Neural adaptation plays an important role in multistable perception, but its effects are difficult to discern in sequences of perceptual reversals. Investigating the multistable appearance of kinetic depth and binocular rivalry displays, we introduce *cumulative history* as a novel statistical measure of adaptive state. We show that *cumulative history*—an integral of past perceptual states, weighted toward the most recent states—significantly and consistently correlates with future dominance durations: the larger the *cumulative history* measure, the shorter are future dominance times, revealing a robust effect of neural adaptation. The characteristic time scale of *cumulative history,* which may be computed by Monte Carlo methods, correlates with average dominance durations, as expected for a measure of neural adaptation. When the *cumulative histories* of two competing percepts are balanced, perceptual reversals take longer and their outcome becomes random, demonstrating that perceptual reversals are fluctuation-driven in the absence of adaptational bias. Our findings quantify the role of neural adaptation in multistable perception, which accounts for approximately 10% of the variability of reversal timing.

*vice versa*. However, numerous studies have failed to find such negative correlations between past and future dominance periods (Borsellino, De Marco, Allazetta, Rinesi, & Bartolini, 1972; Fox & Herrmann, 1967; Lehky, 1995; Walker, 1975). Instead, sequential correlations between dominance periods are typically weak and only significant for some displays (van Ee, 2009). Indeed, the most compelling evidence for a history dependence of multistable perception was obtained after adaptation to non-ambiguous displays and not from the normal multistable dynamics (Blake et al., 1990; Nawrot & Blake, 1989; Petersik, 2002; Wolfe, 1984).

*cumulative history*(an integral of past perceptual states, weighted toward the most recent states) as a novel and more sensitive measure of adaptive states during multistable perception. With this measure, linear correlations between past history and future dominance durations are consistently significant and approximately twice as large as with conventional measures. We also extract from reversal sequences the characteristic time constant of

*cumulative history*. Finally, we show evidence that perceptual transitions are driven by stochastic fluctuations when the adaptive states (as measured by

*cumulative history*) of competing percepts are balanced.

^{2}for the kinetic depth and 19 cd/m

^{2}for the binocular rivalry displays. Anaglyph glasses (red/cyan) were used for the dichoptic presentation.

*σ*= 0.057°) and a maximal luminance of 63 cd/m

^{2}. The sphere was centered at fixation and rotated around the vertical axis with a period of 4 s. As front and rear surface dots were indistinguishable, the orthographic projection was perfectly ambiguous and consistent with either a clockwise or a counterclockwise rotation around the axis. Observers perceive a three-dimensional sphere, which reverses its direction of rotation from time to time.

*T*

_{ i }, we computed the mean dominance time, coefficient of variation, and autocorrelation as

*T*

_{dom}), the coefficient of variation (

*C*

_{v}), and the autocorrelation (

*c*

_{ T }) were computed. The standard error for each statistical measure was then computed as the standard deviation across the 1000 bootstrapped values.

*T*

_{dom},

*C*

_{v}, and

*c*

_{ T }were obtained by averaging the appropriate values over all presentations.

*cumulative history*,

*H*

_{ x }(

*t*) (see Figure 2). Let

*S*

_{ x }(

*t*) be a record of perceptual experience

*x*as a function of time

*t*, defined as unity while percept

*x*dominates, 0.5 during a mixed or patchy percept, and zero when percept

*x*is suppressed. The cumulative history

*H*

_{ x }(

*t*) computed using a leaky integrator (Tuckwell, 2006) is then given by

*τ*

_{ H }is a time constant to be determined empirically. This definition assumes that (i) the contribution of past experience decays exponentially, (ii) multiple contributions of the same percept combine additively, and (iii) there is no contribution (not even a negative one) from competing percepts.

*H*

_{l}and

*H*

_{r}for two alternative percepts (e.g., left- or rightward rotation, left- or right-eye grating) from a sequence of dominance periods up to time

*t,*we computed linear correlation coefficients with the immediately following dominance period

*T*

^{l}or

*T*

^{r}. Specifically, if

*t*

_{ i }marks the beginning of dominance period

*T*

_{ i }

^{ x }, we computed linear correlations between

*H*

_{ x }(

*t*

_{ i }) and ln(

*T*

_{ i }

^{ x }) for all four possible combinations of history and percept (

*H*

_{l}×

*T*

^{l},

*H*

_{l}×

*T*

^{r},

*H*

_{r}×

*T*

^{r}, and

*H*

_{r}×

*T*

^{l}). The resulting four values were then combined into an average absolute correlation.

*T*

_{ i }and obtained the residual (“detrended”) values. This ensured that

*c*

_{ H }→ 0 as

*τ*

_{ H }→ ∞. Next, we used the “detrended”

*T*

_{ i }to compute average absolute correlations for values of

*τ*

_{ H }ranging from 0.001 s to 60 s. The maximal correlation obtained was taken as the value of

*c*

_{ H }, and the

*τ*

_{ H }yielding this maximal correlation was taken as the definitive value of

*τ*

_{ H }(see Figure 3).

*c*

_{ H }and

*τ*

_{ H }were obtained by averaging the appropriate values over all presentations.

*T*) that is explained by cumulative history may be defined with respect to the linear regression:

*R*

^{2}:

*T*(bin width of 300 ms) and

*cumulative history H*(bin width of 0.05) and by comparing the joint distribution

*P*(ln

*T*,

*H*) with the marginal distributions

*P*(ln

*T*) and

*P*(

*H*):

*T*

_{dom}= 11.4 ± 7.6 s for KD and

*T*

_{dom}= 2.5 ± 1.05 s for BR. The patchy appearance (only BR) lasted for 1.05 ± 0.42 s on average.

*C*

_{v}= 0.67 ± 0.18 (KD) and

*C*

_{v}= 0.48 ± 0.12 (BR). Correlations between successive dominance periods of the same appearance were fairly small (

*c*

_{ T }= 0.09 ± 0.1 for KD and

*c*

_{ T }= 0.15 ± 0.05 for BR) and reached significance (

*p*< 0.05) for one (of 8) observer of the KD display and for four (of 11) observers of the BR display. Because sequential correlations are weak and often fail to reach significance, multistable perception has sometimes been considered a “memoryless” process (Blake, Fox, & McIntyre, 1971; Borsellino et al., 1972; Fox & Herrmann, 1967; Leopold & Logothetis, 1999; Levelt, 1965; Walker, 1975).

*cumulative histories*. Specifically, we convolve the record of dominance reports,

*S*

_{ x }(

*t*) with an exponential kernel (Drew & Abbott, 2006; Tuckwell, 2006) to obtain a cumulative measure of perceptual history up to time

*t*:

*x*∈ {right/left, red/green} denotes a uniform percept and

*τ*

_{ H }is a time constant.

*S*

_{ x }(

*t*) takes values of 1 for dominance, 0.5 for patchy dominance (BR only), and 0 for non-dominance. The cumulative history

*H*

_{ x }(

*t*) reflects both how long and how recently a given percept has dominated in the past. Figure 2 illustrates the time courses of

*S*

_{ x }(

*t*) and

*H*

_{ x }(

*t*), for two representative series of dominance reports. Note that, in the absence of “patchy” appearances, for example, for KD stimulus, the cumulative histories of two competing percepts approach unity (

*H*

_{left}+

*H*

_{right}≈ 1).

*τ*

_{ H }(see below), we find that a

*measure of the past*—cumulative history

*H*(

*t*)—is a statistically reliable

*predictor of the future*—the next dominance period

*T*

_{ i }(Figures 3A and 3B). Specifically, the more a percept has dominated in the past, particularly recently (larger

*H*value), the

*shorter*the same percept (and the

*longer*the other percept) dominates in the immediate future. These correlations were comparatively large and reached significance for 17 of 19 observers (

*c*

_{ H }= 0.24 ± 0.1 with

*p*< 0.001 for KD displays and

*c*

_{ H }= 0.30 ± 0.09 with

*p*< 0.0001 for BR displays). Thus, correlations with cumulative history were not only more consistently significant than but also twice as large as sequential correlations of dominance periods (Figure 4A). Parenthetically, similar results may be obtained with other kernels (e.g., half-Gaussian kernels that weighs the recent past more heavily than the distant past).

*τ*

_{ H }equals roughly half the mean dominance time

*T*

_{dom}(Figures 3C and 3D). Defining

*γ*

_{ H }≡

*τ*

_{ H }/

*T*

_{dom}, the KD results showed maximal correlations for

*γ*

_{ H }= 0.54 ± 0.21, while the BR results exhibited peak correlations for

*γ*

_{ H }= 0.56 ± 0.28. The values obtained for

*τ*

_{ H }are robust and not due to selection bias: when the data for each observer are divided in half and only one half is used to optimize

*τ*

_{ H }, virtually identical correlation coefficients are obtained from the other half (not shown).

*T*

_{dom}. As shown in Figure 4B, the inferred time constant

*τ*

_{ H }of neural adaptation correlates only loosely with the average dominance period

*T*

_{dom}. For both displays and all observers, the linear correlation coefficient was 0.825 (

*p*< 0.001). However, when each display is considered separately, the correlation drops to 0.64 (

*p*= 0.09) for KD and to 0.36 (

*p*= 0.28) for BR.

*forward transition*). Occasionally, however, a “patchy” period merely interrupted a uniform appearance, which afterward resumed (

*return transition*; Brascamp et al., 2006; Hollins, 1980). Both

*forward*and

*return*transitions were profoundly affected by the balance of cumulative histories (Figure 5): “Patchy” periods lasted longer, and the

*return*probability was far higher, when cumulative histories were approximately balanced and Δ

*H*≈ 0. In some observers, transitional states (which are often ignored in the calculation of sequential correlations) even predict something about the subsequent dominance period. This is evident from the fact that some

*c*

_{ H }values are larger than zero for very small values of

*τ*

_{ H }(Figures 3C and 3D). Note that, for

*τ*

_{ H }= 0, the value of “cumulative history” corresponds directly to the identity of the preceding state (

*H*= 0, 0.5, or 1).

*cumulative history*—to analyze serial reversals in the appearance of multistable displays. The results obtained with this measure go beyond earlier findings in several ways. First, they reveal a consistently robust and statistically significant correlation between past perceptual history and future dominance duration, which does not become evident with conventional measures such as sequential correlations of dominance durations (Borsellino et al., 1972; Fox & Herrmann, 1967; Lehky, 1995; van Ee, 2009; Walker, 1975). Second, they demonstrate that neural adaptation of the dominant percept raises reversal probability even when an ambiguous display is viewed continuously. All previous evidence on this point involved prolonged adaptation to non-ambiguous displays (Blake et al., 1990; Kang & Blake, 2010; Nawrot & Blake, 1989; Petersik, 2002; Wolfe, 1984). Third, they reveal the characteristic time constant of neural adaptation. Taken together, this constitutes the most compelling evidence so far that neural adaptation contributes to multistable dynamics.

*cumulative history*than with the immediately preceding dominance periods (Figure 4A). Moreover, correlations with

*cumulative history*were significant in 17 of 19 observers, whereas correlations with earlier dominance periods were significant in only 5 of 19 observers.

*Cumulative history*is a more informative measure because it integrates over several preceding dominance periods, taking into account both how long and how recently a particular percept has dominated in the past. Due to its short time constant (see below),

*cumulative history*assumes intermediate values only after one or more short dominance periods. In these (comparatively rare) situations,

*cumulative history*is expected to be particularly predictive.

*cumulative history,*the effect of neural adaptation is comparatively weak and linear correlation coefficients do not much exceed 0.4. To properly gauge the influence of neural adaptation, it is helpful to compute its effect on reversal timing: cumulative history accounts for (on average) approximately 9% of the entropy and approximately 8.5% of the variance of reversal timing. These values should be considered a lower bound, because

*cumulative history*is merely an estimate (and not an equivalent) of the true state of neural adaptation.

*cumulative history*. To this end, one compares the observed sequence with several hypothetical

*cumulative history*time series, each computed with a different time constant. If at least some of these time series correlate significantly with the observed sequence, then the best-correlated time series provides a characteristic time constant for the observed sequence. Typically, the resulting time constants are comparable to the average dominance time, with a linear correlation coefficient of 0.825 (see Figure 4B), suggesting that

*cumulative history*does indeed capture the time evolution of neural adaptation. Note, however, that most computational models of multistable perception (Laing & Chow, 2002; Lankheet, 2006; Moreno-Bote, Rinzel, & Rubin, 2007; Noest, van Ee, Nijs, & van Wezel, 2007; Shpiro, Moreno-Bote, Rubin, & Rinzel, 2009; Wilson, 2003) predict a strictly monotonic relation between average dominance times and the time constant of neural adaptation (sampling error apart). Our results are not consistent with such a strict correlation, even allowing for sampling error. Apparently, factors other than the time constant of

*cumulative history*introduce additional variance in the mean dominance times. While there are many conceivable reasons for this partial dissociation, one intriguing possibility is that the collective dynamics of neural representations is partially uncoupled from the time constants of individual neural components (Braun & Mattia, 2010; Gigante, Mattia, Braun, & Del Giudice, 2009). Further work is needed to understand the full implications of this observation.

*cumulative history*. It has long been understood that the transition between two distinct appearances is informative about the driving forces of multistable dynamics (Brascamp et al., 2006; Hollins, 1980). In our binocular rivalry display, observers frequently reported transitional states in which both gratings were visible (“patchy” or “fused” appearances). In the kinetic depth display, transitional states were too rare to be analyzed. To ascertain the influence of neural adaptation on transitional states, we compared both the duration and the outcome of such transitional states with the

*cumulative history*measure (Figure 5). We observed approximately 50% longer transition phases and approximately 200% more “return transitions” (i.e., transitions leading back to the preceding percept) when the difference between the two

*cumulative histories*was close to zero, compared to the overall average values. The implication is, of course, that at other times—when neural adaptation of one percept exceeds that of the other—transitional states quickly make way for the less adapted percept. Thus, adaptive state, as captured by

*cumulative history,*clearly plays a causal role in perceptual reversals.

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