When an ambiguous stimulus is viewed for a prolonged time, perception alternates between the different possible interpretations of the stimulus. The alternations seem haphazard, but closer inspection of their dynamics reveals systematic properties in many bistable phenomena. Parametric manipulations result in gradual changes in the fraction of time a given interpretation dominates perception, often over the entire possible range of zero to one. The mean dominance durations of the competing interpretations can also vary over wide ranges (from less than a second to dozens of seconds or more), but finding systematic relations in how they vary has proven difficult. Following the pioneering work of W. J. M. Levelt (1968) in binocular rivalry, previous studies have sought to formulate a relation in terms of the effect of physical parameters of the stimulus, such as image contrast in binocular rivalry. However, the link between external parameters and “stimulus strength” is not as obvious for other bistable phenomena. Here we show that systematic relations readily emerge when the mean dominance durations are examined instead as a function of “percept strength,” as measured by the fraction of dominance time, and provide theoretical rationale for this observation. For three different bistable phenomena, plotting the mean dominance durations of the two percepts against the fraction of dominance time resulted in complementary curves with near-perfect symmetry around equi-dominance (the point where each percept dominates half the time). As a consequence, the alternation rate reaches a maximum at equi-dominance. We next show that the observed behavior arises naturally in simple double-well energy models and in neural competition models with cross-inhibition and input normalization. Finally, we discuss the possibility that bistable perceptual switches reflect a perceptual “exploratory” strategy, akin to foraging behavior, which leads naturally to maximal alternation rate at equi-dominance if perceptual switches come with a cost.

*f*

_{A}, can vary over the entire possible range of 0 ≤

*f*

_{A}≤ 1 (or most of it)—that is, alternations occur not only when the competing percepts are approximately equally probable but also when they are largely unbalanced. Here, we refer the percept that dominates for a larger fraction of the time as the “prevalent” or the “strongest” percept.

*A*and

*B*, the mean dominance durations of each of the percepts, denoted

*T*

_{A}and

*T*

_{B}, may change independently of each other. Thus, although

*f*

_{A}and

*f*

_{B}must sum to 1, the system has another degree of freedom (in principle), specified by

*T*

_{A}or

*T*

_{B}. The value of one determines the other (for a given

*f*

_{A}), since they are related through

*T*

_{A}/[

*T*

_{A}+

*T*

_{B}] ≅

*f*

_{ A }. (Note that both the fractions of dominance and the mean dominance durations have been found to be stable over time for both binocular rivalry and ambiguous motion displays and can therefore serve as reliable experimental measures; Hupé & Rubin, 2003; Mamassian & Goutcher, 2005; Merk & Schnakenberg, 2002; Rubin & Hupé, 2004.)

*f*

_{ A }=

*f*

_{B}= 0.5; we will refer to the percepts as “equi-dominant.” Assume that the mean duration is

*T*

_{eq}; we therefore have

*T*

_{A}=

*T*

_{B}=

*T*

_{eq}. Figure 1a shows a schematic representation of the time course of perception as observers view this stimulus, alternating between equal durations of percepts A and B (we use regular alternations for illustration purposes). Since these particular values of

*f*and

*T*(0.5 and

*T*

_{eq}, respectively) were obtained for a particular parametric setup, changing the value of one (or more) of the parameters of the stimulus may therefore change them. Let us denote by

*μ*

_{B}a parameter that affects perception such that reducing its value causes a decrease in

*f*

_{B}. Now, if

*T*

_{B}indeed represents an additional degree of freedom of the system, then the decrease in

*f*

_{B}could come about in many ways, corresponding to different combinations of

*T*

_{A}and

*T*

_{B}that leads to the same reduction in

*f*

_{B}. Three specific possibilities are of particular interest here, and they are illustrated in the three panels of Figure 1b: in the top panel, the decrease in

*f*

_{B}takes place via a decrease in

*T*

_{B}with no change to

*T*

_{A}; in the middle panel, there is both a decrease in

*T*

_{B}and an increase in

*T*

_{A}; and in the bottom panel,

*T*

_{A}increases while

*T*

_{B}remains unchanged. What determines which of these cases (or an intermediate situation) will, in fact, occur?

*T*

_{A}and

*T*

_{B}change as the parameters of the stimulus change. Interestingly, his results suggest that the answer depends on the particular ways the parameters are manipulated. Specifically, Levelt varied the strength of the image presented to one eye (e.g., by changing its contrast) without changing the strength of the other eye's image and found that this manipulation affected differentially the mean duration observers spent perceiving each eye's image. Furthermore, the direction of change was somewhat counterintuitive. Levelt's (1968) manipulation corresponded to reducing

*f*

_{B}by decreasing the contrast of the image shown to eye B (rather than increasing eye A's contrast). Referring back to Figure 1b, the findings were closest to the bottom panel: i.e., reducing the contrast of eye B primarily affected (increased) the mean duration of the epochs observers spend in eye A, with little or no effect on the mean durations of eye B. The solid red and green lines in the bottom panel of Figure 1c illustrates the same result by plotting

*T*

_{A}and

*T*

_{B}as a function of the manipulated parameter, denoted

*μ*

_{B}, over the full range of possible values of

*f*

_{B}below the equi-dominance point (i.e., for all

*f*

_{B}< 0.5). For the sake of completion, the top and middle panels in Figure 1c represent the other two limiting cases considered in Figure 1b (top and middle panels, respectively) in terms of the (putative) changes to

*T*

_{A}and

*T*

_{B}as a function of

*μ*

_{B}.

*f*

_{B}from the equi-dominance point, caused by a decrease in

*μ*

_{B}. However, it is legitimate to describe it as an increase in

*f*

_{B}(from a value <0.5) due to an increase in

*μ*

_{B}. This must be how Levelt (1968) conceptualized the manipulation, since his Proposition II states: “Increasing the stimulus strength in one eye will not affect the average duration of dominance in that eye.” Note that in this phrasing, there is no special consideration of the equi-dominance point. Therefore, Levelt's (1968) phrasing of Proposition II implies that the further increase of

*μ*

_{B}(beyond the equi-dominance point) would simply extend the solid red and green lines in Figure 1c (lower panel), as shown by the two corresponding dashed lines. However, in reality, only changes in the range

*f*

_{B}< 0.5 were tested directly by Levelt (1968), as well as by several more recent replications (e.g., Bossink, Stalmeier, & De Weert, 1993; Leopold & Logothetis, 1996; Mueller & Blake, 1989).

*f*that a given percept is dominant as a measure of its strength. Crucially, this measure does not depend on the physical parameters being varied in the stimulus, nor on the arbitrarily chosen units of those parameters. As a natural extension, we defined the stronger (weaker) percept as that with the larger (smaller)

*f*.

*Parametric manipulations that affect the fraction of dominance of the competing percepts will change the mean dominance duration of the stronger percept more than that of the weaker percept*.” This formulation is consistent with those put forward by Brascamp et al. (2006) and Klink et al. (2008), while generalizing them to more bistable stimuli. Furthermore, when plotted against our measure of percept strength (fraction of dominance), the mean dominance times of the two competing percepts show near-perfect symmetry around equi-dominance (the point where each percept dominates half the time). Consequently, the alternation rate (the number of perceptual states reported per unit of time) reaches a maximum at equi-dominance. These results imply that the alternation rate reaches a maximum when the stimulus is maximally ambiguous. Measures of the ambiguity of the stimulus, such as the entropy, are shown to closely correlate with the alternation rate. Finally, we study the behavior of double-well energy models in which the depth of the wells are affected anti-symmetrically by the parameter manipulations, as well as more realistic rate-based models with input normalization, and show that they naturally account for the experimentally observed behaviors.

^{2}was also added in the center of the two gratings. The contrast (8.6%), the wavelength (0.37°), and speed (1.84°/s) of the grating projected to one eye were fixed, and it moved 45° counterclockwise from the vertical line. Identical parameters were used for the grating moving orthogonally to the other one and presented to the contralateral eye (45° clockwise from the vertical line), except that its contrast varied pseudo-randomly from trial to trial with the values: 3.5, 5.2, 6.9, 8.6, 13.8, 24.5, 53.5, and 100%. The mean luminance of the gratings was 45 cd/m

^{2}, which was identical to the luminance of the screen everywhere else. Subjects adjusted their distance to the monitor in order to achieve fusion of the concentric annuli and the fixation point at each grating. Visual angles above were calculated from a subject who sat 57 cm from the screen. Each trial was repeated four times, the fixed parameter grating was presented half of the trials in one eye and the other half in the contralateral eye, in a randomized order.

^{2}, and that of the background was 76 cd/m

^{2}. The gratings have duty cycle equal to 0.2 and wavelength of 2.7°, and move with a speed equal to 5.4°/s. The positions at which the two gratings overlaid (bars intersections) had a luminance of 15 cd/m

^{2}to favor transparent motion (lower luminance that than that of the bars; Stoner, Albright, & Ramachandran, 1990). The angles between the directions of motion of the two gratings were

*α*= 10, 70, 90, 110, 130, 145, 165, 175, 179°. In half of the trials, the plaid moved upward, and in the other half, it moved downward, in a randomized order.

*λ*= 2.7°, while the wavelength of the other grating took one of the following values in each trial:

*λ*= 0.9, 1.35, 1.8, 2.25, 2.7, 3.24, 4.05, 5.4, 8.1°. The luminance of the bars (and intersections between grating bars) was 40 cd/m

^{2}, and that of the background was 89 cd/m

^{2}. Other parameters of the gratings were identical to those in Experiment 2. The global directions of motion of the two gratings were randomized (up-right, up-left, down-right, and down-left; always ±80° from the vertical each; the global directions of motion did not produce any significant effect).

^{2}. A circular fixation point (radius 0.18°, luminance 58 cd/m

^{2}) was overlaid on a small homogeneous circular region (radius 0.9°, luminance 0.2 cd/m

^{2}) that covered the center of the display. All lines were anti-aliased (i.e., intermediate luminance values were used for the pixels at their edges). Observers sat at a distance of 57 cm from the screen.

*T*

_{ j,k }

^{ i }, defined as the

*j*th dominance duration of percept

*i*within a particular trial

*k*. The index

*i*takes the values 1 or 2 (two percepts, either A or B), and

*k*takes the values of the repeated trials (1 to 8) for the same stimulus parameters. Index

*j*is random, as the number of dominances observed varies from trial to trial.

*i*dominated is defined as

*f*

_{ i }= (the cumulative time percept

*i*was reported as dominant)/(the total time that either of the percepts was reported as dominant). This fraction is a number between zero and one, with a value of 0.5 indicating that the two possible percepts were equally likely. More explicitly, the fraction of dominance of percept

*i*is computed as

*i*is the mean value of the dominance durations of that percept averaged over trials, and it is calculated as

*n*

_{ i }is the total number of recorded durations for percept

*i*. Note that the fractions and mean dominance durations are approximately related through

*f*

_{ i }≅

*T*

_{ i }/[

*T*

_{A}+

*T*

_{B}], where

*T*

_{ i }is the mean dominance duration of percept

*i*. Means and error bars for the mean dominance durations are computed, respectively, as the mean and standard error of the dominance durations across durations in trials and across subjects. The same qualitative results to those described in the main text were observed in binocular rivalry for each subject, for each eye and when the cutoff durations were made shorter (see Figures SM1, SM2, and SM3, respectively, in the Supplementary material). The same qualitative features were also present in the other two experiments for each subject and were also largely insensitive to the chosen cutoff durations (not shown).

*T*is the total accumulated time (8 trials × 60 s = 480 s). Alternation rates and error bars are calculated as the means and standard error, respectively, of the alternation rates across trials and subjects. Note that the alternation rate cannot be expressed directly in terms of the fraction of dominance and the mean dominance durations of each percept, since it is possible to have two or more consecutive epochs with the same percepts. Therefore, the alternation rate should be then considered as a measure independent of the fractions of dominance and mean dominance durations. However, as an approximation,

*Rate*≅ 1/[

*T*

_{A}+

*T*

_{B}].

*r*(representing, e.g., the difference in firing rate of two competing populations) is described. The variable

*r*obeys

*τ*= 10 ms is the timescale of the dynamics, the currents

*I*

_{A}and

*I*

_{B}measure the stimulus strength in favor of percept A or B, respectively, and

*n*(

*t*) is a noise term. Equation 4 has two stationary solutions close to

*r*= ±1. Dominance of percept A corresponds to the case

*r*∼ 1, while dominance of percept B corresponds to the case

*r*∼ −1. A transition occurs when

*r*crosses zero. The dynamics of Equation 4 can be viewed as a noisy descent over the energy landscape (Figure 9a). The effect of increasing stimulus strength for, e.g., percept B is to add a straight line with positive slope to the energy landscape, increasing the energy well for percept B while reducing the energy well for percept A. This leads to an increase of the mean dominance duration of percept B and a reduction of the mean dominance duration of percept A.

*r*

_{A}and

*r*

_{B}, respectively. The firing rates obey coupled differential equations with input noise and firing rate adaptation, as described in the Supplementary material. Two models that work in different regimes are considered. In the first one, perceptual switches occur because of the presence of strong adaptation currents (competition neuronal model with direct cross-inhibition), while in the second one perceptual switches occur as a consequence of noise (attractor model with indirect cross-inhibition), as described in Moreno-Bote et al. (2007). For the two models, the state with large activity of population A and low activity of population B corresponds to dominance of the percept encoded by the population A. Percept B dominates if the reversed activity configuration occurs. We compare the dynamics of the model when the inputs to populations A and B are the stimulus strengths

*I*

_{A}and

*I*

_{B}, respectively, and when the inputs are normalized:

*i*= A, B) where

*I*

_{bg}= 0.01 represents background activity present in the network irrespective of the external inputs and

*s*is a scaling coefficient. This equation implements a normalization of the stimulus

*evidence*(strengths) supporting each percept.

*f*

_{B}. The dependence is highly nonlinear, with a large effect of variations of the contrast at low values, and relatively smaller effect on variations of the contrast at higher values. As a function of the contrast,

*T*

_{A}decreases and

*T*

_{B}increases. There is a point in which the mean dominance duration curves meet (

*T*

_{A}≅

*T*

_{B}), which occurs approximately when the fraction of dominance of each percept is close to 0.5 (this point is defined as the equi-dominance point). When the contrast of grating B is lowered from the equi-dominance point,

*T*

_{A}changes more abruptly than

*T*

_{B}does (this result is consistent with Levelt's second proposition). However, when the contrast is increased from the equi-dominance point,

*T*

_{A}changes very little and

*T*

_{B}largely increases, contrary to Levelt's Proposition II (Brascamp et al., 2006).

*T*

_{coh}and

*T*

_{trans}) as a function of the angle (

*α*) between the directions of motion of the constituent drifting gratings. As a function of the angle,

*T*

_{coh}decreases and

*T*

_{trans}increases. For this experiment, the angle at which both dominance durations happen to have approximately the same value is around 120°. Figure 3a shows that the fraction of dominance of coherency decreases monotonically as a function of the angle. At a point around 120°, the fraction of dominance of coherency or transparency is close to 0.5, and therefore, it corresponds to the equi-dominance point. Thus, the mean dominance durations are close to each other at the equi-dominance point, as shown in Figure 3b. A reduction of the angle below the equi-dominance angle produces a large variation of

*T*

_{coh}, while the variation of

*T*

_{trans}is rather modest. If the angle is increased beyond its equi-dominant value,

*T*

_{coh}barely changes, while

*T*

_{trans}varies by a large amount, both in an absolute and a relative sense.

*T*

_{1,beh}and

*T*

_{2,beh}; the index indicates which grating is perceived as being behind) strongly depend on the wavelength of the first grating,

*λ*

_{1}, related to that in the second grating (Moreno-Bote et al., 2008). As a function of the wavelength,

*λ*

_{1},

*T*

_{2,beh}increases and

*T*

_{1,beh}decreases. When

*λ*

_{1}equals the value of the wavelength of the second grating,

*λ*

_{2}= 2.7°, the gratings are identical apart from its motion direction, and therefore, they expend equal amounts of time as being behind. At this point, both mean dominance durations are equal, and the fraction of dominance for each percept is near 0.5, as shown in Figure 4a. This point corresponds to the equi-dominance point of the stimulus. As the wavelength of grating 1 is reduced from the equi-dominance point,

*T*

_{1,beh}changes largely, while

*T*

_{2,beh}changes very little. If the wavelength is increased from the equi-dominance point,

*T*

_{2,beh}is the mean duration more sensitive to the stimulus parameter variation.

*α*) on plaid transparent motion. Turning to examine the latter case (Figure 3a), we find a large range of values over which

*α*has a gradual and near-linear effect on the fraction of time the transparent percept is dominant (roughly 90°–170°), but also a large range over which that linear trend breaks down. The question therefore arises whether it is possible to make any direct comparisons of the effects of these two parameters (contrast and

*α*) on the mean dominance durations of the competing percepts of the corresponding bistable phenomena (binocular rivalry and plaid global motion), when these parameters have such divergent effects on the relative strength of the competing percepts. (Similar questions can be posed with regard to the third bistable phenomenon we studied, gratings' depth reversals.)

*T*

_{A}and

*T*

_{B}in the three experiments—we first needed to transform the scale the parameter used in each of the three experiments so as to put them on a common footing. We further reasoned that the most natural transformation to use is one in which constant changes of the transformed parameter would yield constant increments in the value of

*f*

_{B}—in other words, to transform the scale of the parameter so that it has a linear effect on

*f*

_{B}. Formally, this is equivalent to plotting

*T*

_{A}and

*T*

_{B}against

*f*

_{B}itself.

*f*= 0.5). Since the stronger percept is defined as that having the largest fraction of dominance, then we can summarize the dependence of the mean dominance duration of each percept on the stimulus manipulation as follows: “

*The mean dominance duration of the stronger percept changes more than that of the weaker percept under stimulus parameter manipulations.*”

*f*

_{coh}= 0.5, Coherency is the stronger percept, and Transparency is the weaker percept. The mean dominance duration of Coherency changes dramatically, while the mean dominance duration of Transparency changes very little. To the left of

*f*

_{coh}= 0.5, Transparency is the stronger percept, and the effects on the dominance durations is the reversed. Therefore, the proposition that the stronger percept has the most sensitive dominance durations is true. The same conclusion can be drawn from Figure 6a or 6c for binocular rivalry and gratings' depth reversals.

*f*= 0.5) and symmetry around it. We will use these features to distinguish between possible models of perceptual bistability in the next section.

*f*of a percept is an indication of the degree of belief that the brain should deposit on a particular interpretation of the stimulus. If

*f*is close to one, the interpretation should be believed to be more likely, and if

*f*is close to zero, the interpretation should be believed to be unlikely. Since perceptual switches are stochastic (Borsellino, De Marco, Allazetta, Rinesi, & Bartolini, 1972; Fox & Herrmann, 1967; Lehky, 1995), the probability of finding the brain having one percept will be close to

*f*for that percept. As a crude approximation, the brain can be considered as a binary machine with two states, one with probability

*f*and the other with probability 1 −

*f*. The entropy of such a binary system is (Cover & Thomas, 2006)

*r*. This variable might correspond to the difference in the firing rates of two competing neuronal populations (see description of the rate-based model in Supplementary material). The energy has two minima, one close to

*r*∼ 1, corresponding to dominance of percept A, and another at

*r*∼ −1, corresponding to dominance of percept B. Double-well energy models have been shown before to reproduce Levelt's propositions (Moreno-Bote et al., 2007), and here they are adapted to account for the new observed dependences of the mean dominance durations and alternation rate on stimulus parameter manipulations. When the two percepts are equi-dominant, the depth of the energy wells (energy barriers) for the two states are identical (black line). As described in the Methods section, when the strength of, e.g., percept B is made larger than the strength of percept A, then the energy landscape is tilted clockwise because of the addition of a straight line with positive slope (gray lines). This has the effect of increasing the energy barrier for percept B, while decreasing the energy barrier of percept A by a similar amount. As a result, the fraction of dominance of percept B (respectively, A) increases (respectively, decreases). Because of the symmetry of the double-well energy model, the model naturally leads to a symmetrical dependence of the mean dominance durations on the fraction of dominance of an arbitrarily chosen state (Figure 9b, left). Furthermore, because of the nonlinearity intrinsic to the system, the magnitude of the increase of the mean dominance durations of the stronger percept is larger than the magnitude of the reduction of the mean dominance durations of the weaker percept, and as a consequence, the alternation rate displays a maximum at equi-dominance (gray line, right panel). Interestingly, the alternation rate vs.

*f*curve virtually matches the scaled entropy of a binary system with probabilities

*f*and 1 −

*f*,

*H*(

*f*) (black line).

*f*curves are clearly asymmetrical, and the alternation rate has a maximum very far from the equi-dominance point. This result is typical of networks with direct inputs: when the stimulus strength to, e.g., population B is increased, its input is intensified, leading to an increase of its mean dominance duration and fraction of dominance, while leading to a very little change of the mean dominance duration of the competing population A (except when the stimulus strength is very large; see Moreno-Bote et al., 2007). Introducing gain normalization into the models produces a symmetrical mean dominance duration vs. fraction curve, and a rate vs. fraction curve with inverted U-shape, peaking at the equi-dominance point and being symmetrical around it, therefore reproducing the experimental results. The reason why the model with input normalization features this property can be understood as follows. With gain normalization, increasing the stimulus strength of one percept leads to both an increase in the inputs of its associated population and to a reduction in the inputs to the competing population (see Equation 5). This has a similar effect to that of tilting the energy landscape in the double-well energy model and leads to the symmetric effects of parameters on the mean dominance durations. Finally, it is noteworthy that the entropy (black lines) closely follows the alternation rate in the models with gain normalization, but not if the inputs are not normalized.

*f*= 0.5), revealing that the two percepts are treated equally by the visual system, regardless of their identity. The definition of strength of the percept as its fraction of dominance

*f*allowed us to define the stronger percept uniformly across paradigms (i.e., the stronger percept is the one having the largest

*f*). Furthermore, we could summarize the results by the following proposition: “

*The mean dominance duration of the stronger percept changes more than that of the weaker percept under stimulus parameter manipulations*.” This proposition is consistent to those suggested previously (Bossink et al., 1993; Brascamp et al., 2006; Levelt, 1968; Mueller & Blake, 1989; van Boxtel et al., 2007). While here we have defined the strength of the percept as its fraction of dominance, Klink et al. (2008) has recently formulated a very similar proposition where the strength of the percept was implicitly identified with the “stimulus strength” (for instance, the contrast of one grating in binocular rivalry). As we have explained above, the fraction of dominance constitutes a more natural measure of strength of a percept, as it is independent of the physical parameter that was manipulated in the experiment. More importantly, it allows describing the effects of parameter manipulations on the mean dominance durations in cases where the notion of stimulus strength is not well defined, as in our ambiguous plaid motion stimulus, where the angle between the gratings cannot be naturally associated with the stimulus strength of a particular percept.

*maximum*

*alternation rate occurs at equi-dominance*. Moreover, the alternation rate vs. fraction of dominance curve is symmetric around the equi-dominance point. These results might have important consequences for the understanding of the role of perceptual bistability in visual processing. Rather than being a mere curiosity, we have recently proposed (Moreno-Bote et al., 2008) that perceptual bistability plays a functional role in vision by allowing a faster matching between correct interpretation and stimulus compared to a hypothetical case without alternations. Under conditions of high ambiguity or noise in the stimulus, retaining the most likely percept forever would be harmful, because it is still possible that one of the other, less probable, percepts is the correct interpretation of the stimulus. For instance, missing a predator in the rain by confusing it with a mate could have devastating consequences. It is then tempting to think that perceptual switches have evolved as a necessary consequence for perceptual exploratory behavior that allows a correct and faster matching between interpretation and stimulus under conditions of highly ambiguous stimulation or noise.

*f*be the probability that one percept is correct; hence 1 −

*f*is the probability that the competing percept is correct. Assume that each percept is “chosen” with the same probability that it is expected to be correct (i.e., either

*f*or 1 −

*f*), and that the cost of switching is proportional to the square of the alternation rate

*r*. If the initial state is the one with probability

*f*, then the probability of transitioning to the other percept and that it is the correct one is (1 −

*f*)

*r*. This happens with probability

*f*(1 −

*f*)

*r*. If the initial state is the one with probability 1 −

*f*, then the probability of transitioning to the other percept and that it is the correct one is

*fr*. This situation happens with probability (1 −

*f*)

*fr*. For the sake of simplicity, we assume that exploration occurs mainly during the initial periods after the transitions, rather than continuously throughout the whole dominance epochs. Therefore, the expected gain because of exploration is, on average, 2

*f*(1 −

*f*)

*r*. After subtracting from it the cost of exploration, the total expected gain per unit time after each transition is

*a*is a constant. For the cost of transitions, we chose the squared rate instead of linear or other nonlinear dependences because it is the simplest case that leads to nontrivial results (i.e., not choosing always either the most likely or less likely percept). Note that when the alternation rate is zero, the reward obtained from exploration is zero. Exploring with a large alternation rate is very costly, because of the square in the cost term, and it is not optimal either. There is a value of alternation rate for which the expected reward per unit time reaches a maximum, and this is attained when

*f*, it is very similar to the entropy, and it has a maximum at

*f*= 1/2 (not shown). This simplistic model illustrates, consistently with experimental results, that perceptual alternations can lead to maximization of reward, and that the brain can pay the higher cost of increasing the alternation rate if the sensory input is highly ambiguous. Although the hypothesis that perceptual bistability is a form of exploration is consistent with the experimental observations, further research is required to determine its adequacy.

*N*> 2). With multiple stable interpretations, the strongest percept does not need to have a fraction of dominance

*f*larger than one half, as in perceptual bistability, but rather it just need to have the largest

*f*. Similarly, equi-dominance is defined as the point at which the

*f*s for all percepts are equal to 1/

*N*, rather than one half. A default hypothesis is that the above propositions will hold true even for the more general case of multistable perception, but its confirmation awaits further research.

*f*curves and the maximum of alternation rate at equi-dominance point are not easily obtained in neuronal network models of perceptual bistability. In fact, most common models based on direct inputs (i.e., each interpretation strength is modeled as input strength to a particular neuronal population) typically show rather asymmetrical mean dominance durations and alternation rate vs.

*f*curves (Figure 9). We have shown that if inputs are normalized, then the curves become symmetrical. Input normalization can be interpreted as producing a normalization of the

*evidence*supporting each percept. In fact, by normalizing the input, each of the competing populations no longer represents the absolute value of evidence supporting it, but rather the relative evidence in relation to any other source of information in the stimulus that supports other interpretations.

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*α*increases the frequency of the transparent-motion percept. However, does this come about via a strengthening of inputs to the neural population(s) encoding motion transparency? Or via a weakening of inputs to the population(s) representing the coherent interpretation? Or perhaps both? In addition, are these questions even valid, in terms of the neural organization of the system? This example therefore illustrates how insights from the general mechanisms of bistability could contribute to our understanding of visual processing.

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