W.J.M. Levelt systematized the influence of stimulus strength on binocular rivalry dynamics in several formal propositions. His counterintuitive 2nd proposition states that mean dominance duration of one eye's stimulus depends not on the strength of that stimulus but, instead, on the strength of the stimulus viewed by the other eye. Some studies have reported results consistent with this proposition but others have found violations of the proposition. This paper examines the dynamics of binocular rivalry by changing the size of rival stimuli and the tracking instructions during rivalry tracking periods in which the contrasts of the two rival stimuli are varied independently. Levelt's 2nd proposition was validated when those stimuli were large, but it was violated when the rival stimuli were small, suggesting that the dynamics of binocular rivalry are spatiotemporal in nature. A simple energy model with coupling among neighboring areas of rivalry can account for these findings. Other dynamics depending on the size of rival stimuli are discussed.

- as the contrast level of the other eye's stimulus increases, dominance durations of one eye's stimulus decrease on average, and
- as the contrast level of that stimulus increases, dominance durations of a given eye's stimulus do not vary on average.

^{2}) through a mirror stereoscope placed 90 cm from the monitor.

*y*-axis plots the mean dominance duration of the rival stimulus I shall term the ipsilateral stimulus, the

*x*-axis designates the contrast of that ipsilateral stimulus, and the three separate lines in each panel refer to the contrast values of the other, contralateral rival stimulus. The contrast values of the ipsilateral and contralateral stimuli are specified in terms of multiples of the lowest contrast level tested for those stimuli. The following results are based on analyses of actual dominance durations collected over an entire tracking period. However, I also analyzed these data in two other ways: by transforming all dominance durations to their log (Hupé & Rubin, 2003) and by eliminating perceptual dominance durations during the first 10 sec of the tracking period (Logothetis et al., 1996). These alternative ways of treating the data did not change the pattern of results described below.

*F*(2,14) = 18.12,

*p*< 0.001]. This is not surprising considering that increases in the size of rival stimuli increase the incidence and duration of mixed dominance (Blake et al., 1992; O'Shea et al., 1997), thereby reducing the durations of exclusive dominance. In addition, a perceptual switch within a local region of a rival figure tends to propagate to neighboring regions (Wilson et al., 2001). Combining these two facts about rivalry, the probability of a spontaneous perceptual switch within a local region of a rival target should increase with larger sized rival stimuli, and thus a perceptual switch within any local region can spread to produce perceptual switches over the entire figure. This can account for the dominance durations measured in the LP condition being shorter than those measured in the SW condition.

*F*(1,7) = 24.76,

*p*< 0.01 for the SW condition;

*F*(1,7) = 25.82,

*p*< 0.01 for LP condition;

*F*(1,7) = 37.79,

*p*< 0.001 for the LW condition]. This result is consistent with the first part of Levelt's 2nd proposition.

*F*(1,7) = 19.96,

*p*< 0.01], consistent with the findings of Brascamp et al. (2006). This pattern of results is particularly conspicuous when the contra-contrast is low [interaction;

*F*(1,7) = 8.46,

*p*< 0.05]. This outcome is incompatible with the contrast-invariant property of Levelt's 2nd proposition. But for the LW condition the dominance durations remain invariant irrespective of the ipsi-contrast [

*F*(1,7) = 2.08,

*p*= 0.19]. The interaction between the ipsi-contrast and contra-contrast was not statistically significant either [

*F*(1,7) < 1,

*p*> 0.5]. This result for the LW condition is consistent with the contrast-invariant property of Levelt's 2nd proposition, suggesting that this property emerges in consequence of spatial interactions associated with large rival stimuli.

*F*(1,7) = 3.48,

*p*= 0.10 for LP condition] and the interaction between the ipsi-contrast and contra-contrast was not statistically significant either [

*F*(1,7) = 3.33,

*p*= 0.11 for LP condition]. Other than the expected differences in the incidence of mixed dominance in LP and LW conditions, the behavior of the dynamics of rivalry within a limited region of a large rival stimulus are comparable to the behavior of the dynamics of that entire stimulus. This comparability leads me to conclude that the size of rival stimuli is the major factor determining the dynamics of binocular rivalry in terms of their dominance durations—periods of mixed dominance are not critical in producing the contrast-invariant property of Levelt's 2nd proposition for large rival stimuli.

*r*represents the difference in firing rates of the two competing populations. This energy function has two local minima and each local maximum determined by the input strength parameters

*g*

_{ A}and

*g*

_{ B}, respectively. In the context of this energy-based formalism, it is more difficult for a system to escape from a state with increasing depth of that state or increasing energy barrier between the two states. This increased difficulty produces, in average, longer dominance durations during binocular rivalry [see Brascamp et al. (2006), Kim et al. (2006), and Moreno-Bote et al. (2007) for a detailed discussion of the dynamics of local rivalry with the energy model]. The dynamics of local rivalry satisfy

*τ*

*ηX*(

*E*

_{ j},

*E*

_{ i}) and the noise term

*ωn*

_{ i}(

*t*) are added as shown in Equation 2. As defined in Equation 3,

*X*(

*E*

_{ j},

*E*

_{ i}) governs the interaction between the perceptual states of the two local rivalries such that local rivalry

*i*only interacts with the nearest other local rivalries

*j*(

*NB*

_{ i}indicates the set of the nearest neighbors of the local rivalry

*i*).

*K*

_{ i}is the normalization factor, which corresponds to the number of neighboring local rivalries connected to the local rivalry

*i*(but additional simulation without this normalization factor produced qualitatively similar results). [

*E*

_{ i}] represents the perceptual state of the given local rivalry

*i,*which is either +1 or −1. This ±1 value is used because the energy function has a local minimum at

*r*∼ ±1. Therefore, if the perceptual states of the two adjacent local rivalries are the same, this interaction does not influence Equation 2. The coupling strength of the network model is determined by

*η*in Equation 2 and this

*η*equals 0 for SW model (details about the model parameters and the simulation procedures are described in 1).

*g*

_{ A},

*g*

_{ B}, and

*η*). In Figure 4, line style (width and solid/dashed) indicates the dominance durations at a given contra-contrast

*g*

_{ A}whereas color indicates the coupling strength

*η*as shown at the right of Figure 4c. First,

*g*

_{ A}and

*g*

_{ B}were varied among 0.1, 0.2, and 0.4 for the SW model. In the model, they correspond to the contrasts of the two rival stimuli. The mean dominance durations produced by the simulation of the single local rivalry were very similar to the experimental results ( Figure 4a). This is consistent with the simulation of Moreno-Bote et al. (2007), confirming that the SW model produces dominance durations whose variation violates the contrast-invariant property of Levelt's 2nd proposition: increasing ipsi-contrast

*g*

_{B}increases the mean dominance duration.

*η*parameter inherent in the LP and LW models produces patterns of results mirroring those obtained from the experiments, especially the contrast-invariant property of Levelt's 2nd proposition. Figures 4b and 4c summarize the dominance durations of the LP and LW model, respectively. When

*η*equals 0, the LP model's dominance durations are identical to the dominance durations of SW model, as they should be. Consistent with experimental results, dominance durations decrease with increasing

*g*

_{ A}at the same coupling strength

*η*for all three models. In addition, as the coupling strength

*η*increases, overall dominance durations decrease for LP and LW models compared to the SW model. Most importantly, with the properly selected coupling strength

*η,*the mean dominance durations remain relatively unchanged at increasing ipsi-contrast

*g*

_{ B}and this pattern of result is particularly conspicuous for LW model. Thus, this simulated behavior of the LW model captures the contrast-invariant property of Levelt's 2nd proposition.

*x*-axis and companion contrast is represented by the color of three lines. Consistent with Brascamp et al. (2006), the FRTs increased with departure contrast and decreased with increasing companion contrast [three-way ANOVA with factor of two experimental conditions (SW and LW) × departure contrast × companion contrast showed significant effect of the departure contrast

*F*(1,7) = 8.45,

*p*< 0.5 and companion contrast

*F*(1,7) = 14.86,

*p*< 0.01]. This observation is most pronounced at the highest departure contrast and the lowest companion contrast. Importantly, the RTs occur in similar proportions for both SW and LW conditions [

*F*(1,7) = 2.75,

*p*= 0.14], suggesting that the incidence of RTs cannot explain the contrast-invariant property of Levelt's 2nd proposition. To provide converging evidence for this tentative conclusion, I performed additional analyses of the dynamics of RTs for these two conditions.

- those associated with RTs, and
- those associated with alternation transitions (ATs), i.e., episodes in which dominance switched from one stimulus to the other.

*F*(1,7) = 58.10,

*p*< 0.001 for the SW condition;

*F*(1,7) = 9.20,

*p*< 0.05 for the LW condition]. This pattern of results is particularly conspicuous at low contra-contrast and high ipsi-contrast levels, accounting for much of the increased predominance shown in Figure 6. However, the mean dominance durations of ATs and RTs, shown in Figures 7c and 7d, change similarly as a function of ipsi-contrast level: when the ipsi-contrast level increases, the mean dominance durations associated with both the ATs and RTs increase for the SW condition but remain unchanged for the LW condition. An ANOVA with three factors (transition type, ipsi-contrast, and contra-contrast) shows no significant interactions between the transition type and ipsi-contrast [

*F*(1,7) = 0.71,

*p*= 0.43 for the SW condition;

*F*(1,7) = 0.08,

*p*> 0.5 for the LW condition]. Thus, the contrast-invariant property of Levelt's 2nd proposition observed with large rival stimuli is not accounted for by mean dominance durations associated with RTs.

*τ*= 10. According to Moreno-Bote et al. (2007), the noise term

*n*

_{i}(

*t*) in Equation 2 follows Ornstein–Uhlenbeck process ṅ

_{i}= −

*n*

_{i}/

*τ*

_{s}+

*σ*

*ξ*

_{i}(

*t*) whose amplitude was increased by

*ω*= 5. In this equation, the time constant

*τ*

_{s}= 100, deviation term

*σ*= 0.7, and

*ξ*

_{i}(

*t*) represents a white noise randomly selected from a normal distribution. Euler's method was used for all numerical integration with time step

*δt*equals 0.1 for 10

^{6}time unit, which means for 10

^{7}iterations. Matlab (Mathworks, MA) running in Machintosh G5 computer (Apple, CA) was used for the simulation.