the dynamics of perceptual rivalry in bistable and...

The dynamics of perceptual rivalry in bistable and tristableperception

Guy Wallis # $

Centre for Sensorimotor Neuroscience, School of HumanMovement Studies and Queensland Brain Institute,

University of Queensland, Australia

Stefanie Ringelhan #Centre for Sensorimotor Neuroscience, School of HumanMovement Studies, University of Queensland, Australia

Human observers are extremely adept at correctlyinterpreting the visual input that they receive despite itsinherent ambiguity. There are, however, situations inwhich no single, valid interpretation exists andperception oscillates. Such situations offer insight intothe processes underlying perception, as they reveal theconditions under which our percept can alter in thepresence of an unchanging physical stimulus. Manystudies have focused on perceptual switching duringbinocular and monocular rivalry, or when viewingambiguous figures. The majority of these studies havefocused on bistable phenomena. In this paper we reportthe study of a range of ambiguous stimuli that provoketristable perception, thereby providing a more exactingtest bed for current models. We find that subjectsspontaneously move from periods of bistable to tristableperception and that this results in a characteristic changein the rate of perceptual switching. In contrast to reportsfor bistable stimuli, we find only weak evidence forconsistency in the relative switch rates of individualsacross our test stimuli and explain why our results areinconsistent with the theory of interhemisphericswitching. We go on to describe how the results are,instead, largely consistent with models of rivalry basedon mutual inhibition and adaptation.

Introduction

Background

One of the great challenges to human perception isthat there is no simple relationship between the imagereceived by our eyes and the objects and events in theworld that produced that image. Although we areusually able to build a single, correct impression of anexternal stimulus, there are situations in which inherent

stimulus ambiguity or visual conflict makes thisimpossible. The perceptual vacillation that results inthese cases is particularly evident when viewingambiguous figures such as the Necker Cube (Necker,1832), or during binocular, monocular, and stimulusrivalry (Breese, 1899; Campbell & Howell, 1972;Helmholtz, 1867; Levelt, 1967, 1968; Logothetis,Leopold, & Sheinberg, 1996). In all four cases, theappearance of a visual stimulus alters spontaneouslyand often completely, despite the stimulus not under-going any physical change. Hence it is possible for asingle, unchanging stimulus to produce quite differentperceptual experiences in an observer from onemoment to the next. Quite apart from questionssurrounding the basic processes underlying switchingbehavior, stimuli capable of perceptual rivalry havebecome important tools for behavioral scientistsbecause of their ability to elicit perceptual change in thebrain without altering the physical stimulus. This, it ishoped, will provide insights into where perceptualdecisions take place and, beyond that, the neural basisof consciousness (see e.g., Blake & Logothetis, 2002).

Although ambiguous and rivalrous stimuli generallyinduce bistable perception (with percepts oscillatingbetween two possibilities), they can, in a few instances,also produce three or more. Figures 1 and 2 provideexamples of visual stimuli that provoke tristableperception, with the latter based on monocular rivalry.Other examples of visual stimuli evoking multistableperception include the combination of interocularfusion with binocular rivalry (Suzuki & Grabowecky,2002) and the motion-induced disappearance of dots(Bonneh, Cooperman, & Sagi, 2001). Apart from visualstimuli, it is also possible to experience multistabilityfor auditory cues (Warren, 1961; Warren & Gregory,1958). Although intrinsically more complicated thanbistable phenomena, tristable and multistable rivalry is

Citation: Wallis, G., & Ringelhan, S. (2013). The dynamics of perceptual rivalry in bistable and tristable perception. Journal ofVision 13(2):24, 1–21, http://www.journalofvision.org/content/13/2/24, doi:10.1167/13.2.24.

Journal of Vision (2013) 13(2):24, 1–21 1http://www.journalofvision.org/content/13/2/24

doi: 10 .1167 /13 .2 .24 ISSN 1534-7362 � 2013 ARVOReceived April 11, 2012; published February 20, 2013

http://hms.health.uq.edu.au/vislab/

http://hms.health.uq.edu.au/vislab/

mailto:[email protected]

mailto:[email protected]

http://www.strategy.wi.tum.de/people/phds-and-post-docs/stefanie-ringelhan/

http://www.strategy.wi.tum.de/people/phds-and-post-docs/stefanie-ringelhan/

Figure 1. Example of a depth-related tristable stimulus. (a) The original version of the stimulus, adapted from a paper by Poston and

Stewart (1978). We tried this stimulus but many subjects failed to report the third possible percept (of a cube in front of a larger

cube), and so it was not used in the experiments described here. (b) We have since created a new version that accentuates the third

percept by making this interpretation correct in terms of visual perspective. It nonetheless remains possible to see the other two

more obvious interpretations: a large cube with one corner cut away, or a misshapen room with a cube sitting in the corner. Note that

there is, in theory, a valid fourth interpretation (of an inverted room within a larger room), but for the observers we tested, this

interpretation turned out to be too improbable for it to ever win the perceptual tug-of-war.

Figure 2. Example of a tristable, monocular rivalry stimulus similar to that used in Naber et al. (2010). (a) The stimulus consists of

three superimposed sinusoidal gratings, each offset from the others by 1208 and each with a unique hue (corresponding to red,

green, and blue in this case). In our case we also chose to ensure than none of the gratings were oriented more closely than 158 to a

cardinal axis. With a little tinkering in a dark room, it is possible for most observers to experience reliable rivalry, such that perception

spontaneously shifts between the three constituent gratings b, c, and d. In practice we were unable to obtain stable switching across

all of our subjects. Switching tended to be rapid and therefore hard for them to track, and all subjects described periods of mixed

percepts or traveling waves of the type described by Wilson, Blake, and Lee (2001) in binocular rivalry. We therefore chose not to use

this stimulus in the experiments described here.

Journal of Vision (2013) 13(2):24, 1–21 Wallis & Ringelhan 2

of especial interest because it offers a more exacting testof models of switching behavior, as we demonstrate inthis paper.

Searching for commonalities

One key to understanding the various forms ofperceptual rivalry is to understand commonalities anddifferences. On the face of it, binocular, stimulus, andmonocular rivalry do share a number of similarities. Allinvolve periods of disappearance of one or another partof the stimulus, and all are capable of producingintermediate/hybrid percepts. On the other hand, thereare differences. The longstanding debate about theneural locus of binocular rivalry (Blake & Logothetis,2002) has revealed differences between stimulus andbinocular rivalry (Bonneh, Sagi, & Karni, 2001; Lee &Blake, 1999), which has, in turn, led to conjecture thatthey occur at different levels of the visual hierarchy(Freeman, 2005; Wilson, 2003). Nonetheless, detailedstudies of suppression and the effects of stimulusproperties including color have confirmed that many ofthe similarities run deep (Andrews & Purves, 1997;O’Shea, Parker, La Rooy, & Alais, 2009) and may wellreveal functional linkages between these three types ofrivalry (Pearson & Clifford, 2005).

Despite these commonalities it is less clear to whatextent ambiguous figures are related to the other threephenomena. Ambiguous patterns do not, for example,generally involve perceptual disappearance of all orpart of the stimulus. On further reflection, however,some similarities do exist. The switching from onepercept to another appears to defy fully consciouscontrol, and this paper considers one example, at least,for which changes in interpretation do lead to theappearance or disappearance of parts of the stimulus,albeit of illusory occluders (see Hidden Square inFigure 4). Recent years have seen a growing number ofattempts to unite all four types of multistableperception in the hope of identifying a commonmechanism behind perceptual switching. One com-monly cited piece of evidence for such a link is that therelative rate of perceptual switching in an individual isconsistent across a range of stimuli—i.e., someone whoexperiences rapid switching for one type of stimulususually experiences rapid switching for other stimuli aswell. High correlations have been reported betweenswitch rates in binocular rivalry, multistable motioninduced blindness, and bistable plaids (Carter &Pettigrew, 2003; Sheppard & Pettigrew, 2006). By wayof clarification one should add that the link is onlycorrelational and describes switch rates of individualsrelative to other observers. Hence, the precise rate atwhich interpretations switch can and does vary fromstimulus to stimulus (Andrews & Purves, 1997; Carter

& Pettigrew, 2003; Sheppard & Pettigrew, 2006).Although first brought to the vision science commun-ity’s attention by Carter and Pettigrew (2003), the resulthas since been investigated by many groups across uni-and multimodal domains. The results appear largelyconsistent within certain modalities such as vision andaudition, but not consistently across multimodaldomains, such as visuo-haptic (Carter, Konkle, Wang,Hayward, & Moore, 2008; see also Schwartz, Grimault,Hupe, Moore, & Pressnitzer, 2012, for a brief review).In a recent study which included a large number ofsubjects and genetic markers in its analysis, Kondo,Kitagawa, Kitamura, Nomura, and Kashino (2012)reported evidence for consistencies across numerousstimuli, with evidence of a link between switch rate andserotonin (for visual shape) and dopamine (for sound).This result mirrors earlier work by Carter andcolleagues who described a link between switch ratesand activation of 5HT receptors, a receptor familyclosely linked to the role of serotonin in regulatingneural activity (Carter et al., 2005; Carter et al., 2007).

The picture is, however, not as clear when oneconsiders the possible role of attention. Studies ofattentional control suggest that there are differences inthe effect of attention on different types of ambiguousfigure (Taddei-Ferretti et al., 2008). Some authors haveargued that ambiguous figures are much more amena-ble to personal control than binocularly rivaling stimuli(Meng & Tong, 2004). On the other hand, other studieshave revealed that attention can have a sizeable effecton switch rates and stimulus dominance in binocularrivalry (Chong, Tadin, & Blake, 2005). Overall, it seemsthat all stimuli are amenable to a degree of personalcontrol, but the precise level of control and effects ofattending are specific to both the stimulus and stimulustype (van Ee, van Dam, & Brouwer, 2005). Attempts toalter normal switching behavior using transcranialmagnetic stimulation have shown that it can affectbinocular rivalry (Miller et al., 2000; Pearson, Tadin, &Blake, 2007) and although this was not reported forambiguous-figure rivalry, caloric stimulation did workfor both (Miller et al., 2000).

Models of switching

Despite the tantalizing links across the differenttypes of ambiguous stimuli, few researchers haveattempted to trace all of the effects to a singleunderlying source. One notable exception has beenPettigrew and colleagues (Carter & Pettigrew, 2003;Sheppard & Pettigrew, 2006; Miller et al., 2000) whohave proposed that switching is driven by a single,external oscillator. The basis for this oscillator, theyargue, is a subcortical signal that cyclically promotesrival interpretations across the two cortical hemi-


spheres (Carter & Pettigrew, 2003; Miller et al., 2000;Miller et al., 2010). More mainstream models attemptto reconcile evidence of higher-level, object-basedrivalry (Leopold & Logothetis, 1999) with lower-level,eye-based effects (Tong, 2001), leading theoreticians topostulate the presence of numerous competitive neuralprocesses extending through a hierarchy of stages(Blake & Logothetis, 2002; Crewther et al., 2005;Dayan, 1998; Freeman, 2005; Gomez, Argandona,Solier, Angulo, & Vazquez, 1995; Hupe & Rubin, 2003;Laing & Chow, 2002; Leopold & Logothetis, 1996;Wilson, 2003). That said, many authors argue thatbinocular rivalry may reflect the actions of a quiteseparate system to those underlying other forms ofrivalry (Meng & Tong, 2004; Polonsky, Blake, Braun,& Heeger, 2000). To some extent, what is unusualabout binocular rivalry is that it is something weexperience throughout our everyday lives, and hencethe visual system may well have evolved a specificmechanism for automatically supressing diplopic re-gions (Arnold, Grove, & Wallis, 2007; Shimojo &Nakayama, 1990, 1994).

Multistability

Models of perceptual rivalry have traditionallyfocused on trying to explain the switching process inbinocular rivalry or other bistable phenomena. Inpractice, the oscillatory behavior of bistable systemscan be simulated via a myriad of flip-flop oscillatorynetworks, making it difficult to isolate the mostappropriate model. Suzuki and Grabowecky (2002)sought a more complex stimulus that might providedeeper insights into the mechanisms involved inperceptual rivalry. They adopted stimuli that involvethe simultaneous combination of monocular andbinocular rivalry (see Diaz-Caneja, 1970; Kovacs,Papathomas, Yang, & Feher, 1996), providing fourpossible percepts. The authors described long periodsof bistable switching interleaved with a suddenrecombination of the monocular cues, a phenomenathey referred to as ‘‘perceptual trapping.’’ One meansof explaining this effect, based on neural adaptation, isthat disappearance of part of the stimulus promotes itslater, complete reappearance after the complimentaryform has experienced a period of dominance andsubsequent adaptation. Indeed, the authors were ableto provide a general model of this behavior basedalong these lines. Unfortunately, what is not clear isthe extent to which trapping is a peculiarity of stimuliin which elements cyclically appear and disappear.More recently, two groups presented preliminaryresults from studies looking at tristable perceptualphenomena (Naber, Gruenhage, & Einhauser, 2009;Wallis & Ringelhan, 2009). This paper describes the

Wallis and Ringelhan (2009) data in more detail,providing results from the study of several stimuli thatoffer a more powerful and yet tractable data set forinvestigating the explanatory and predictive power ofvarious models.

As a motivation for what follows, it is instructive toconsider how various models of rivalry might accom-modate tristable perception. The interhemisphericswitching model can, in a broad sense, explain bistableswitching, but the question occurs as to what happenswhen there are three possible interpretations. Onepossibility is that one hemisphere represents oneinterpretation and the other hemisphere the other two.That immediately offers a strong and testable predic-tion: Switching between two of the percepts will occurrarely, if at all, and one percept will dominate on asmany occasions as the sum of the other two. Foradaptation the story is more complex. It has beenknown for some time that if a rivaling stimulus isinterrupted (e.g., by blinking), the previously perceivedstimulus tends to dominate again (Leopold, Wilke,Maier, & Logothetis, 2002; Orbach, Zucker, & Olson,1966). It is conceivable that switching away frompercept A to B (for example) would serve the purposeof an interruption of percept A and hence promote itsreoccurrence after B—suggesting that ABA or bistableswitches would be more common than ABC or tristableones. On the other hand, a model in which the neuralpopulation associated with percept A had not yet fullyrecovered from initial adaptation at the momentpercept B wanes would promote takeover by percept C,suggesting that ABC type switches would be morelikely—see Figure 3. If this doesn’t happen, itconstrains the recovery of neurons to be practicallycomplete within the period of dominance of asubsequent perceptual state.

Experiment

Methods

Participants

Twenty-four participants with corrected-to-normalvision were tested. All participants were recruitedthrough the School of Psychology’s paid participantregister at the University of Queensland. The experi-ment was conducted in accordance with University ofQueensland Ethics guidelines and with approval of theUniversity’s Behavioural and Social Sciences EthicalReview Committee. Informed consent was obtainedfrom all participants. One additional participant wasdropped from the experiment for failing to achieve therequired stimulus selection criteria (see Procedure).


Stimuli

A quick survey of the literature on rivalry andmultistable perception suggests that there are relativelyfew stimuli that evoke tristable perception. Weuncovered five and selected three for further testing.The other two were not used here for reasons given inthe figure legends (see Figures 1 and 2).

Adelson-Movshon grid

Adelson and Movshen (1986) described a stimulus aspart of their investigations into the aperture problemand border ownership (Adelson & Movshon, 1982,1986; Fennema & Thompson, 1979; Stumpf, 1911). Forthe purposes of this investigation we will refer to this asthe ‘‘Adelson-Movshon grid’’ (AMGrid). This stimulusconsists of three sets of lines moving at 1208 to oneanother (see Figure 4a). Although the three sets of linesmove in separate directions, the percept is of one set oflines moving in one direction and diamonds moving inthe opposite direction. In the case of our stimulus thethree interpretations were: (a) Line motion down andright at 158 to vertical, (b) line motion right and up at

458, and (c) line motion up and left at 158 to thehorizontal. The impression then swaps to one of theother three directions as time progresses. (Stimulusparameters: aperture 108, stripe speed 0.88/s, stripespacing 2.08, stripe width 0.48.) The whole stimulus wasrotated by 158 clockwise from vertical to avoid aninnate preference observers have demonstrated forperceiving motion along the cardinal axes (see Andrews& Schluppeck, 2000).

Hidden square

Another stimulus used here has been investigatedby several authors looking at illusory contours andedge integration (Lorenceau & Shiffrar, 1992; Shiffrar& Pavel, 1991; Figure 4b). The stimulus has beendescribed as bistable in the past; however, withtraining it is possible to see three distinct arrange-ments of the four moving lines: (a) All four lines canbe grouped to form a diamond oscillating left andright behind illusory occluding bars; (b) Neighboringpairs on each side can be grouped, resulting in theimpression that they are moving up and down; and (c)

Figure 3. Tristable switching patterns. Tristable switching permits more complex switching dynamics to emerge than for bistable

phenomena. (a) A neural adaptation model might suggest that cyclical switching (ABC) is more likely than sequences involving a

backward switch (ABA). (b) Likewise, interhemispheric switching might predict that switches between the two states preferred by one

hemisphere (B and C) are only possible after perceiving the state preferred by the opposite hemisphere (A), resulting in hub-based

switching, characterized by very lopsided transition probabilities.

Figure 4. Static renderings of the four stimuli used in these experiments. The first induced tristable perception in our subjects, and the

fourth induced bistable perception. (a) The Adelson-Movshon grid (AMGrid). (b) The hidden square (HS). (c) The triangle array (Tri).

(d) A bistable plaid motion pattern (Plaid).


Opposite pairs can be grouped, resulting in the sensethat they slide back and forth in depth. As with all ofour stimuli, we used training with disambiguatedforms of the stimulus to ensure subjects were aware ofall three percepts. For the purposes of this paper thestimulus will be referred to as the Hidden Square (HS)illusion. (Stimulus parameters: stripe length 2.08,stripe width 0.48, amplitude of stripe motion 2.08,stripe speed 28/s.)

Triangle array

If tristable perception is mentioned at all, authorsusually refer to a stimulus consisting of an array ofequilateral triangles (e.g., Attneave, 1971), usually

arranged at regular intervals to form a larger triangle(see Figure 4c). Observers tend to group the triangles,describing them as all pointing upward or at an angle of61208 to the vertical. The perceived direction thenalternates over time. In our stimulus the choicescorresponded to (a) up and left, (b) right, or (c) downand left

Plaid

As well as the three stimuli described above, we alsotested one stimulus that provoked bistable perceptionin our observers, to act as a reference to literature onbistable perception. A modern take on the two-orientation version of the AMGrid stimulus involves

Movie 4.

Movie 5. Movie 6.

Movie 3.Movie 1. Movie 2.

Movie 7. Movie 8.

Movie 9. Movie 10. Movie 11.

Movies 1–11. Animated versions of the dynamic stimuli used (Movies 1, 5, and 9), along with various disambiguated versions used

during training to highlight the possible alternative interpretations.


use of transparent lines which enhances the duration oftwo competing percepts (Hupe & Rubin, 2003; seeFigure 4d). In our case the two percepts were (a)diamonds with a single, coherent direction of motion,or (b) transparent lines moving in two distinctdirections. Of particular relevance to the issues beingstudied in this paper, the stimulus has been used inprevious studies of relative switch rate (Sheppard &Pettigrew, 2006). (Stimulus parameters: aperture 108,stripe speed 0.88/s, stripe spacing 2.48, stripe width0.68.)

The parameters given above represent those usedand for which our subjects were best able to perceive allof the potential interpretations without periods ofuncertainty or hybrid percepts emerging.

Animated versions of the stimuli are supplied asmovies for the three dynamic stimuli (Movies 1–11). Asone of the reviewers of this paper pointed out, anotherstimulus capable of eliciting tristable perception is theNecker Cube. Although generally regarded as producingbistable perception, in its static form the cube canappear to be composed of an array of abutting oroverlapping flat, outlined shapes (triangles, diamonds,or rhomboids), especially when viewed from specificangles (Taddei-Ferretti, Musio, & Santillo, 1995). Asecond reviewer also brought our attention to the factthat the plaid motion rivalry stimulus used as ourbistable reference can also produce tristable rivalry sincethe depth order of the separate lines can swap (Hupe &Pressnitzer, 2012). That said, our subjects were in-structed to only report whether they saw coherentmotion or separate lines and received appropriatetraining to that effect (see supplementary materials).Also they enhanced the effect by coloring their two setsof gratings in opposite colors (red and green) rather thanusing a single colour as we did. More to the point, thestimulus has been successfully used in the past in studiesmatching bistable switching behavior across multiplevisual stimuli and forms of rivalry, and hence remains auseful baseline to which switching behavior can becompared (e.g., Sheppard & Pettigrew, 2006).

Display

Participants sat 60 cm from a 24 00 Sony Trinitronmonitor (Sony Corporation, Tokyo, Japan) observingthe stimulus which subtending an angle of approxi-mately 108 · 108. The moving stimuli were rendered ata frame rate of 24 Hz, which was sufficient to give theimpression of smooth motion of the three movingstimuli.

Procedure

The four stimuli were tested for 24 min each overtwo sessions split over 2 days. The order of testing was

counterbalanced across subjects. Each 24-min testsession was broken down into four 6-min blocks with a2-min pause between blocks. Each session was precededby a period of subject familiarization using a disam-biguated version of the relevant stimulus (Movies 1–11). The familiarization process included a period oftraining using the up, down, and left arrow responsekeys on a standard QWERTY keyboard to indicatewhich of the three possible interpretations was cur-rently being perceived. To ensure rapid and accurateresponses, training included a 5-min response selectiontest, during which unambiguous versions of thestimulus were displayed in a rapidly and unpredictablyalternating stream of images (stimulus duration 3000 6500 ms). Subjects were only allowed to proceed to themain part of the experiment if they were able to selectthe appropriate arrow key quickly enough to matchtheir selection to the stimulus displayed for 85% of thetotal time the stimulus appeared. This proved to be ademanding task, as much of the 15% error can beattributed to reaction time. Many subjects requiredmultiple attempts to reach criteria. One subject failed toreach criteria after four attempts and so was droppedfrom the experiment.

Results

Basic switching patterns

The first two predictions to test are those of hub-based and cyclical switching. Figure 5 plots theprobability of cyclical switching. In order to estimatethe significance of these results, it is important to try toestimate what one would expect by chance. Giveninnate preferences or biases in the subjects towardsparticular interpretations,one would actually predict animbalance between the proportion of ABA and ABCswitches on the basis of this bias alone. In order tocompensate for this effect, we calculated the probabilityof a nonrepeating sequence of three states (ABC, BAC,CBA, etc.) based on the assumption of quasi-indepen-dence of states (see the Appendix). Figure 5 includesthese expected levels estimated on the basis of randomsequences generated to match the first-order and lengthcharacteristics of each individual. In practice themajority of subjects do show a preference for cyclicalswitching. This is consistent with a neural adaptationmodel in which neural recovery after a period ofdominance is of a similar duration to the duration ofthe current percept. In other words, neurons corre-sponding to percept A are still recovering by the timeactivity associated with percept B declines, tending topromote percept C next, rather than A.

Figure 6 provides summary data for how often aparticular interpretation was reported. The hub-basedswitching model would predict that one percept shoulddominate as often as the sum of the other two percepts


http://www.journalofvision.org/content/suppl/2013/02/20/13.2.24.DC1/JOV-03117-2012-s1.pdf

(computed as the number of visits, not perceptduration). There is no evidence for this. Figure 7reports the period of dominance duration acrosssubjects, ranked by switch rate. The durations areclearly quite variable across subjects, but one positiveconsequence of this is that no single percept appears todominate across all subjects. What also emerges fromthe data is that subjects who saw all three perceptsequally often also experienced the fastest switch rates, aresult consistent with work on stimuli evoking bistableperception (Moreno-Bote, Shapiro, Rinzel, & Rubin,2010).

Results from the two graphs are combined in Figure9, which shows the link between the duration of apercept and the probability of it occurring (i.e., theproportion of switches to that state). All three stimulireveal a degree of correlation between the twomeasures, suggesting that subjects often had a preferredinterpretation of the stimulus (switched to more often)that was also stable for longer periods or, equally, a lesspreferred interpretation (switched to less often) thatwas only briefly perceived: AMGrid (r2¼ 0.44, p-slope, 0.01**), HS (r2¼ 0.17, p-slope , 0.01**), andTriangle (r2¼0.42, p-slope , 0.01**). One consequenceof this relationship is that the duration of ABA

switches should, on average, be longer than ABCswitches.

To investigate this prediction further, Figure 8reports the duration of an ABA versus ABC switch forall 24 subjects and all three stimuli. There is a cleartendency for ABA switches to take longer than ABCswitches in any one individual. A paired t-test analysisconfirmed that the differences were statistically signif-icant for all three stimuli: AMGrid t23 ¼�5.03, p ,0.01**; HS t23¼�2.22, p , 0.05*; Tri t23¼�2.37, p ,0.05*. As described above, this link is predicted by therelationship between preference for a state and itsaverage duration, established in Figure 9. It isconceivable that other effects are at work too, but thisappears to be a sufficient explanation for the effect.Irrespective of its source, this relation is a testableprediction of any model of switching behavior. Note wealso carried out extensive analyses of switch ratedurations. For those interested, please see thesupplementary materials section entitled: ‘‘Analysis ofswitch-rate distribution.’’

Stochastic process analysis of transitions

With bistable perception, the study of state transi-tions during prolonged, uninterrupted viewing is

Figure 5. Probabilities of tristable versus bistable switching. The probability of each subject producing a forward (ABC), as opposed to

reverse switch (ABA), is recorded for the three tristable stimuli. The green lines indicate the probability of such a switch pattern based

on a quasi-independent model derived from an individual’s perceptual state probabilities. The dotted lines are confidence intervals

derived from simulations of the model matched in length to the sequence lengths of the subjects. Overall there is a tendency for

subjects to produce more ABC type switches than would be expected by chance. This is largely consistent with an adaptation model

that promotes tristable over bistable switching.


http://www.journalofvision.org/content/suppl/2013/02/20/13.2.24.DC1/JOV-03117-2012-s1.pdf

Figure 7. Relative duration of perceptual dominance of the three states for all 24 subjects. For a description of the individual states

see the description of the test stimuli in the Methods section (the colors correspond to states A [blue], B [green], and C [purple],

respectively). Subjects are ranked by switch rate along the horizontal axis lowest to highest for each of the three tristable percepts.

Interesting to note is that as switch rate increases (left to right), dominance duration for all three percepts becomes more similar.

Figure 6. Ratios of the total number of visits to each perceptual state for each stimulus. The colored bars correspond to each of the

three perceptual states. For a description of the individual states, see the description of the test stimuli in the Methods section (the

colors correspond to states A [blue], B [green], and C [purple], respectively). A possible prediction of the interhemispheric switching

hypothesis is that the number of transitions to one state is will be equal to the sum of transitions to the other two. There is no

evidence for this.


limited to their timing. With the inclusion of a thirdperceptual state, a range of new questions emerge. Forexample, one can ask about transition probabilities,i.e., the extent to which the choice of subsequent statesis predicted by the current state. It is possible that thereare preferences to move from one state to another orthat a state will be revisited after a set number ofintermediate states. Any patterning may reveal detailsof the underlying mechanisms involved.

Following the example of Attneave (1959), Gottmanand Roy (1990) offer an approach for assessingdeterminacy in sequences based on information theoryin which one can gain an estimate of the ‘‘surprise’’associated with the distribution of observed occur-rences of sequences of progressively increasing length.As Attneave (1959) points out, this approach is onlyasymptotically equivalent to the chi-squared approachthat it is imitating, and with the advent of fastcomputers, simulations and explicit calculations arenow tractable, obviating the need for these forms ofapproximation. Unfortunately, neither approach isreally valid for what is being asked in this case. Allsequential analysis based on chi-squared contingencytables relies on the fact that the expected values at allpoints in the table are nonzero, but this is not true inour case because state repetitions cannot occur. Inother words, AA or BB is not possible since we are onlyconsidering transitions from state to state. Bishop,Fienberg, and Holland (1975) review methods foranalyzing incomplete contingency tables, but it turnsout that for the special case that only the leading

diagonal entries are missing/zero, an exact model exists(Goodman, 1968, see his table 11). An appropriatemodification to the information theory approach is alsodescribed by Chatfield and Lemon (1970), but as theypoint out, for small samples the approximationsinvolved may be violated. Given that many of thesequences in the experiments described here werearound 100 steps long, the approach is probably notsuitable; hence we adopted the Goodman (1968)approach to search for first-order relations. Figure 10displays the results of the analysis. There does appearto be evidence for first-order determinacy for theTriangle stimulus, but not for the other two.

But what about higher-order relationships? TheGoodman (1968) approach is only useful for testingfirst order relations. Naber, Gruenhage, and Einhauser(2010) describe a method for determining whethersecond- or higher-order relations in the switchsequences exist. Modern techniques based on maxi-mum-likelihood estimation are another alternative andhave the advantage that they can pick apart the detailsof any relations. Buhlmann and Wyner (1999) describethe basis for an approach they call Variable-LengthMarkov Chain modelling. According to Machler (2009,personal communication), variable length markovchains (VLMC) has the advantage that it does not relyon the assumption of independence of the perceptualstates (c.f. Bishop et al., 1975). For the relatively shortsequences produced by subjects in these experiments, itis debatable whether there is sufficient power to identifypatterns definitively. To control for this, random

Figure 9. Link between the probability of selecting a percept

and its relative duration. In all three stimuli, the more likely it

was for a subject to switch to a particular percept, the longer

that percept persisted. The link was significant for all three

stimuli, although the overall effect was relatively weak.

Figure 8. Tristable transition times. Comparison of the time

taken to complete an ABC type transition (tristable) compared

with an ABA type transition (bistable) for each stimulus (each

point represents an individual subject). Note that for the vast

majority of cases the time to complete an ABA switch is longer.

This link is predicted by the relationship between preference for

a state and its average duration, established in Figure 8.


sequences of states were generated matched to the stateprobabilities of the subjects and the sequence lengthsthey produced. Preliminary investigations using animplementation of VLMC, written by Machler andBuhlmann (2004) in the statistics language ‘‘R,’’ arereported here. The first step of the analysis involvesdetermining a criterion level of a sequence order limitparameter that they refer to as ‘‘K.’’ In their paper theauthors provide a method for estimating this parameterbased on Aikaike’s C (AIC), which was the methodemployed here. The procedure was applied to each ofthe 24 subjects’ individual switch patterns for the threetypes of stimuli and formed a minimum at a cutoffcriterion centred around K ¼ 4 (see Machler andBuhlmann, 2004). Using this value the analysis was runfor each individual’s data for each of the three stimuli.The results appear in Figure 11. Note that the newanalysis takes a different approach to that offered bythe Goodman. His analysis is looking for discrepanciesfrom a nonrepeating sequence, whereas this analysis islooking for discrepancies from a repeating sequence,which is why first order relations almost always emerge.One means of testing the significance of any higher-order relations is to compare them with what onewould expect by chance. For that reason, the figure alsoprovides 95% confidence intervals derived from simu-lation of random sequences whose lengths werematched to the lengths of sequences generated by thesubjects. Note that one might expect the order of arandom sequence should be zero, but as describedabove, because the ‘‘random’’ sequences in this case do

not repeat (AA, BB, or CC), the states are notindependent. The sequences are also of limited length,which has the potential to generate spurious low-orderdependencies.

Because the VLMC approach looks for discrepanciesfrom true randomness, it is not possible to decide thesignificance of the first order relations, since similarlength nonrepeating random sequences also containsignificant first order relationships. The Goodmananalysis is best suited for that purpose. Although first-order relationships are part and parcel of any non-repeating sequence, higher-order relationships are not.The VLMC analysis indicates that some subjectsexhibit higher-order relations that deviate significantlyfrom chance. Looking across subjects and stimuli, thepattern is not very consistent. The only slight consis-tency is a tendency for subjects with higher switch ratesto produce higher-order patterns. Unfortunately, thismay simply be due to the fact that these subjectsproduced longer sequences of switches for the analysisto search through. In the end, the sequences areprobably too short to tell us definitively when higher-order relations do not exist. Instead they should be seenas a lower bound on the order which, for some subjectsat least, is already significantly higher than one.

Periodicity of bistable and tristable switching

Another question one can ask about the pattern ofswitching is whether there are blocks of time duringwhich subjects experience tristable switching followed

Figure 10. First-order state transition analysis. The three tristable stimuli were tested for evidence of first order (i.e., single state

transition or Markov) determinacy using a suitably corrected chi-squared analysis (AMGrid: blue bars, HiddenSquare: green bars,

Triangle: red bars). There is some evidence that switching in the triangle stimulus (red bars) departs from random switching because

the chi-squared test rejected the null hypothesis of quasi-independence in 12 of 24 subjects ( p , 0.05). However, no such trend

emerged for either of the other two stimuli, and correcting for multiple comparisons sees the number of cases which achieve

significance drop to 5 of 24 ( p , 0.0021).


by periods in which they experience bistable switching.In some sense this is related to the stochastic processanalysis, but not trivially so because the precise orderof states is less important than whether they resulted ina bistable or tristable switch. To test for periodicity oftwo-step switches, the switch types of each individualwere reduced to a stream of switch types 1 ¼ tristable(ABC) and�1¼ bistable (ABA). The stream of eventswas then analyzed using autocorrelation. An appro-priate baseline for the autocorrelation might be thoughtto be 0.0, but this is not the case because theprobabilities of a bistable versus tristable switch are notequal and they vary across subject. In order to estimatea baseline, the probabilities of each transition type wereused to generate 1,000 sequences of a length matched toeach individual and their specific probabilities of ABCversus ABA switching. The values at each statetransition lag from the average of the simulatedsequences were then used to normalize the values fromthe real data generated for each individual bysubtracting the simulated data mean from the actualautocorrelation figure calculated. The results appear inFigures 12 and 13.

As an alternative to the cross-correlation approach,one can also simply estimate the probability of a switchof the same type as the current one (be it bistable ortristable) at differing time lags. To ensure that the

cross-correlation approach was not introducing anyspurious effects, we also ran the analysis in thismanner, using simulation to predict the probability of asame transition type (be it bistable or tristable) onewould expect by chance for each individual. Thedifference between the chance value and that obtainedfrom each subject was then used as dependent measurefor a group-level t-test and nonparametric (Wilcoxon)equivalent. The results were basically indistinguishableto those obtained using the cross-correlation approach.The only minor difference was that the significanteffects became marginal at lags of 1 and 3 for the Tristimulus on t test (lag 3, p¼ 0.057) or nonparametric(Wilcoxon) test (lag 1, p¼ 0.052). In the forerunninganalysis, the effects of bistable and tristable repetitionsare combined. To identify the source of these effects interms of the two types of repetition, a further analysiswas conducted on the two types of repetition inde-pendently. Analysis using both t tests and the Wilcoxontest singled out the same lags as the cross-correlationanalysis as being the only lags at which repetitions ofboth bistable and tristable switches were significantlyabove chance.

To summarize these effects, it seems that whichevertype of switch observers have just experienced (be ittristable or bistable), they are likely to experience thesame type of switch in the following two transitions.

Figure 11. Variable Length Markov Chain analysis. The error bars represents the mean and corrected 95% confidence interval derived

from random sequences matched to the lengths of sequences generated by each of the 24 subjects and their individual discrete state

probabilities. For a minority of subjects there is clear evidence of divergence from a simple quasi-independent process (order 2 and

above) and these deviations cannot be explained by random fluctuation, state probability, or sequence length. However, for the

majority of subjects only a first order relation is present, which cannot be distinguished from chance behavior in this analysis.


Beyond that point, the transitions appear largelyrandom. One interpretation of this result is that, onaverage, subjects experienced brief periods in which thestates were seen approximately equally often, leading toconsistent tristable switching, followed by periods overwhich one stimulus became suppressed, resulting inconsistent bistable switching. This is, in turn, consistentwith a periodic (presumably internally generated) signalcyclically promoting tri- and bistable switching. Itshould be added at this point however, that we cannotsay for certain whether there are any specific patterns ofswitching (e.g., ABC rather than BCA, or ABA ratherthan CBC etc.) underlying the results, or if thepatterning is generic to any type of tristable andbistable switch. Future research should consider this inmore detail by gathering a larger number of switchevents.

Stimulus onset effects

The response of any oscillatory system to an externalstimulus is a function of both the input and its currentstate. For example, if the system is assumed to beinitially inactive, stimulus onset will produce a periodof increasing system activity that may well result indifferent patterns of switching from those seen duringsteady-state switching. Switching patterns peculiar tostimulus onset have indeed been reported for bistableperception (Hupe & Rubin, 2003). In the particularcase of binocular rivalry featuring an in-built contrastbias, Mamassian and Goutcher (2005) were able toshow an initial preference for the high-contrast imagethat waned over time. Of more direct relevance to thestimuli used here, Hupe and Pressnitzer (2012) detected

Figure 12. Autocorrelation analysis for concurrent ABA or ABC

type switches (individuals). The purpose of the analysis is to see

if the likelihood of performing an ABA or ABC switch is affected

by switching at previous time steps (lags) from the current

switch. Significant deviations from zero indicate deviations from

correlations predicted by chance. The Normalized Autocorrela-

tion measure is derived from the autocorrelation values for a

particular subject minus the value one would expect by chance

given the relative probabilities of ABA and ABC type switches

for that individual. For all three stimuli the first two lags

produced higher than expected correlations.

Figure 13. Autocorrelation analysis for concurrent ABA or ABC

type switches (population). The graph summarizes the same

data presented for individuals in the previous figure. For all

three stimuli the first two lags produced higher than expected

correlations, suggesting that the current type of switch (ABA or

ABC) predicts the type of switch that will be seen next and the

one subsequent to that (or equally the two previous switches).

Beyond two switches the links are more sporadic. Error bars

represent the 95% confidence intervals, and asterisks indicate

significant deviation from chance for both t test and the

Wilcoxon test at p , 0.05.


onset transients for the Plaid stimulus used here. Intheir study, the authors reported an increased durationof the first stable percept relative to subsequentdurations, an effect they refer to as the ‘‘inertia of thefirst percept.’’ Their subsequent analysis was based onthis finding as all measures were related back to this‘‘maximum.’’ In order to test whether an equivalentform of perceptual ‘‘inertia’’ was detectable in our data,we simply plotted the difference in log-transformedduration of the percept immediately subsequent to thecurrent percept (see Figure 14). We obtained aconsistent pattern, but it was actually in the oppositedirection to that described by Hupe and Pressnitzer(2012). The reason for the discrepancy is not clear. Weare using slightly different parameter values for thePlaid (e.g., a single color for both sets of lines). We alsodid not attempt to equate the long-term perceptualdominance of the two percepts as those authors did,and we included data from all trials, not just those inwhich subjects reported the coherent percept first.Nonetheless, it seems unlikely that minor alterations tothe stimulus or analysis would reverse this highlyconsistent effect. It is also striking that the effect isconsistent across all four of the stimuli and all fourblocks of trials. One potential problem with askingsubjects to report state changes as and when they sawthem is that we cannot be sure if their responses arecorrect. Some subjects took several seconds to maketheir first response, and so we cannot rule out thepossibility that their first response was different to thefirst percept they had. Nonetheless, the training session(with strict timing criteria) trained them to respond asquickly and accurately as possible, and the consistencyof behavior across stimuli and subjects suggests thateven slow subjects were behaving in a mannerconsistent with those subjects who responded morerapidly. More work will be required, but suffice it tosay that there is evidence for unusual dynamics at themoment of stimulus onset. In our case the effect is forthe first percept to be shorter than the subsequentpercept, a result that is consistent with preliminarysimulations of a tristable oscillator in our lab (Wallis &Ringelhan, 2009). The reason for this appears to be thatall states accumulate activity initially due to a lack ofadaptation in any of the populations. This then allowsthe first dominant state to be usurped sooner thanduring steady-state activity when recovery from adap-tation prevents a second state from rising too quickly,and the dominant state is firing fully—thereby inhib-iting the other neural populations more strongly.

In their paper on bistable and tristable stimuli, Hupeand Pressnitzer (2012) report a strong preference intheir subjects to see the coherent version of the Plaid atstimulus onset. For sake of comparison we analyzedour own data for all four stimuli across all four blocksand found similar biases. In order of the description of

each of their perceptual states in the Methods section

above: AMGrid (62.5%, 10%, 27%), HS (67%, 16.5%,

16.5%), Tri (46.5%, 7%, 46.5%), and Plaid (89.5%,

10.5%). The latter pattern is largely consistent with the

bias reported by Hupe and Pressnitzer (2012). Note,

however, that these figures are only a first indicator of

the full story. A full analysis would need to take

Figure 14. Relative duration of pairs of consecutive percepts. To

check for onset transient behaviour the first five perceptual

states were analyzed in terms of their duration relative to one

another. Relative duration was measured as the difference

between the log-transformed durations of the two percepts in

seconds. The horizontal axis lists the pair of perceptual

durations being compared (‘‘1–2’’ indicates the first stable

percept versus the second, etc.). Negative numbers indicate

that the earlier percept was shorter in duration than the later

percept, and positive numbers that they were longer. Note that

there was a strong tendency across all subjects and stimuli for

the first percept to be shorter in duration than the second

percept. There was also a slight tendency for the second

percept to last longer than the third, but beyond that point no

obvious pattern emerges. This appears to provide some

evidence for transient stimulus onset behavior. Each bar

corresponds to a single subject with their data averaged across

the four blocks of trials.


account of inherent perceptual biases for each stimulusand each individual.

We also checked for evidence of increased variabilityin perceptual duration at onset. A test across subjectsof the duration variability over four switches (measuredas standard deviation of the log-transformed durationsaveraged across all four switches and all four blocks)yielded no significant change from the first fourswitches to the next four for any of the stimuli.

Effects of transition order on percept duration

In their paper on tristability, Naber et al. (2010)described links between perceptual duration andpreceding or subsequent perceptual state. We repeatedthe analysis they conducted on median-normalizeddominance durations (see figure 4 of their paper) tolook for evidence that the durations are influenced bythe preceding state. Results from our analysis appear inFigure 15. As is clear from the figure, we did indeedfind evidence for an increased duration of percept Awhen transitioning from C, as well as an increasedduration of percept C when transitioning from A, ascompared to transitions from B to A or B to C.However, this asymmetry was only apparent for theAMGrid.

One of the things to note about the AMGrid is that itis closely allied to the two stimuli used by Naber et al.(2010), both of which involved two sets of rotationallyoffset parallel lines. As a result, both of their stimuliproduced the impression of lines oriented upwards,leftwards and/or rightwards, leaving open the possi-

bility that the types of asymmetry reported in theirpaper and for the AMGrid here, are limited to thisfamily of stimuli. One reason for thinking this might bethe case is that we know from Andrews and Schlup-peck’s (2000) study that observers have a preference forseeing the AMGrid moving vertically or horizontallyrather than at an oblique orientation—possibly due toeye movements, the oblique-effect (Andrews &Schluppeck, 2000), or the oblique plaid effect (Hupe &Rubin, 2004). We deliberately oriented our stimulussuch that no perceived motion ran exactly vertically orhorizontally, but this still resulted in two percepts (Aand C) being closer to the ordinal axes than the third(B). Looking at dominance durations for the AMGridin Figure 7, the vast majority of subjects spent less timeseeing percept B than the other two. Similarly, lookingat Figure 6, they also transitioned to it less frequently,all of which points to an orientation-specific effect. Thefact that the link between previous state and durationfails to occur for the other two stimuli tested hereprovides further grounds for thinking that the link maybe unique to those patterns containing oriented lines.

Switch rate correlations

One of the main driving forces behind attempts tounite all switching phenomena into one model is theconsistency of relative switch rates for individualsacross stimuli. Some studies have reported r2 values ashigh as 0.7 between switch rates for binocular and plaidrivalry stimuli (Sheppard & Pettigrew, 2006). Correla-tional analysis was conducted between the four stimuli

Figure 15. Median dominance duration for the 24 subjects. Data have been prenormalized to individual median durations. Data are

broken down by previous percept—in the case of the current interpretation being ‘‘A,’’ responses are split into transitions from ‘‘B’’and transitions from ‘‘C.’’ Any discrepancies in the two neighboring distributions are indicative that the previous percept impacts the

dominance duration of the current percept. The error bars indicate upper and lower quartiles and asterisks indicate a statistically

significant difference using the Wilcoxon rank sum test (*p , 0.05, **p , 0.01).


under investigation. Results appear in Table 1, in whichthe data are reported for all switches, for bistableswitches alone, and for tristable switches alone.Correlations were relatively low (r2 � 0.2) except forthe two stimuli which were the most similar inappearance: the Plaid and AMGrid (r2¼ 0.6). Asterisksindicate a significant value of slope of above zeromagnitude (p , 0.05). The analysis revealed weak butsignificant correlations in switching for three of thefour stimuli (all except the HS stimulus).

In earlier studies by Carter and Pettigrew (2003) andSheppard and Pettigrew (2006), the high correlationsreported were obtained with larger numbers of subjects(up to two or three times the number tested here). It isquite possible that higher correlations would have beenobtained by testing larger numbers of subjects,although current evidence suggests that correlationswould not have been as high as in their studies evenwith the addition of more subjects.

Another important consideration to bear in mindwhen interpreting these correlations in switch rate is thefact that the relative dominance of the percepts variedacross subject and stimulus. As mentioned above, bothour results and those of Moreno-Bote et al. (2010) haveshown that switch rate is affected by the degree ofperceptual ‘‘balance’’ between the percepts, and it maywell be the case that failure to balance the relativeduration of the three percepts affects individualsdifferently across stimuli, leading to the reported loss ofcorrelation (Schwartz et al., 2012). In future experi-ments one option might be to tailor the parameters ofthe stimuli to each subject so as to obtain matcheddominance durations, as proposed by Schwartz et al.(2012). That said, in the Sheppard and Pettigrew (2006)paper, the authors also chose not to balance thedominance durations of their stimuli (see their figure 3)and were nonetheless able to report highly correlatedswitch rates across stimuli.

Modelling tristable switching

It has been evident for many years that one way tosimulate multistable perceptual switching would bethough some form of simple oscillatory circuit (e.g.,Attneave, 1971). One of the earliest models to capture

the ‘‘all-or-nothing’’ nature of the perceptual experiencein binocular rivalry was Matsuoka (1984). While hiswork continues to be cited today, it is often used as anexample of a naive forerunner of more sophisticatedmodels which integrate eye- and object-based influ-ences. Nonetheless, a recent model by Noest, van Ee,Nijs, & Wezel, (2007) has had success in simulatingnumerous aspects of rivalry, including the effects ofinterruption (Leopold et al., 2002; Orbach et al., 1966),based closely on Matsuoka’s models.

Of relevance to the work on tristable perceptiondescribed here, and something many researchers inperceptual rivalry might not know, Matsuoka went onto study more general situations in which more thantwo states interact. In a pair of papers he describedseveral higher-order models including a fully connectedthree-way oscillator (Matsuoka, 1985, 1987). In ourown preliminary studies based on his model, we havediscovered evidence for a number of interestingphenomena (Wallis & Ringelhan, 2009). For example,when the system enters a period of bistable switching(e.g., ABAB), the overall duration of each perceptlengthens (i.e., switch rate drops) relative to that seenduring tristable switching. We also find that activationof the population associated with the interpretation nolonger perceived (e.g., C), synchronises (phase-locks)with that of one of the dominant percepts (e.g., A),resulting in a lengthened predominance of that staterelative to the other perceived state (e.g., B), but at alower level of activation. These results are onlypreliminary, but point to other ways in which the studyof tristable stimuli can produce a range of complex,testable predictions.

Conclusions

Although much progress has been made, theprocesses and neural substrates underlying perceptualrivalry remain poorly understood. This paper hasstudied switching behavior for a family of stimuli thatevoke tristable perceptual rivalry in observers. We haveargued that such stimuli can help discriminate com-peting theories of rivalry but also unravel what the

Switch mode AMGrid-HS AMGrid-Tri AMGrid-Plaid HS-Tri HS-Plaid Tri-Plaid

All 0.0025 0.1334* 0.6029* 0.0421 0.0096 0.1765*

ABA (bistable) 0.0001 0.1766* 0.6651* 0.0534 0.0007 0.2199*

ABC (tristable) 0.0576 0.0722 0.6060* 0.0754 0.0231 0.1694*

Table 1. Correlation coefficients for individual switch rates across all six possible pairings of the four test stimuli (reported as r2).

Notes: Three correlation coefficients are quoted in each case: ‘‘All’’ the average rate across all switch types, ‘‘ABA’’ only thoseswitches in which the third state matched the first (bistable switch), ‘‘ABC’’ only switches in which all three states are visited in asequence (tristable switch). Note that for the Plaid stimulus, a tristable switch rate does not exist. In this case the bistable switch ratewas compared with the tristable rate of the other stimulus.


underlying mechanisms might be. Compared to bista-ble perception, tristable perception possesses complexand yet tractable dynamics which yield bolder and, insome cases, contradictory predictions from competingmodels of perceptual rivalry.

One of the initial findings was that in contrast topredictions of interhemispheric switching, there was noevidence for hub-based switching, in which onetransition (e.g., B to C) never occurs, or in which thesum of visits to one state is equal to the sum of visits tothe other two. Although not an insurmountableobjection to the hypothesis, it does appear to demandmore complex dynamics than the current character-ization of the model would permit. On a more positivenote for this and other hypotheses, there was someevidence for carry-over of relative switch rates betweenstimuli. However, the effects were surprisingly weak.

A more mainstream alternative to the interhemi-spheric switching model is one based on neuraladaptation combined with mutual inhibition. Theresults described in this paper are broadly consistentwith such a model in so far as cyclical switching wasseen to occur more often that chance in all subjectsacross all stimuli. What this result also tells us is thatthe recovery period for a neural population immedi-ately after a period of dominance is of the same orderas the dominance duration of the subsequent state—since this imparts an advantage for the third state todominate when the second state wanes. Given thatstimulus interruption promotes reoccurrence of theprevious state (Leopold et al., 2002; Orbach et al.,1966) it was conceivable that the opposite might betrue—there would be a preference for reverse ABA typeswitching. That this does not happen places constraintson the decay time of a previously dominant state and itsrecovery time, both relative to the dominance of theimmediately subsequent percept. It may also indicatethat neural states associated with the oscillator do notcut off as cleanly or suddenly as the perceptualexperience—suggesting a higher-level thresholding de-cision process at work.

Regarding previous studies of tri and quadri-stableperception, we were unable to replicate the findings ofNaber et al. (2010) regarding links between dominanceduration and subsequent state choice, except in thespecial case of a stimulus very similar in appearance tothose which they used. We would argue that more workneeds to be done before one can feel confident that thetypes of relationship they describe apply to patternsother than oriented parallel lines. We would likewiseargue that the perceptual trapping described by Suzukiand Grabowecky (2002) is likely a function of thecombined monocular/binocular stimuli they used.Given the absence of higher order patterns in thestochastic process analysis conducted here, we wouldargue that no discernible long-term perceptual trapping

emerged for the stimuli we tested, although on a shortertimescale we did find evidence for being ‘trapped’ intocycles of tristable and then bistable switching.

Despite their long history and appeal, adaptationmodels of perceptual rivalry have received criticismover a number of years. For one, they predict that thetime for which a percept is suppressed should correlatewith the time for which it is subsequently activefollowing reversal. In fact, periods of dominance andsuppression of a particular percept are uncorrelated(Taylor & Aldridge, 1974; Walker, 1975). Modernmodels accommodate this issue by applying liberalamounts of randomizing (decorrolating) ‘‘noise’’ at theinput or internal stages. This may seem a little fishy, butit currently remains the best means available forcapturing the complexities of attentional and memorybased top-down influences and bottom-up influences offixation location and eye-movements (if only duringpursuit).

Another prediction of adaptation models to causeproblems in the past is the idea that if neuraladaptation is taking place, a concurrent stimulusdetection task that is reliant on the same neuralpopulation should also adapt. To test this, detectionthresholds have been measured for briefly interjectedstimuli presented to either the dominant or nondom-inant eye during binocular rivalry. Inconveniently foradaptation models, no evidence for changes in thresh-old was found (Fox & Check, 1972; Norman, Norman,& Bilotta, 2000). That was, until very recently, whenAlais, Cass, O’Shea, & Blake (2010) took a differentapproach, normalizing adaptation times to the dura-tion of the percept, rather than predicting thresholds onthe basis of time since the last perceptual switch. Withthe help of these recent discoveries, all current evidenceseems to be pointing inexorably to a model based onmutual inhibition and adaptation. The power of thesemodels was further emphasized by Noest et al.’s (2007)elegant wrinkle on the models of Matsuoka and others.As mentioned earlier, they were able to explain how anumber of well-established peculiarities of perceptualrivalry can be encapsulated in the model itself, withoutrecourse to external memory or other processes. Indeedit may well be the case that the shunting inhibitionmodel that they introduced can explain why timerelative to dominance duration is the controllingvariable in Alais et al.’s (2010) study, rather than timeitself.

Whatever the mechanisms are that underlie percep-tual rivalry in humans; tristable perception appears topresent an exciting and exacting situation in which totest actual neural behavior. The model described herealso offers concrete predictions in terms of increasedvariability at stimulus onset, rate changes betweenperiods of tristable and bistable switching, and phaselocking of a dominant and nondominant population


during periods of bistable perception. Hence tri andquadristable perception, and adaptive models of thetype described in this paper, appear well placed to helpneuroscientists identify both the mechanisms andneural substrates of perceptual rivalry in all its guises.

Keywords: rivalry, multistable perception, bistableperception, tristable perception

Acknowledgments

We are grateful to two anonymous reviewers fordetailed and expert comments that helped improvemany aspects of the manuscript. GW was supported byQEII and Future Fellowships from the AustralianResearch Council (DP0343522, FT100100020), and SRwas supported by a Program Grant from the HumanFrontier Science Program held by GW (HFSP-GrantRGP 03/2006).

Commercial relationships: none.Corresponding author: Guy Wallis.Email: [email protected]: Centre for Sensorimotor Neuroscience,School of Human Movement Studies and QueenslandBrain Institute, University of Queensland, Australia.

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Appendix

In order to assess the effects of inherent biases insubject perceptual preferences, it is necessary toestablish a measure of expectation associated with theswitch sequences they produced. The method employedhere involved creating random sequences matched interms of length and overall state probability to eachindividual. Because we know a priori that the statescannot repeat, the calculations for this are not trivial,but they are relatively straightforward to derive. If theprobabilities for the three states (A, B, or C) aredenoted p(A), p(B) and p(C), for a switch sequence oflength n with state given by si for the ith state, theformula for any particular state is simply:

pðAÞ ¼ 1

n

Xn

ðsi ¼ AÞ

To determine the probability of a particular three statesequence si, siþ1, and siþ2, we need to calculate theappropriate conditional probabilities. Taking the ex-ample sequence ABC, the probability of the sequenceoccurring p(ABC) is given by

pðABCÞ ¼ pðAÞpðBjAÞpðCjBÞwhere p(B j A) denotes p(siþ1¼ B j si ¼ A).

The formulae for the conditional probabilities needto take account of the quasi-independence of thesequences (no same-state transitions). To do this weneed to establish selection probabilities p(a), p(b), andp(c),which correspond to the states A, B, and C. Whengenerating a sequence, if state A is selected, theprobability that it is a same-state transition is p(a).p(a)¼ p(a)2, representing a measure of state transitions‘‘lost’’ due to the constraint that same-state transitionsare not possible. These corrected selection rates are inexact proportion to the state probabilities p(A), p(B),and p(C):

pðAÞ ¼ kbpðBÞ ¼ kcpðCÞ

pðaÞ � pðaÞ2 ¼ kb

�pðbÞ � pðbÞ2

�

¼ kc

�pðcÞ � pðcÞ2

�

Hence we can derive expressions of the form

pðBÞ�pðaÞ � pðaÞ2

�¼ pðAÞ

�pðbÞ � pðbÞ2

�

Since p(A) þ p(B) þ p(C) ¼ 1, we can derive exactexpressions for the selection probabilities:

pðaÞ

¼ 4pðAÞpðBÞ � 4pðAÞ � 2pðAÞ þ 4pðBÞ2 þ 1

4pðAÞ2 þ 4pðAÞpðBÞ � 4pðAÞ þ 4pðBÞ2 � 4pðBÞ þ 1

pðbÞ

¼ 4pðAÞpðBÞ � 4pðBÞ � 2pðAÞ þ 4pðAÞ2 þ 1

4pðAÞ2 þ 4pðAÞpðBÞ � 4pðAÞ þ 4pðBÞ2 � 4pðBÞ þ 1

pðcÞ ¼ 1� pðaÞ � pðbÞThese selection probabilities are used to governselection of states in the data simulation, providingrandom sequences which match the empirically derivedstate probabilities p(A), p(B) and p(C) under theconstraint that p(A j A), p(B j B) and p(C j C) ¼ 0.

The corrected conditional probabilities can also bederived, producing equations of the form:

pðBjAÞ ¼ pðbÞpðbÞ þ pðcÞ

pðCjBÞ ¼ pðcÞpðaÞ þ pðcÞ

Hence from the expression given above for p(ABC):

pðABCÞ ¼ pðAÞpðcÞpðbÞ�pðaÞ þ pðcÞ

��pðbÞ þ pðcÞ

�


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