values however this pattern subsides and reverses Theproposition therefore applies only in a restricted portion of the con-trast domain namely where the fixed contrast is high Outside thisdomain the opposite of the proposition becomes true

Figure 4 Relation between the FRTand mean dominance durationof the departure eye in various contrast conditions The width andheight of the rhombi symbolize ipsilateral and contralateral contrastrespectively Dominance durations and the FRT show a positivelinear correlation as indicated by the regression line p is ANOVAp value F is ANOVA F ratio These data point toward a commonunderlying variable responsible for trends in both departure domi-nance durations and the FRT

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1247

models can reproduce our findings we tested their predic-tions on dominance durations transition durations and theFRT for a matrix of ipsilateralcontralateral input strengthsTo adequately cover transitions we chose two distinct

classes of models each addressing one of the two types oftransitions that may occur (see also the Discussion section)The Wilson model and the Kalarickal and Marshall modelcover superposition transitions during which both imagesare seen superimposed (Burke Alais amp Wenderoth 1999Liu Tyler amp Schor 1992) whereas the Stollenwerk andBode model covers piecemeal transitions during whichparts of both images are seen in complementary regions ofthe stimulus (Blake et al 1992 Lee Blake amp Heeger2005 OrsquoShea Sims amp Govan 1997 Silver amp Logothetis2004 Wilson et al 2001)The models by Wilson and by Kalarickal and Marshall

comprise a single oscillator formed by the two perceptsrsquorepresentations interacting via adaptation and inhibitionPiecemeal transitions are beyond their scope as they haveno spatial dimension We defined superposition phases asthose when none of the representations strongly dominatesbut instead both are intermediately activeThe model by Stollenwerk and Bode involves oscillators

similar to those described above but several of them linkedtogether in a 2D network corresponding to neighboringzones in visual space Neighboring oscillators are coupledsuch that they tend to follow each otherrsquos dominance statesso that once a dominance change emerges in one location apiecemeal transitionmay occurWe defined transition periodsas those during which less than 100 of all oscillators werein the same dominance stateFigures 5AndashC display simulation results at the original

papersrsquo parameter settings From left to right the charts ineach panel show dominance durations transition durationsand the FRT as a function of input strength The white barsin the left charts reproduce the original papersrsquo demonstra-tions of agreement with Leveltrsquos second proposition Theyalso illustrate how we complemented the original inputstrength combinations to form 4 4 matrices Aside fromthe white bars the models show a striking lack of concor-dance underscoring the added constraints posed by thepresent characterization The charts show marked devia-tions from our data First all models predict transitions tobe much shorter than dominance phases rather than of thesame order and also underestimate the FRT Second con-trast dependence of particularly transition durations andthe FRT is not correctly reproducedTo see if model predictions would improve at other pa-

rameter settings we performed simulations at an extensiverange of values confirming the shortcomings discussedabove There was one parameter region for the Kalarickaland Marshall model however (Panel D) where the con-trast dependence of all three variables was in qualitativeagreement with our data although both transition dura-tions and FRT were overestimated Interestingly at theseparameter settings the system is in a mode entirely differ-ent from the one that was originally intended (see the

Figure 5 Validation of three existing models The left charts of eachpanel show simulated dominance durations as a function ofipsilateral (ips) and contralateral (cont) input strength the middleand right ones show transition durations and the FRTas a functionof departure (dep) and destination (dest) input strength (theWilson model produces no return transitions as it is noise-free)Durations are given in seconds for theWilsonmodel and in arbitraryunits (au) for the remaining ones Panels AndashC show results at theoriginal parameter settings with the white bars reproducing theoriginal demonstrations of agreement with Leveltrsquos second propo-sition Although all three models support the proposition (for a lim-ited range of inputs) their behavior diverges at most other pointsIn addition there are marked deviations from our data First themodels incorrectly predict at least an order difference betweendominance and transition durations and underestimate the FRTSecond none of the models reproduces the found patterns ofcontrast dependence of transition durations and the FRT (D) Insimulations at other parameter settings we found one parameterregion for the Kalarickal and Marshall model where the resultswere reminiscent of our data Here contrast dependencies of allthree variables were qualitatively correct but were accompaniedby an overestimation of both transition durations and the FRT Fur-ther analysis (see the Appendix) shows that in this region modeldynamics are essentially stochastic Deterministic forces keep thesystem in a state intermediate between both dominance statesand it is noise that causes incidental excursions into either domi-nance percept

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1248

Appendix) Its deterministic dynamics have settled at inter-mediate activities for both representations and it is purelynoise that causes the oscillations manifested in the figure

Discussion

Noise and rivalry dynamics

Our most striking finding is that current ideas on thesystem underlying rivalry do not agree with the observeddynamics Broadly speaking we show that there is muchmore to the alternation cycle than current models can ex-plain and no single adjustment will likely bridge this gapMore specifically however there is one aspect whereinthe deviation between models and data is particularly strongand which to us indicates one necessary model additionThis aspect is the behavior of transitions These are pres-ently treated as brief and irreversible switches betweentwo dominance periods whereas the data show that theytake considerable time and that return transitions are com-mon We will argue that the features of transitions in cur-rent models follow from an overemphasis on the role ofslow adaptation and that our results point to a system inwhich part of this role is taken over by stochastic varia-tions (ie noise) in the system componentsThis is illustrated in Figure 6 schematically showing

perceptual dynamics of rivalry as motions across energylandscapes (eg Billock amp Tsou 2003) The system state(ball) always develops toward lower energy thus minimain the landscapes are fixed points (attractors) correspond-ing to left-eye and right-eye dominance Changes in thelandscapes are due to changes in adaptation state tem-porarily modifying the relative strength of the attractorsSequence AYBYC-I shows the situation as many currentmodels treat it A The ball occupies the left attractor in-dicating for example left-eye dominance AYBYC-ISlow adaptation destabilizes the occupied attractor (blackarrow) while recovery from adaptation deepens the otherone (white arrow) until the left attractor disappears (C-I)causing the ball to move (dashed arrow) through the tran-sition region (gray) to the remaining attractor This tran-sition takes place quickly and not until considerable timehas passed allowing the left attractor to reappear may thesystem return to its previous state This separation of time-scales between landscape changes and ball movement isnecessary for these models to oscillate yet it is clear thatthe long transition durations and particularly the existenceof return transitions corresponding to the ball moving half-way between both attractors and then returning argueagainst this type of system In Panels C-II and C-III weshow two (not mutually exclusive) alternatives to Panel C-Ithat do agree with our data C-II In this scenario transi-tions are not initiated by destruction of the occupied at-tractor but by noise (curved arrow) tossing the system into

the transition region near the maximum that separates theleft and right domain of attraction Here the slope isshallow that is deterministic forces are weak and may bepositive or negative depending on the actual system stateThis may explain both the long duration of transitions andthe frequent occurrence of returns C-III Here the slope inthe transition region is so low (that is the deterministicdynamics here are so slow) that the attractor may reappearbefore the transition is over Again this is in line with longtransitions and provides an opportunity for noise to tip thesystem over in either direction We think that this latterscenario is particularly likely at lower contrasts (see below)Both scenarios point to a strong stochastic term as a keyingredient missing from present thinking

Figure 6 Schematic representation of rivalryrsquos dynamics ascurrently understood (AYBYC-I) and as our data imply(AYBYC-II and AYBYC-III) Dynamical modes of the systemare shown as energy landscapes The twominima are stable states(attractors) corresponding to left-eye and right-eye dominance (A)The ball occupies the left attractor indicating left-eye dominance(AYBYC-I) Adaptation (black arrow) destabilizes the occupiedattractor as recovery from adaptation (white arrow) deepens theother one Panel C-I shows transitions as treated by current mod-els As the occupied attractor disappears due to adaptation thesystem moves to the remaining one (dashed arrow) It cannot re-turn to the left attractor until (recovery from) adaptation has changedthe landscape to the mirror image of Panel C-I This scenario isincompatible with the slow and reversible transitions we observewhich point toward scenarios such as sketched in Panels C-II andC-III (C-II) Transitions are initiated by noise (curved arrow) driv-ing the system out of the still present attractor to a location in thetransition region (gray) near the separation between both attractordomains Here deterministic forces are small and the system maydevelop in either direction (C-III) The system remains in the at-tractor until it is gone but the deterministic forces in the transitionregion are so small that the attractor may reappear before thetransition is over Again noise may tip the system to the left or tothe right Both scenarios imply a crucial role for noise

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1249

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

steps Contrast conditions were distributed randomly overtrials Sessions consisted of four 5-min experimental trials(see the Appendix for remarks on this duration) and a 2-mincontrol trial each The first minute of each trial was dis-carded Control trials in which contrasts were always 100100 Michelson were compared over sessions to verifysubjectsrsquo constant performance leading a fifth subjectrsquos datato be excluded from analysis Each subject produced an av-erage of about 180 dominance durations per condition pereye and the accompanying transition durations

Results

Figure 2 shows the results for a typical subject (Panel A)and averaged over all four (Panel B) The top and middlecharts show dominance and transition durations the bottomones show the FRT that is the fraction of transition pe-riods after which dominance returned to the previously dom-inant eye instead of crossing over Return transitions werenot included in the calculation of transition durations achoice that did not notably affect any of our conclusionsNone of the subjects showed a significant eye preferencethus data could be pooled over eyes Consequently regard-ing dominance durations contrasts are given for the ipsi-lateral and contralateral eye that is for the eye of whichdurations are plotted and for the other eye respectivelyConcerning transitions we use the terms Bdeparture[ anddestination contrast that is in the eye that was dominantbefore the transition and in the other one respectivelyFigure 2 shows that both dominance and transition du-

rations are on the order of seconds with dominance phasestaking up the most time Furthermore return transitions arealmost absent in some conditions but in other conditionsthey make up as much as about half of all transitions start-ing from a given eye Dominance durations (top) and theFRT (bottom) show similar patterns of contrast depen-dence both being enhanced by a high ipsilateraldeparturecontrast and attenuated by a high contralateraldestinationcontrast In addition both values increase slightly whencontrast is lowered symmetrically in both eyes but this riseis small compared to the off-diagonal effects Transition du-rations (middle) show a different pattern of contrast de-pendence with the roles of departure and destination contrastbeing largely equivalent As a result transition durations arelargest when both contrasts are minimal (cf Hollins 1980)

A restriction to Leveltrsquos second proposition

A hallmark of current ideas on rivalry known as Leveltrsquossecond proposition is the notion that a change in one eyersquoscontrast has a strong effect on the other eyersquos dominancedurations and only a weak effect (or none in the originalformulation) on those of the eye itself This has been shownin experiments in which one eyersquos contrast was fixed while

the other one was varied To test if our data support thisnotion we reconstructed four such experimental regimesfrom our data as shown in Figure 3 The bottom-rightinset depicting a schematic top view of a chart such as inFigure 2 illustrates this Taking a section through thematrix at a given ipsilateral contrast (dashed line egpurple) and combining it with the section at the samecontralateral contrast (solid line same color) yields thedata for one such regime The four reconstructions differ

Figure 2 Dominance durations transition durations and the FRTas a function of the two eyesrsquo contrasts for one subject (A) andaveraged over all four (B) Ipsilateral (ips) and contralateral (cont)refer to the eye whose dominance durations are plotted and to theother eye respectively Departure (dep) and destination (dest)refer to the eye that was dominant before a transition started andthe other eye respectively Contrasts were customized for eachsubject with lsquolsquoMinrsquorsquo meaning near detection threshold and lsquolsquoMaxrsquorsquomeaning 100 Michelson Durations in Panel B were normalizedper subject relative to the dominance duration in the 100100contrast condition (12 09 16 and 09 s for these subjects) henceproportional relations between dominance and transition durationsare preserved The figure shows that dominance and transitiondurations are both on the order of seconds and that the FRT variesbetween about 0 and as much as 05 depending on condition Allthree quantities show a systematic dependence on contrast whichis similar for dominance durations and the FRT but different fortransition durations

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1246

in the level of the fixed contrast (dotted arrows inPanels AndashD) Panels AndashD depict the reconstructions fromour across-subject averages with colors corresponding tothose in the inset Panels A and B show the classic result Asthe contrast in one eye is altered dominance durationschange mainly in the other eye However Panels C andespecially D show that the pattern subsides and actuallyreverses for lower values of the fixed contrast Here themain effect is in the eye in which the contrast is changedThe validity of the proposition therefore depends entirelyon the subsection of the data one considers valid at highvalues of the fixed contrast but invalid at lower ones

Return transitions and dominance durations

Figure 2 showed similar contrast dependencies for domi-nance durations and the FRT Here we examine this simi-larity more closely Figure 4 displays across-subject averagesof the FRT as a function of mean dominance duration in thedeparture eye that is it shows how transitions starting fromdominance for a given eye are related to dominance du-rations of that eye The horizontal extent and the verticalextent of the 16 rhombi symbolize ipsilateral and contra-lateral contrast respectively with smaller values correspond-ing to lower contrast There is a clear positive correlationbetween dominance durations and the FRT as quantifiedby the regression line It turns out that there is no directcausal link between the occurrence of a return transition andthe occurrence of a long-lived percept (see the Appendix)that is one does not cause the other Instead the correla-tion stems from a common dependence on an underlyingvariable strongly influenced by contrast In addition irre-spective of contrast our data showed a tendency for sub-jects with long dominance durations to experience manyreturn transitions strengthening the notion that both quan-tities are connected

Comparison with existing models

We performed simulations (see the Appendix) with threeprevailing models based on adaptation and inhibition whichagree with experiments to date (Kalarickal amp Marshall 2000Stollenwerk amp Bode 2003 Wilson 2005) To verify if these

Figure 3 Leveltrsquos second proposition In its current form this prop-osition states that a change in one eyersquos contrast primarily affectsthe other eyersquos dominance durations rather than those of the eyeitself It is based on experiments where one eyersquos contrast wasfixed and the other one was systematically varied From our datawe took four subsets corresponding to such experimental re-gimes as shown in the bottom-right inset schematically depictingour 4 4 ipsilateralcontralateral contrast matrix Each color de-notes one subset with the solid lines indicating the data pointsfor the variable-contrast eye (ie contralateral contrast fixed) andthe dashed lines those for the fixed-contrast eye (ie ipsilateralcontrast fixed) Panels AndashD show across-subject averages usingthe same colors as in the inset (A and B) At high values of thefixed contrast (dotted arrows) we see the classic pattern (C and D)At lower values however this pattern subsides and reverses Theproposition therefore applies only in a restricted portion of the con-trast domain namely where the fixed contrast is high Outside thisdomain the opposite of the proposition becomes true

Figure 4 Relation between the FRTand mean dominance durationof the departure eye in various contrast conditions The width andheight of the rhombi symbolize ipsilateral and contralateral contrastrespectively Dominance durations and the FRT show a positivelinear correlation as indicated by the regression line p is ANOVAp value F is ANOVA F ratio These data point toward a commonunderlying variable responsible for trends in both departure domi-nance durations and the FRT

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1247

models can reproduce our findings we tested their predic-tions on dominance durations transition durations and theFRT for a matrix of ipsilateralcontralateral input strengthsTo adequately cover transitions we chose two distinct

classes of models each addressing one of the two types oftransitions that may occur (see also the Discussion section)The Wilson model and the Kalarickal and Marshall modelcover superposition transitions during which both imagesare seen superimposed (Burke Alais amp Wenderoth 1999Liu Tyler amp Schor 1992) whereas the Stollenwerk andBode model covers piecemeal transitions during whichparts of both images are seen in complementary regions ofthe stimulus (Blake et al 1992 Lee Blake amp Heeger2005 OrsquoShea Sims amp Govan 1997 Silver amp Logothetis2004 Wilson et al 2001)The models by Wilson and by Kalarickal and Marshall

comprise a single oscillator formed by the two perceptsrsquorepresentations interacting via adaptation and inhibitionPiecemeal transitions are beyond their scope as they haveno spatial dimension We defined superposition phases asthose when none of the representations strongly dominatesbut instead both are intermediately activeThe model by Stollenwerk and Bode involves oscillators

similar to those described above but several of them linkedtogether in a 2D network corresponding to neighboringzones in visual space Neighboring oscillators are coupledsuch that they tend to follow each otherrsquos dominance statesso that once a dominance change emerges in one location apiecemeal transitionmay occurWe defined transition periodsas those during which less than 100 of all oscillators werein the same dominance stateFigures 5AndashC display simulation results at the original

papersrsquo parameter settings From left to right the charts ineach panel show dominance durations transition durationsand the FRT as a function of input strength The white barsin the left charts reproduce the original papersrsquo demonstra-tions of agreement with Leveltrsquos second proposition Theyalso illustrate how we complemented the original inputstrength combinations to form 4 4 matrices Aside fromthe white bars the models show a striking lack of concor-dance underscoring the added constraints posed by thepresent characterization The charts show marked devia-tions from our data First all models predict transitions tobe much shorter than dominance phases rather than of thesame order and also underestimate the FRT Second con-trast dependence of particularly transition durations andthe FRT is not correctly reproducedTo see if model predictions would improve at other pa-

rameter settings we performed simulations at an extensiverange of values confirming the shortcomings discussedabove There was one parameter region for the Kalarickaland Marshall model however (Panel D) where the con-trast dependence of all three variables was in qualitativeagreement with our data although both transition dura-tions and FRT were overestimated Interestingly at theseparameter settings the system is in a mode entirely differ-ent from the one that was originally intended (see the

Figure 5 Validation of three existing models The left charts of eachpanel show simulated dominance durations as a function ofipsilateral (ips) and contralateral (cont) input strength the middleand right ones show transition durations and the FRTas a functionof departure (dep) and destination (dest) input strength (theWilson model produces no return transitions as it is noise-free)Durations are given in seconds for theWilsonmodel and in arbitraryunits (au) for the remaining ones Panels AndashC show results at theoriginal parameter settings with the white bars reproducing theoriginal demonstrations of agreement with Leveltrsquos second propo-sition Although all three models support the proposition (for a lim-ited range of inputs) their behavior diverges at most other pointsIn addition there are marked deviations from our data First themodels incorrectly predict at least an order difference betweendominance and transition durations and underestimate the FRTSecond none of the models reproduces the found patterns ofcontrast dependence of transition durations and the FRT (D) Insimulations at other parameter settings we found one parameterregion for the Kalarickal and Marshall model where the resultswere reminiscent of our data Here contrast dependencies of allthree variables were qualitatively correct but were accompaniedby an overestimation of both transition durations and the FRT Fur-ther analysis (see the Appendix) shows that in this region modeldynamics are essentially stochastic Deterministic forces keep thesystem in a state intermediate between both dominance statesand it is noise that causes incidental excursions into either domi-nance percept

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1248

Appendix) Its deterministic dynamics have settled at inter-mediate activities for both representations and it is purelynoise that causes the oscillations manifested in the figure

Discussion

Noise and rivalry dynamics

Our most striking finding is that current ideas on thesystem underlying rivalry do not agree with the observeddynamics Broadly speaking we show that there is muchmore to the alternation cycle than current models can ex-plain and no single adjustment will likely bridge this gapMore specifically however there is one aspect whereinthe deviation between models and data is particularly strongand which to us indicates one necessary model additionThis aspect is the behavior of transitions These are pres-ently treated as brief and irreversible switches betweentwo dominance periods whereas the data show that theytake considerable time and that return transitions are com-mon We will argue that the features of transitions in cur-rent models follow from an overemphasis on the role ofslow adaptation and that our results point to a system inwhich part of this role is taken over by stochastic varia-tions (ie noise) in the system componentsThis is illustrated in Figure 6 schematically showing

perceptual dynamics of rivalry as motions across energylandscapes (eg Billock amp Tsou 2003) The system state(ball) always develops toward lower energy thus minimain the landscapes are fixed points (attractors) correspond-ing to left-eye and right-eye dominance Changes in thelandscapes are due to changes in adaptation state tem-porarily modifying the relative strength of the attractorsSequence AYBYC-I shows the situation as many currentmodels treat it A The ball occupies the left attractor in-dicating for example left-eye dominance AYBYC-ISlow adaptation destabilizes the occupied attractor (blackarrow) while recovery from adaptation deepens the otherone (white arrow) until the left attractor disappears (C-I)causing the ball to move (dashed arrow) through the tran-sition region (gray) to the remaining attractor This tran-sition takes place quickly and not until considerable timehas passed allowing the left attractor to reappear may thesystem return to its previous state This separation of time-scales between landscape changes and ball movement isnecessary for these models to oscillate yet it is clear thatthe long transition durations and particularly the existenceof return transitions corresponding to the ball moving half-way between both attractors and then returning argueagainst this type of system In Panels C-II and C-III weshow two (not mutually exclusive) alternatives to Panel C-Ithat do agree with our data C-II In this scenario transi-tions are not initiated by destruction of the occupied at-tractor but by noise (curved arrow) tossing the system into

the transition region near the maximum that separates theleft and right domain of attraction Here the slope isshallow that is deterministic forces are weak and may bepositive or negative depending on the actual system stateThis may explain both the long duration of transitions andthe frequent occurrence of returns C-III Here the slope inthe transition region is so low (that is the deterministicdynamics here are so slow) that the attractor may reappearbefore the transition is over Again this is in line with longtransitions and provides an opportunity for noise to tip thesystem over in either direction We think that this latterscenario is particularly likely at lower contrasts (see below)Both scenarios point to a strong stochastic term as a keyingredient missing from present thinking

Figure 6 Schematic representation of rivalryrsquos dynamics ascurrently understood (AYBYC-I) and as our data imply(AYBYC-II and AYBYC-III) Dynamical modes of the systemare shown as energy landscapes The twominima are stable states(attractors) corresponding to left-eye and right-eye dominance (A)The ball occupies the left attractor indicating left-eye dominance(AYBYC-I) Adaptation (black arrow) destabilizes the occupiedattractor as recovery from adaptation (white arrow) deepens theother one Panel C-I shows transitions as treated by current mod-els As the occupied attractor disappears due to adaptation thesystem moves to the remaining one (dashed arrow) It cannot re-turn to the left attractor until (recovery from) adaptation has changedthe landscape to the mirror image of Panel C-I This scenario isincompatible with the slow and reversible transitions we observewhich point toward scenarios such as sketched in Panels C-II andC-III (C-II) Transitions are initiated by noise (curved arrow) driv-ing the system out of the still present attractor to a location in thetransition region (gray) near the separation between both attractordomains Here deterministic forces are small and the system maydevelop in either direction (C-III) The system remains in the at-tractor until it is gone but the deterministic forces in the transitionregion are so small that the attractor may reappear before thetransition is over Again noise may tip the system to the left or tothe right Both scenarios imply a crucial role for noise

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1249

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

in the level of the fixed contrast (dotted arrows inPanels AndashD) Panels AndashD depict the reconstructions fromour across-subject averages with colors corresponding tothose in the inset Panels A and B show the classic result Asthe contrast in one eye is altered dominance durationschange mainly in the other eye However Panels C andespecially D show that the pattern subsides and actuallyreverses for lower values of the fixed contrast Here themain effect is in the eye in which the contrast is changedThe validity of the proposition therefore depends entirelyon the subsection of the data one considers valid at highvalues of the fixed contrast but invalid at lower ones

Return transitions and dominance durations

Figure 2 showed similar contrast dependencies for domi-nance durations and the FRT Here we examine this simi-larity more closely Figure 4 displays across-subject averagesof the FRT as a function of mean dominance duration in thedeparture eye that is it shows how transitions starting fromdominance for a given eye are related to dominance du-rations of that eye The horizontal extent and the verticalextent of the 16 rhombi symbolize ipsilateral and contra-lateral contrast respectively with smaller values correspond-ing to lower contrast There is a clear positive correlationbetween dominance durations and the FRT as quantifiedby the regression line It turns out that there is no directcausal link between the occurrence of a return transition andthe occurrence of a long-lived percept (see the Appendix)that is one does not cause the other Instead the correla-tion stems from a common dependence on an underlyingvariable strongly influenced by contrast In addition irre-spective of contrast our data showed a tendency for sub-jects with long dominance durations to experience manyreturn transitions strengthening the notion that both quan-tities are connected

Comparison with existing models

We performed simulations (see the Appendix) with threeprevailing models based on adaptation and inhibition whichagree with experiments to date (Kalarickal amp Marshall 2000Stollenwerk amp Bode 2003 Wilson 2005) To verify if these

Figure 3 Leveltrsquos second proposition In its current form this prop-osition states that a change in one eyersquos contrast primarily affectsthe other eyersquos dominance durations rather than those of the eyeitself It is based on experiments where one eyersquos contrast wasfixed and the other one was systematically varied From our datawe took four subsets corresponding to such experimental re-gimes as shown in the bottom-right inset schematically depictingour 4 4 ipsilateralcontralateral contrast matrix Each color de-notes one subset with the solid lines indicating the data pointsfor the variable-contrast eye (ie contralateral contrast fixed) andthe dashed lines those for the fixed-contrast eye (ie ipsilateralcontrast fixed) Panels AndashD show across-subject averages usingthe same colors as in the inset (A and B) At high values of thefixed contrast (dotted arrows) we see the classic pattern (C and D)At lower values however this pattern subsides and reverses Theproposition therefore applies only in a restricted portion of the con-trast domain namely where the fixed contrast is high Outside thisdomain the opposite of the proposition becomes true

Figure 4 Relation between the FRTand mean dominance durationof the departure eye in various contrast conditions The width andheight of the rhombi symbolize ipsilateral and contralateral contrastrespectively Dominance durations and the FRT show a positivelinear correlation as indicated by the regression line p is ANOVAp value F is ANOVA F ratio These data point toward a commonunderlying variable responsible for trends in both departure domi-nance durations and the FRT

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1247

models can reproduce our findings we tested their predic-tions on dominance durations transition durations and theFRT for a matrix of ipsilateralcontralateral input strengthsTo adequately cover transitions we chose two distinct

classes of models each addressing one of the two types oftransitions that may occur (see also the Discussion section)The Wilson model and the Kalarickal and Marshall modelcover superposition transitions during which both imagesare seen superimposed (Burke Alais amp Wenderoth 1999Liu Tyler amp Schor 1992) whereas the Stollenwerk andBode model covers piecemeal transitions during whichparts of both images are seen in complementary regions ofthe stimulus (Blake et al 1992 Lee Blake amp Heeger2005 OrsquoShea Sims amp Govan 1997 Silver amp Logothetis2004 Wilson et al 2001)The models by Wilson and by Kalarickal and Marshall

comprise a single oscillator formed by the two perceptsrsquorepresentations interacting via adaptation and inhibitionPiecemeal transitions are beyond their scope as they haveno spatial dimension We defined superposition phases asthose when none of the representations strongly dominatesbut instead both are intermediately activeThe model by Stollenwerk and Bode involves oscillators

similar to those described above but several of them linkedtogether in a 2D network corresponding to neighboringzones in visual space Neighboring oscillators are coupledsuch that they tend to follow each otherrsquos dominance statesso that once a dominance change emerges in one location apiecemeal transitionmay occurWe defined transition periodsas those during which less than 100 of all oscillators werein the same dominance stateFigures 5AndashC display simulation results at the original

papersrsquo parameter settings From left to right the charts ineach panel show dominance durations transition durationsand the FRT as a function of input strength The white barsin the left charts reproduce the original papersrsquo demonstra-tions of agreement with Leveltrsquos second proposition Theyalso illustrate how we complemented the original inputstrength combinations to form 4 4 matrices Aside fromthe white bars the models show a striking lack of concor-dance underscoring the added constraints posed by thepresent characterization The charts show marked devia-tions from our data First all models predict transitions tobe much shorter than dominance phases rather than of thesame order and also underestimate the FRT Second con-trast dependence of particularly transition durations andthe FRT is not correctly reproducedTo see if model predictions would improve at other pa-

rameter settings we performed simulations at an extensiverange of values confirming the shortcomings discussedabove There was one parameter region for the Kalarickaland Marshall model however (Panel D) where the con-trast dependence of all three variables was in qualitativeagreement with our data although both transition dura-tions and FRT were overestimated Interestingly at theseparameter settings the system is in a mode entirely differ-ent from the one that was originally intended (see the

Figure 5 Validation of three existing models The left charts of eachpanel show simulated dominance durations as a function ofipsilateral (ips) and contralateral (cont) input strength the middleand right ones show transition durations and the FRTas a functionof departure (dep) and destination (dest) input strength (theWilson model produces no return transitions as it is noise-free)Durations are given in seconds for theWilsonmodel and in arbitraryunits (au) for the remaining ones Panels AndashC show results at theoriginal parameter settings with the white bars reproducing theoriginal demonstrations of agreement with Leveltrsquos second propo-sition Although all three models support the proposition (for a lim-ited range of inputs) their behavior diverges at most other pointsIn addition there are marked deviations from our data First themodels incorrectly predict at least an order difference betweendominance and transition durations and underestimate the FRTSecond none of the models reproduces the found patterns ofcontrast dependence of transition durations and the FRT (D) Insimulations at other parameter settings we found one parameterregion for the Kalarickal and Marshall model where the resultswere reminiscent of our data Here contrast dependencies of allthree variables were qualitatively correct but were accompaniedby an overestimation of both transition durations and the FRT Fur-ther analysis (see the Appendix) shows that in this region modeldynamics are essentially stochastic Deterministic forces keep thesystem in a state intermediate between both dominance statesand it is noise that causes incidental excursions into either domi-nance percept

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1248

Appendix) Its deterministic dynamics have settled at inter-mediate activities for both representations and it is purelynoise that causes the oscillations manifested in the figure

Discussion

Noise and rivalry dynamics

Our most striking finding is that current ideas on thesystem underlying rivalry do not agree with the observeddynamics Broadly speaking we show that there is muchmore to the alternation cycle than current models can ex-plain and no single adjustment will likely bridge this gapMore specifically however there is one aspect whereinthe deviation between models and data is particularly strongand which to us indicates one necessary model additionThis aspect is the behavior of transitions These are pres-ently treated as brief and irreversible switches betweentwo dominance periods whereas the data show that theytake considerable time and that return transitions are com-mon We will argue that the features of transitions in cur-rent models follow from an overemphasis on the role ofslow adaptation and that our results point to a system inwhich part of this role is taken over by stochastic varia-tions (ie noise) in the system componentsThis is illustrated in Figure 6 schematically showing

perceptual dynamics of rivalry as motions across energylandscapes (eg Billock amp Tsou 2003) The system state(ball) always develops toward lower energy thus minimain the landscapes are fixed points (attractors) correspond-ing to left-eye and right-eye dominance Changes in thelandscapes are due to changes in adaptation state tem-porarily modifying the relative strength of the attractorsSequence AYBYC-I shows the situation as many currentmodels treat it A The ball occupies the left attractor in-dicating for example left-eye dominance AYBYC-ISlow adaptation destabilizes the occupied attractor (blackarrow) while recovery from adaptation deepens the otherone (white arrow) until the left attractor disappears (C-I)causing the ball to move (dashed arrow) through the tran-sition region (gray) to the remaining attractor This tran-sition takes place quickly and not until considerable timehas passed allowing the left attractor to reappear may thesystem return to its previous state This separation of time-scales between landscape changes and ball movement isnecessary for these models to oscillate yet it is clear thatthe long transition durations and particularly the existenceof return transitions corresponding to the ball moving half-way between both attractors and then returning argueagainst this type of system In Panels C-II and C-III weshow two (not mutually exclusive) alternatives to Panel C-Ithat do agree with our data C-II In this scenario transi-tions are not initiated by destruction of the occupied at-tractor but by noise (curved arrow) tossing the system into

the transition region near the maximum that separates theleft and right domain of attraction Here the slope isshallow that is deterministic forces are weak and may bepositive or negative depending on the actual system stateThis may explain both the long duration of transitions andthe frequent occurrence of returns C-III Here the slope inthe transition region is so low (that is the deterministicdynamics here are so slow) that the attractor may reappearbefore the transition is over Again this is in line with longtransitions and provides an opportunity for noise to tip thesystem over in either direction We think that this latterscenario is particularly likely at lower contrasts (see below)Both scenarios point to a strong stochastic term as a keyingredient missing from present thinking

Figure 6 Schematic representation of rivalryrsquos dynamics ascurrently understood (AYBYC-I) and as our data imply(AYBYC-II and AYBYC-III) Dynamical modes of the systemare shown as energy landscapes The twominima are stable states(attractors) corresponding to left-eye and right-eye dominance (A)The ball occupies the left attractor indicating left-eye dominance(AYBYC-I) Adaptation (black arrow) destabilizes the occupiedattractor as recovery from adaptation (white arrow) deepens theother one Panel C-I shows transitions as treated by current mod-els As the occupied attractor disappears due to adaptation thesystem moves to the remaining one (dashed arrow) It cannot re-turn to the left attractor until (recovery from) adaptation has changedthe landscape to the mirror image of Panel C-I This scenario isincompatible with the slow and reversible transitions we observewhich point toward scenarios such as sketched in Panels C-II andC-III (C-II) Transitions are initiated by noise (curved arrow) driv-ing the system out of the still present attractor to a location in thetransition region (gray) near the separation between both attractordomains Here deterministic forces are small and the system maydevelop in either direction (C-III) The system remains in the at-tractor until it is gone but the deterministic forces in the transitionregion are so small that the attractor may reappear before thetransition is over Again noise may tip the system to the left or tothe right Both scenarios imply a crucial role for noise

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1249

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

models can reproduce our findings we tested their predic-tions on dominance durations transition durations and theFRT for a matrix of ipsilateralcontralateral input strengthsTo adequately cover transitions we chose two distinct

classes of models each addressing one of the two types oftransitions that may occur (see also the Discussion section)The Wilson model and the Kalarickal and Marshall modelcover superposition transitions during which both imagesare seen superimposed (Burke Alais amp Wenderoth 1999Liu Tyler amp Schor 1992) whereas the Stollenwerk andBode model covers piecemeal transitions during whichparts of both images are seen in complementary regions ofthe stimulus (Blake et al 1992 Lee Blake amp Heeger2005 OrsquoShea Sims amp Govan 1997 Silver amp Logothetis2004 Wilson et al 2001)The models by Wilson and by Kalarickal and Marshall

comprise a single oscillator formed by the two perceptsrsquorepresentations interacting via adaptation and inhibitionPiecemeal transitions are beyond their scope as they haveno spatial dimension We defined superposition phases asthose when none of the representations strongly dominatesbut instead both are intermediately activeThe model by Stollenwerk and Bode involves oscillators

similar to those described above but several of them linkedtogether in a 2D network corresponding to neighboringzones in visual space Neighboring oscillators are coupledsuch that they tend to follow each otherrsquos dominance statesso that once a dominance change emerges in one location apiecemeal transitionmay occurWe defined transition periodsas those during which less than 100 of all oscillators werein the same dominance stateFigures 5AndashC display simulation results at the original

papersrsquo parameter settings From left to right the charts ineach panel show dominance durations transition durationsand the FRT as a function of input strength The white barsin the left charts reproduce the original papersrsquo demonstra-tions of agreement with Leveltrsquos second proposition Theyalso illustrate how we complemented the original inputstrength combinations to form 4 4 matrices Aside fromthe white bars the models show a striking lack of concor-dance underscoring the added constraints posed by thepresent characterization The charts show marked devia-tions from our data First all models predict transitions tobe much shorter than dominance phases rather than of thesame order and also underestimate the FRT Second con-trast dependence of particularly transition durations andthe FRT is not correctly reproducedTo see if model predictions would improve at other pa-

rameter settings we performed simulations at an extensiverange of values confirming the shortcomings discussedabove There was one parameter region for the Kalarickaland Marshall model however (Panel D) where the con-trast dependence of all three variables was in qualitativeagreement with our data although both transition dura-tions and FRT were overestimated Interestingly at theseparameter settings the system is in a mode entirely differ-ent from the one that was originally intended (see the

Figure 5 Validation of three existing models The left charts of eachpanel show simulated dominance durations as a function ofipsilateral (ips) and contralateral (cont) input strength the middleand right ones show transition durations and the FRTas a functionof departure (dep) and destination (dest) input strength (theWilson model produces no return transitions as it is noise-free)Durations are given in seconds for theWilsonmodel and in arbitraryunits (au) for the remaining ones Panels AndashC show results at theoriginal parameter settings with the white bars reproducing theoriginal demonstrations of agreement with Leveltrsquos second propo-sition Although all three models support the proposition (for a lim-ited range of inputs) their behavior diverges at most other pointsIn addition there are marked deviations from our data First themodels incorrectly predict at least an order difference betweendominance and transition durations and underestimate the FRTSecond none of the models reproduces the found patterns ofcontrast dependence of transition durations and the FRT (D) Insimulations at other parameter settings we found one parameterregion for the Kalarickal and Marshall model where the resultswere reminiscent of our data Here contrast dependencies of allthree variables were qualitatively correct but were accompaniedby an overestimation of both transition durations and the FRT Fur-ther analysis (see the Appendix) shows that in this region modeldynamics are essentially stochastic Deterministic forces keep thesystem in a state intermediate between both dominance statesand it is noise that causes incidental excursions into either domi-nance percept

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1248

Appendix) Its deterministic dynamics have settled at inter-mediate activities for both representations and it is purelynoise that causes the oscillations manifested in the figure

Discussion

Noise and rivalry dynamics

Our most striking finding is that current ideas on thesystem underlying rivalry do not agree with the observeddynamics Broadly speaking we show that there is muchmore to the alternation cycle than current models can ex-plain and no single adjustment will likely bridge this gapMore specifically however there is one aspect whereinthe deviation between models and data is particularly strongand which to us indicates one necessary model additionThis aspect is the behavior of transitions These are pres-ently treated as brief and irreversible switches betweentwo dominance periods whereas the data show that theytake considerable time and that return transitions are com-mon We will argue that the features of transitions in cur-rent models follow from an overemphasis on the role ofslow adaptation and that our results point to a system inwhich part of this role is taken over by stochastic varia-tions (ie noise) in the system componentsThis is illustrated in Figure 6 schematically showing

perceptual dynamics of rivalry as motions across energylandscapes (eg Billock amp Tsou 2003) The system state(ball) always develops toward lower energy thus minimain the landscapes are fixed points (attractors) correspond-ing to left-eye and right-eye dominance Changes in thelandscapes are due to changes in adaptation state tem-porarily modifying the relative strength of the attractorsSequence AYBYC-I shows the situation as many currentmodels treat it A The ball occupies the left attractor in-dicating for example left-eye dominance AYBYC-ISlow adaptation destabilizes the occupied attractor (blackarrow) while recovery from adaptation deepens the otherone (white arrow) until the left attractor disappears (C-I)causing the ball to move (dashed arrow) through the tran-sition region (gray) to the remaining attractor This tran-sition takes place quickly and not until considerable timehas passed allowing the left attractor to reappear may thesystem return to its previous state This separation of time-scales between landscape changes and ball movement isnecessary for these models to oscillate yet it is clear thatthe long transition durations and particularly the existenceof return transitions corresponding to the ball moving half-way between both attractors and then returning argueagainst this type of system In Panels C-II and C-III weshow two (not mutually exclusive) alternatives to Panel C-Ithat do agree with our data C-II In this scenario transi-tions are not initiated by destruction of the occupied at-tractor but by noise (curved arrow) tossing the system into

the transition region near the maximum that separates theleft and right domain of attraction Here the slope isshallow that is deterministic forces are weak and may bepositive or negative depending on the actual system stateThis may explain both the long duration of transitions andthe frequent occurrence of returns C-III Here the slope inthe transition region is so low (that is the deterministicdynamics here are so slow) that the attractor may reappearbefore the transition is over Again this is in line with longtransitions and provides an opportunity for noise to tip thesystem over in either direction We think that this latterscenario is particularly likely at lower contrasts (see below)Both scenarios point to a strong stochastic term as a keyingredient missing from present thinking

Figure 6 Schematic representation of rivalryrsquos dynamics ascurrently understood (AYBYC-I) and as our data imply(AYBYC-II and AYBYC-III) Dynamical modes of the systemare shown as energy landscapes The twominima are stable states(attractors) corresponding to left-eye and right-eye dominance (A)The ball occupies the left attractor indicating left-eye dominance(AYBYC-I) Adaptation (black arrow) destabilizes the occupiedattractor as recovery from adaptation (white arrow) deepens theother one Panel C-I shows transitions as treated by current mod-els As the occupied attractor disappears due to adaptation thesystem moves to the remaining one (dashed arrow) It cannot re-turn to the left attractor until (recovery from) adaptation has changedthe landscape to the mirror image of Panel C-I This scenario isincompatible with the slow and reversible transitions we observewhich point toward scenarios such as sketched in Panels C-II andC-III (C-II) Transitions are initiated by noise (curved arrow) driv-ing the system out of the still present attractor to a location in thetransition region (gray) near the separation between both attractordomains Here deterministic forces are small and the system maydevelop in either direction (C-III) The system remains in the at-tractor until it is gone but the deterministic forces in the transitionregion are so small that the attractor may reappear before thetransition is over Again noise may tip the system to the left or tothe right Both scenarios imply a crucial role for noise

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1249

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

Appendix) Its deterministic dynamics have settled at inter-mediate activities for both representations and it is purelynoise that causes the oscillations manifested in the figure

Discussion

Noise and rivalry dynamics

Our most striking finding is that current ideas on thesystem underlying rivalry do not agree with the observeddynamics Broadly speaking we show that there is muchmore to the alternation cycle than current models can ex-plain and no single adjustment will likely bridge this gapMore specifically however there is one aspect whereinthe deviation between models and data is particularly strongand which to us indicates one necessary model additionThis aspect is the behavior of transitions These are pres-ently treated as brief and irreversible switches betweentwo dominance periods whereas the data show that theytake considerable time and that return transitions are com-mon We will argue that the features of transitions in cur-rent models follow from an overemphasis on the role ofslow adaptation and that our results point to a system inwhich part of this role is taken over by stochastic varia-tions (ie noise) in the system componentsThis is illustrated in Figure 6 schematically showing

perceptual dynamics of rivalry as motions across energylandscapes (eg Billock amp Tsou 2003) The system state(ball) always develops toward lower energy thus minimain the landscapes are fixed points (attractors) correspond-ing to left-eye and right-eye dominance Changes in thelandscapes are due to changes in adaptation state tem-porarily modifying the relative strength of the attractorsSequence AYBYC-I shows the situation as many currentmodels treat it A The ball occupies the left attractor in-dicating for example left-eye dominance AYBYC-ISlow adaptation destabilizes the occupied attractor (blackarrow) while recovery from adaptation deepens the otherone (white arrow) until the left attractor disappears (C-I)causing the ball to move (dashed arrow) through the tran-sition region (gray) to the remaining attractor This tran-sition takes place quickly and not until considerable timehas passed allowing the left attractor to reappear may thesystem return to its previous state This separation of time-scales between landscape changes and ball movement isnecessary for these models to oscillate yet it is clear thatthe long transition durations and particularly the existenceof return transitions corresponding to the ball moving half-way between both attractors and then returning argueagainst this type of system In Panels C-II and C-III weshow two (not mutually exclusive) alternatives to Panel C-Ithat do agree with our data C-II In this scenario transi-tions are not initiated by destruction of the occupied at-tractor but by noise (curved arrow) tossing the system into

the transition region near the maximum that separates theleft and right domain of attraction Here the slope isshallow that is deterministic forces are weak and may bepositive or negative depending on the actual system stateThis may explain both the long duration of transitions andthe frequent occurrence of returns C-III Here the slope inthe transition region is so low (that is the deterministicdynamics here are so slow) that the attractor may reappearbefore the transition is over Again this is in line with longtransitions and provides an opportunity for noise to tip thesystem over in either direction We think that this latterscenario is particularly likely at lower contrasts (see below)Both scenarios point to a strong stochastic term as a keyingredient missing from present thinking

Figure 6 Schematic representation of rivalryrsquos dynamics ascurrently understood (AYBYC-I) and as our data imply(AYBYC-II and AYBYC-III) Dynamical modes of the systemare shown as energy landscapes The twominima are stable states(attractors) corresponding to left-eye and right-eye dominance (A)The ball occupies the left attractor indicating left-eye dominance(AYBYC-I) Adaptation (black arrow) destabilizes the occupiedattractor as recovery from adaptation (white arrow) deepens theother one Panel C-I shows transitions as treated by current mod-els As the occupied attractor disappears due to adaptation thesystem moves to the remaining one (dashed arrow) It cannot re-turn to the left attractor until (recovery from) adaptation has changedthe landscape to the mirror image of Panel C-I This scenario isincompatible with the slow and reversible transitions we observewhich point toward scenarios such as sketched in Panels C-II andC-III (C-II) Transitions are initiated by noise (curved arrow) driv-ing the system out of the still present attractor to a location in thetransition region (gray) near the separation between both attractordomains Here deterministic forces are small and the system maydevelop in either direction (C-III) The system remains in the at-tractor until it is gone but the deterministic forces in the transitionregion are so small that the attractor may reappear before thetransition is over Again noise may tip the system to the left or tothe right Both scenarios imply a crucial role for noise

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1249

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

As such the notion that noise plays a role in rivalry is notnew and indeed it is obvious that no biological system isnoise-free Particularly stochastic variations in dominancedurations are well known (Brascamp van Ee Pestman ampvan den Berg 2005 Fox amp Herrmann 1967 Levelt 1967)and can be reproduced by current models (Kalarickal ampMarshall 2000 Lehky 1988 Stollenwerk amp Bode 2003Wilson 2005) However to account for the stochastics ofdominance durations on the level of detail that has hith-erto been considered it suffices to add jitter to essentiallydeterministic systems (eg Lehky 1988) The exact na-ture of this jitter is of little importance and even if it isleft out entirely the dynamics are not notably affected(Wilson 2005) Consequently the view of noise that hasdominated the literature so far is of a detail that introducessome unpredictability to a course of events that is itselfgoverned by deterministic forces The role that we pro-pose for noise on the other hand is of quite a differentorder Our data imply stochastic forces that are on a parwith deterministic ones frequently dominating them Rele-vant models therefore require careful consideration of thenature of this noise and will display dynamics that differfundamentally from those of current models To put it an-other way the point is not that noise plays a role in ri-valry the point is that it is a crucial factor for the systemto function the way it doesOne recent study (Kim Grabowecky amp Suzuki 2006)

extensively dealt with noise in rivalry and it is importantto discuss their findings in relation to ours Like those au-thors we are convinced of the importance of noise and wealso endorse their assertion that more empirical constraintslike the ones presented here are required to infer the in-ternal workings of this system In addition Kim et al takesome important steps toward characterizing the relevantnoise components by calculating the amount of externalperturbation that is equivalent to the system noise and byshowing that this amount scales linearly with stimulus con-trast We significantly add to their conclusions by provid-ing compelling evidence that noise is indeed a dominantfactor underlying oscillations in unperturbed rivalry In ad-dition whereas their data did not allow them to distinguishbetween various models (although the data did constrainthe nature of the noise component) our data indicate short-comings in all models we testedGiven the importance of noise we may wonder where in

the system these random fluctuations originate There ispresently no definitive answer to this question but let usconsider the options From studies dealing with noise inthe context of visual detection and discrimination we knowthat signal loss may arise at any level of visual processingfrom retinal photoreceptors and ganglion cells to the cortexitself and both prior to and beyond binocular combination(Pelli 1990 Raghavan 1989) In addition eye movements(both blinks and saccades) can influence the rivalry percept(van Dam amp van Ee 2006 Wade 1975) providing apossible noise source on the input side (but note thatKim et al 2006 report similar results in the absence and

presence of blinks) Finally there is the possibility ofdeterministic chaos (Laing amp Chow 2002 Wilson 2005)which although technically deterministic is equivalent tonoise on the present level of analysis One intriguingpossibility to note on the side is that regardless of itsorigins here noise may represent more than simply aphysical limit on the systemrsquos accuracy In the context ofrivalry it may instead be of functional use Arguablyrivalryrsquos alternations reflect a general property of vision tocontinuously reorganize sensory input to reach a perceptualsolution (Andrews amp Purves 1997 Leopold amp Logothetis1999) In such a framework noise may act to destabilize thepresent organization and prevent the brain from gettingtrapped in a single interpretation while others may havemore survival value (Kim et al 2006)Although we emphasize the role of noise in rivalry we

do not deny the importance of deterministic forces likeadaptation and inhibition Figure 5D is instructive in thisrespect showing results from the Kalarickal and Marshallmodel in a parameter regime (see the Appendix) where thesystem has only one attractory which is permanent that isitself moved around by noise One definitive argumentagainst this system and against most purely stochastic sys-tems is that it produces exponential-like distributions ofdominance durations which is incorrect Rather all evi-dence points to a system characterized by an interactionbetween deterministic and stochastic forces Specificallydeterministic dynamics likely dominate the part of the al-ternation cycle when the occupied attractor is deep whilenoise drives the system in the temporal vicinity of a tran-sition This view is further supported by recent findings(Lankheet 2006) showing that noise added externally(ie to the stimulus) primarily (although not exclusively)has an effect in a small time window near the transition

Leveltrsquos second proposition

A surprising finding of our study is the fact that Leveltrsquossecond proposition a widespread notion considered crucialfor assessing the validity of models is not generally appli-cable It should be emphasized that the present findingsare qualitatively different from other demonstrations of thelimits of the proposition Previous work has already led tothe attenuated view generally accepted to the present daythat the effect of a unilateral contrast change is not exclu-sively but mainly in the fixed-contrast eye (Bossink et al1993 Mueller amp Blake 1989) Our data on the other handshow that under certain conditions the main effect is in thevariable-contrast eye so that one may just as accurately(and just as incompletely) claim the opposite of the propo-sition The reason that this has so far been overlooked seemsto be simply that the proposition had not yet been tested forthe full contrast range that we considered Levelt based theproposition on measurements in which one target was fixedat 89 contrast while the other was adjusted down to 8(Levelt 1966) and to our knowledge no subsequent study

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1250

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

has done the converse that is fix one contrast at a low valueand then increase the other one to near maximum which arethe conditions crucial to our conclusions In other wordsalthough the proposition is correct in the contrast range inwhich Levelt and subsequent authors measured it doesnot generalize beyond this range It is therefore not a gen-eral feature of rivalry and one should act with caution inattaching any particular significance to it In terms of modelconstraints the most appropriate description of the relationbetween contrast and dominance durations is probably notany single proposition but the three-dimensional surface im-plied by Figure 2 As a rule of thumb for experimenters onthe other hand the most accurate alternative for the propo-sition may be the statement that Bchanges to one of the twocontrasts mainly affect dominance durations in the highercontrast eye[

The nature of transitions

As previously noted transitions may be either local(superposition) or spatial (piecemeal) (Blake Zimba ampWilliams 1985 Hollins 1980 Liu et al 1992 Yang et al1992) For the specific conclusion that noise is of greatimportance it is not necessary to distinguish these two typesAlthough slow transitions may partly be explained by aseries of fast local transitions adding up to a longer spatialone the return transitions in our data form a compellingargument for noise regardless of transition type Ultimatelyhowever binocular rivalry cannot be understood without aclear view of the nature of transitions because the two typescorrespond to entirely different system states Specificallythe occurrence of piecemeal transitions points toward par-allel rivalry in a number of separate (but linked) modulesas in the Stollenwerk and Bode model (Blake et al 1992Stollenwerk amp Bode 2003 Wilson et al 2001) A power-ful paradigm to study such transitions in isolation has re-cently been developed (Lee et al 2005 Wilson et al 2001)The occurrence of superposition on the other hand indi-cates that the system is reaching the limits of bistabilityand approaching fusion as noted by Liu et al (1992) whowere able to study this perceptual state in the period di-rectly following stimulus onset Clearly then it is of valueto know which type dominated in our study Because thenature of our main experiment in which subjects trackedtheir percepts in real time did not allow a distinction be-tween the two types which often occur simultaneously orin quick succession we performed a control experiment(see the Appendix) to address this issue Subjects observed10-s binocular rivalry trials wherein a number from 1 to 5is assigned to each trial as a whole with 1 meaning that alltransition percepts in this trial were superposition perceptsand 5 meaning that they were all piecemeal This controlinvolved the same stimuli and subjects as the main experi-ment but we used only the four symmetric contrast condi-tions The outcome was clear-cut Going from high to lowcontrast all subjects had a monotonic decrease in their

scores from mostly piecemeal (41 on average) at high con-trast to mostly superposition (21 on average) at low con-trast These results underscore that both local and spatialtransitions play a role in rivalry with piecemeal perceptsdominating at high contrast and superposition becomingmore prominent as contrast decreases Our findings dovetailnicely with existing literature as our transition durations athigh contrast (about 05 to 1 s) agree well with the lowerlimit predicted in case of pure piecemeal transitions (thetime required for a border between two regions of oppositedominance to sweep over our stimulus calculated based onHorton and Hoyt 1991 Wilson et al 2001) whereas super-position periods at stimulus onset have been shown to beparticularly prominent at low contrast (Liu et al 1992)Note that the changes in the transition phase that

accompany a contrast reduction occur gradually with thefirst effects regarding both their durations and the subjectsrsquoassessment of their nature being apparent at contrasts ashigh as 50 This argues against the notion (Liu et al1992) that a separate neural mechanism is responsible forbinocular summation at low contrast which would besupported for instance by a discontinuity in the transitionsrsquofeatures near the low contrast end A more parsimoniousinterpretation of the present data is that what we observe isa gradual evolution of the binocular rivalry process fromone that produces abrupt perceptual flips to one that hasmore gradual transitions with stable binocular summationbeing a limiting case at near-threshold contrasts Such agradual evolution is a natural property of many oscillatormodels as illustrated by Panel C-III in Figure 6 whichshows that shallowing of the attractors at low contrast canbe accompanied by increased stability of the region inbetween Hence although the possibility of a separatesummation process cannot presently be excluded the datado not force us to invoke any such additional mechanism

Future research

The perspective on rivalry developed here gives rise tonew questions For instance one would like to know hownoise affects rivalry dynamics Conversely rivalryrsquos alter-nation cycle may within the proper interpretative frame-work offer a new handle to assess the nature of internalnoise a lively debated issue in the field of visual detectionand discrimination (eg Gorea amp Sagi 2001 KontsevichChen amp Tyler 2002 Legge amp Foley 1980) We think thatthe two different roles that noise may play in the rivalryprocess as illustrated in Figure 6 offer two natural startingpoints for exploring such issues First (Panel C-II) noiseplays a role in terminating dominance phases by pushingthe system out of an attractor corresponding to exclusivedominance Second (Panel C-III) noise in part determineswhen and in which direction transition phases end by af-fecting the path of the system in the transition region Theobservable outcome of these two processes is formed by dis-tributions of dominance and transition durations respectively

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1251

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

With this in mind we are currently developing models(Brascamp Noest van Ee amp van den Berg 2006 Noestamp van Ee 2006) of dominance and transition phases sep-arately allowing us to interpret the shapes of these dis-tributions in terms of neural interactions and to further ourunderstanding of the rivalry process particularly of therole played by noise

Conclusion

We have investigated binocular rivalryrsquos alternation cyclein terms of both mean dominance and transition duration andthe FRT in relation to stimulus contrast Both dominanceand transition durations were on the order of seconds theFRT varied between about 0 and as much as 05 We foundsystematic patterns of contrast dependence for all threevariables similar for dominance durations and the FRT butdifferent for transition durations Regarding dominance du-rations we show that Leveltrsquos second proposition is validonly in a limited portion of the contrast domain and may bereplaced more generally by the proposition that changes inone eyersquos contrast mainly affect dominance durations in thehigher contrast eye We found a strong correlation betweenthe FRT and mean dominance duration of the departureeye which shows the presence of a common variable under-lying both quantitiesOur data allow inferences regarding rivalryrsquos mechanism

not allowed by previous data and provide more stringentmodel constraints They refute prevailing models driven byadaptation and inhibition as these underestimate both tran-sition durations and the FRT and predict incorrect patternsof contrast dependence The data imply a crucial influenceof stochastic variations in the neural circuitry mediatingrivalry which these models overlook

Appendix

Trial duration and fatigue

Our experimental trials were longer than usual in the lit-erature (5 min) and it is important to rule out the possi-bility that subject fatigue played any role in our resultsFigure A1 shows what Figure 2 would have looked likehad we analyzed only the first 2 min of every trial ratherthan the last 4 min Clearly there are no important dif-ferences between this figure and Figure 2 indicating thattrial duration is not a crucial factor

Dominance durations and return transitions

The correlation between departure dominance durationand FRT shown in Figure 4 can have either of two expla-

nations First both may depend on a common underlyingvariable (an indirect causal link) or second the occur-rence of a return transition may cause a long duration orvice versa (a direct causal link) Figure A2 unconfoundsthese options For every return transition away from andback to a given eye we considered the 10 last dominancedurations (normalized per trial and eye separately) of thateye preceding the return transition and the 10 first onesoccurring afterward rank numbering in time from j10 toj1 and from 1 to 10 This enabled us to calculate theaverage dominance duration per rank number A directcausal link would show up as a positive deflection near thereturn transition in this event-related average Figure A2shows no such deflection thus the correlation is insteaddue to a common factor causing both phenomena Strik-ingly there is even a negative deflection at rank numbersj1 and 1 This may be explained as follows Subjects em-ploy some decision criterion as to the amount of suppressed-eye contamination in a dominant percept that will causethem to report the onset of a transition If as we proposea transition is a gradual errant process then there are oc-casions on which a dominant percept is temporarily dis-turbed by some near-criterion amount of contaminationbefore it restabilizes On some of these occasions subjects

Figure A1 Same as Figure 2 using data from only the first 2 min ofevery trial This figure and Figure 2 are almost identical demon-strating that trial duration did not crucially influence our results

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1252

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

will report a return transition and thereby cut the presentdominance duration into two shorter ones whereas on otheroccasions they will not This would explain the deflection inthe figure

Simulations

The results in Figures 5AndashC are for the parameter set-tings used by the original authors For the input strengthswe took four values equidistant in log space with thesmallest and largest ones based on the original papers 09and 15 for Stollenwerk and Bode 0853 and 1 for Wilsonand 03 and 07 for Kalarickal and Marshall For thesingle-oscillator models we defined transitions as thoseperiods during which both poolsrsquo activities lay within afactor of 4 from each other for the multiple-oscillatormodel the criterion was that less than 100 of the oscil-lators should be in the same dominance state that is havethe strongest activity for the same percept For the single-oscillator models we explored the surrounding parameterspace by varying all parameters from half to twice theiroriginal values in three (Kalarickal and Marshall) or four(Wilson) equal log steps For the multiple-oscillator modelsimulation time did not allow this hence instead we fixedthe two parameters defining the sigmoid nonlinearity in-volved in the model and varied the remaining six variablesfrom 23 to 32 times their base values in two equal logsteps For these latter simulations we reduced the numberof coupled oscillators to 10 10 to fit our stimulus basedon the assumption that one oscillator covers about 01- of

visual angle roughly the maximum stimulus size for whichrivalry is non-piecemeal (Blake et al 1992) Along withthe lattice size we reduced the base value for gt0 the spatialextent of noise by a factor of 2 because Stollenwerk andBode tuned that value for a 20 20 lattice For all simu-lations we used an explicit Euler iteration scheme withstep sizes of 01 (Stollenwerk and Bode) 002 (Kalarickaland Marshall) and 0001 (Wilson) Note that in theKalarickal and Marshall model noise is applied to the adap-tation state rather than to its rate as implied by the originalequations (GJ Kalarickal personal communication)Besides the above simulations we performed simula-

tions at the original settings to verify our correct recon-struction of the models Figure A3 shows the outcomesalong with those read from the original figures The closeagreement confirms that our reconstructions were correct

Kalarickal and Marshall modellsquolsquobestrsquorsquo parameters

As shown in Figure 5D there exist parameter settingsat which the Kalarickal and Marshall model is inqualitative agreement with our data It turns out that atthese settings (W1

+ = W2+ = 0315 W12

j = W21j = 125

c1 = 0005 c2 = 0016 c3 = 0105 s = 0005) the modelhas not two but one attractor and the dynamics are entirelygoverned by noise This situation is illustrated in Figure A4Panel A by a phase plane plot displaying the states thatthe system may occupy in terms of both poolsrsquo activities(x1 and x2) For any system state or x1 x2 combination x1

Figure A2 Event-related average dominance duration for the departure eye relative to the occurrence of a return transition The absence of apositive deflection for dominance durations near the return transition excludes the option that the positive correlation shown in Figure 4 arisesbecause return transitions cause long dominance durations or vice versa Instead the correlation must be due to a common underlyingvariable The negative deflection that is evident instead is not unexpected if we consider a transition as a gradual errant process (see text)Error bars indicate standard errors

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1253

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

and x2 develop as indicated by the flow arrows The twocurves (null-clines) however indicate the locations at whicheither x1 (black) or x2 (gray) does not change hence at anintersection between the null-clines neither changes Atthese parameter settings contrary to the original ones thelines have only one intersection an attractor (plus sign) towhich all flow arrows lead The only reason the systemBoscillates[ is that the attractor itself is displaced by noiseIt is unlikely that in reality binocular rivalry is entirelystochastic as stochastic systems are generally associatedwith monotonically decreasing distributions of dominancedurations instead of the well-known unimodal ones ob-served in rivalry Panel B illustrates this showing a dis-tribution of dominance durations obtained at these settings

Superposition versus piecemeal transitions

In our main experiment we were unable to distin-guish superposition and piecemeal transitions In thiscontrol experiment we investigated their relative impor-tance and how this changes with contrast Our foursubjects viewed the grating stimuli for 10 s at a timeand afterward scored the subjective nature of thetransition percepts observed during the period on a scaleranging from 1 (only superposition) to 5 (only piecemeal)Each report was followed by another 10 s before the startof the next trial preventing afterimages from interferingwith the stimulus on the next trial This was repeated fivetimes per subject per condition randomly interleaving

Figure A3 The original papersrsquo simulation results (squares) and reproductions using our reconstructed models (triangles) The closeagreement confirms the correct reconstruction of the models

Figure A4 The Kalarickal and Marshall model in a monostable regime At these settings the system has not two but one attractor (A) and thedynamics are entirely governed by noise (B) This system produces exponential-like distributions of dominance durations rendering it anunlikely candidate for explaining rivalry along with most other purely stochastic systems

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1254

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

conditions and applying only the four symmetric contrastcombinations As shown in Figure A5 there is a clearshift from mainly superposition at low contrast to mainlypiecemeal percepts at high contrast although note thatboth types of percepts were perceived throughout thecontrast range as neither 1 nor 5 was scored very often

Acknowledgment

The work was supported by a grant from the NetherlandsOrganization for Scientific Research (NWO) assigned toRvE

Commercial relationships noneCorresponding author Jan W BrascampEmail jwbrascampbiouunlAddress Padualaan 8 3584 CH Utrecht The Netherlands

References

Alais D amp Blake R (2005) Binocular rivalry CambridgeLondon MIT Press

Andrews T J amp Purves D (1997) Similarities in normaland binocularly rivalrous viewing Proceedings of theNational Academy of Sciences of the United States ofAmerica 94 9905ndash9908 [PubMed] [Article]

Billock V A amp Tsou B H (2003) Vision-nonlinear dy-namic approaches In R G Driggers (Ed) Encyclope-dia of optical engineering (pp 2917ndash2928) New YorkMarcel Dekker

Blake R (2001) A primer on binocular rivalry includingcurrent controversies Brain and Mind 2 5ndash38

Blake R OrsquoShea R P amp Mueller T J (1992) Spatialzones of binocular rivalry in central and peripheralvision Visual Neuroscience 8 469ndash478 [PubMed]

Blake R Zimba L amp Williams D (1985) Visual motionbinocular correspondence and binocular rivalry Bio-logical Cybernetics 52 391ndash397 [PubMed]

Bossink C J Stalmeier P F amp De Weert C M (1993)A test of Leveltrsquos second proposition for binocularrivalry Vision Research 33 1413ndash1419 [PubMed]

Brascamp J W Noest A J van Ee R amp van den BergA V (2006) Transition phases show the importanceof noise in binocular rivalry [Abstract] Journal ofVision 6(6) 845a httpjournalofvisionorg66845doi10116766845

Brascamp J W van Ee R Pestman W R amp van denBerg A V (2005) Distributions of alternation rates invarious forms of bistable perception Journal of Vision5(4) 287ndash298 httpjournalofvisionorg541doi101167541 [PubMed] [Article]

Burke D Alais D amp Wenderoth P (1999) Determinantsof fusion of dichoptically presented orthogonal gratingsPerception 28 73ndash88 [PubMed]

Fox R amp Herrmann J (1967) Stochastic properties of bin-ocular rivalry alternations Perception amp Psychophysics2 432ndash436

Fox R amp Rasche F (1969) Binocular rivalry and reciprocalinhibition Perception amp Psychophysics 5 215ndash217

Gorea A amp Sagi D (2001) Disentangling signal fromnoise in visual contrast discrimination Nature Neuro-science 4 1146ndash1150 [PubMed] [Article]

Hollins M (1980) The effect of contrast on the complete-ness of binocular rivalry suppression Perception ampPsychophysics 27 550ndash556 [PubMed]

Horton J C amp Hoyt W F (1991) The representation ofthe visual field in human striate cortex A revision ofthe classic Holmes map Archives of Ophthalmology109 816ndash824 [PubMed]

Kalarickal G J amp Marshall J A (2000) Neural modelof temporal and stochastic properties of binocularrivalry Neurocomputing 32ndash33 843-853

Kim Y J Grabowecky M amp Suzuki S (2006) Stochas-tic resonance in binocular rivalry Vision Research 46392ndash406 [PubMed]

Kontsevich L L Chen C C amp Tyler C W (2002)Separating the effects of response nonlinearity andinternal noise psychophysically Vision Research 421771ndash1784 [PubMed]

Laing C R amp Chow C C (2002) A spiking neuronmodel for binocular rivalry Journal of ComputationalNeuroscience 12 39ndash53 [PubMed]

Figure A5 Relative importance of piecemeal and superpositionpercepts as a function of stimulus contrast Both for individualsubjects (black) and averaged over all four (gray) there is a clearshift from mainly superposition at low contrast to mainly piecemealpercepts at high contrast although both types of percepts seem tooccur throughout the contrast range

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1255

Lankheet M J (2006) Unraveling adaptation and mutualinhibition in perceptual rivalry Journal of Vision6(4) 304ndash310 httpjournalofvisionorg641doi101167641 [PubMed] [Article]

Lee S H Blake R amp Heeger D J (2005) Travelingwaves of activity in primary visual cortex during bin-ocular rivalry Nature Neuroscience 8 22ndash23 [PubMed][Article]

Legge G E amp Foley J M (1980) Contrast maskingin human vision Journal of the Optical Society ofAmerica 70 1458ndash1471 [PubMed]

Lehky S R (1988) An astable multivibrator model ofbinocular rivalry Perception 17 215ndash228 [PubMed]

Leopold D A amp Logothetis N K (1999) Multistablephenomena Changing views in perception Trends inCognitive Sciences 3 254ndash264 [PubMed]

Levelt W J (1966) The alternation process in binocularrivalry British Journal of Psychology 57 225ndash238

Levelt W J (1967) Note on the distribution of domi-nance times in binocular rivalry British Journal ofPsychology 58 143ndash145 [PubMed]

Liu L Tyler C W amp Schor C M (1992) Failure ofrivalry at low contrast Evidence of a suprathresholdbinocular summation process Vision Research 321471ndash1479 [PubMed]

Mueller T J (1990) A physiological model of binocularrivalry Visual Neuroscience 4 63ndash73 [PubMed]

Mueller T J amp Blake R (1989) A fresh look at thetemporal dynamics of binocular rivalry BiologicalCybernetics 61 223ndash232 [PubMed]

Noest A J amp van Ee R (2006) Statistical-mechanicsmodeling of rivalrous dominance and transition du-rations Perception 35(Suppl) 53

OrsquoShea R P Sims A J amp Govan D G (1997) Theeffect of spatial frequency and field size on the spread

of exclusive visibility in binocular rivalry VisionResearch 37 175ndash183 [PubMed]

Pelli D G (1990) The quantum efficiency of vision InC Blakemore (Ed) Vision Coding and efficiencyCambridge Cambridge University Press

Raghavan M (1989) Sources of visual noise SyracuseSyracuse University

Shiraishi S (1977) A test of Leveltrsquos model on binocularrivalry Japanese Psychological Research 19 129ndash135

Silver M A amp Logothetis N K (2004) Grouping andsegmentation in binocular rivalry Vision Research44 1675ndash1692 [PubMed]

Stollenwerk L amp Bode M (2003) Lateral neural model ofbinocular rivalry Neural Computation 15 2863ndash2882[PubMed]

van Dam L C amp van Ee R (2006) Retinal image shiftsbut not eye movements per se cause alternations inawareness during binocular rivalry Journal of Vision6(11) 1172ndash1179 httpjournalofvisionorg6113doi1011676113 [PubMed] [Article]

Wade N J (1975) Binocular rivalry between single linesviewed as real images and afterimages Perception ampPsychophysics 17 571ndash577

Watson A B amp Pelli D G (1983) QUEST A Bayesianadaptive psychometric method Perception amp Psycho-physics 33 113ndash120 [PubMed]

Wilson H R (2005) Rivalry and perceptual oscillationsA dynamical synthesis In D Alais amp R Blake (Eds)Binocular rivalry Cambridge London MIT Press

Wilson H R Blake R amp Lee S H (2001) Dynamicsof travelling waves in visual perception Nature 412907ndash910 [PubMed]

Yang Y Rose D amp Blake R (1992) On the variety ofpercepts associated with dichoptic viewing of dissimilarmonocular stimuli Perception 21 47ndash62 [PubMed]

Journal of Vision (2006) 6 1244ndash1256 Brascamp et al 1256