Pools of bistable variables belong to a class of neural representations particularly suited for Bayesian integration of sensory information (Beck et al., 2008; Pouget et al., 2013). In general, summation of activity is equivalent to optimal integration of information, provided that response variability is Poisson-like, and response tuning differs only multiplicatively (Ma et al., 2006; Ma et al., 2008). Pools of bistable variables closely approximate these properties (see Appendix 1, section Quality of representation: Suitability for inference).

The representational accuracy of even a comparatively small number of bistable variables can be surprisingly high. For example, if normally distributed inputs drive the activity of initially inactive pools of bistable variables, pools as used in the present model (${\displaystyle N=25}$, ${\displaystyle w=2.5}$) readily capture 90% of the Fisher information (see Appendix 1, section Quality of representation: Integration of noisy samples).

Any model of BR must represent the conflicting evidence from both eyes (e.g., different visual orientations), which supports alternative perceptual hypotheses (e.g., distinct grating patterns). We assume that conflicting evidence accumulates in two separate pools of ${\displaystyle N=25}$ bistable variables, ${\displaystyle E}$ and ${\displaystyle E^{\mathrm{\prime }}}$, (‘evidence pools,’ Figure 1c). Fractional activations ${\displaystyle e(t)}$ and ${\displaystyle e^{\mathrm{\prime }}(t)}$ develop stochastically following Equation 5 (Materials and methods). Transition rates ${\displaystyle {\nu }_e^{\pm }}$ and ${\displaystyle {\nu }_{e^{\mathrm{\prime }}}^{\pm }}$ vary exponentially with activation energy (Equation 1), with baseline potential ${\displaystyle u_e^0}$ and baseline rate ${\displaystyle {\nu }_e}$. The variable components of activation energy, ${\displaystyle u_e}$ and ${\displaystyle u_{e^{\mathrm{\prime }}}}$, are synaptically modulated by image contrasts, ${\displaystyle c}$ and ${\displaystyle c^{\mathrm{\prime }}}$:

where ${\displaystyle w_{\mathit{v}\mathit{i}\mathit{s}}}$ is a coupling constant and ${\displaystyle I=f(c)\in [0,1]}$ is a monotonic function of image contrast ${\displaystyle c}$ (see Materials and methods).

Once evidence for, and against, alternative perceptual hypotheses (e.g., distinct grating patterns) has been accumulated, reaching a decision requires a sensitive and reliable mechanism for identifying the best supported hypothesis and amplifying the result into a categorical read-out. Such a winner-take-all decision (Koch and Ullman, 1985) is readily accomplished by a dynamical version of biased competition (Deco and Rolls, 2005; Wang, 2002; Deco et al., 2007; Wang, 2008).

We assume that alternative perceptual hypotheses are represented by two further pools of ${\displaystyle N=25}$ bistable variables, ${\displaystyle R}$ and ${\displaystyle R^{\mathrm{\prime }}}$, forming two ‘non-local attractors’ (‘decision pools,’ Figure 1c). Similar to previous models of decision-making and attentional selection (Deco and Rolls, 2005; Wang, 2002; Deco et al., 2007; Wang, 2008), we postulate recurrent excitation within pools, but recurrent inhibition between pools, to obtain a ‘winner-take-all’ dynamics. Importantly, we assume that ‘evidence pools’ project to ‘decision pools’ not only in the form of selective excitation (targeted at the corresponding decision pool), but also in the form of indiscriminate inhibition (targeting both decision pools), as suggested previously (Ditterich et al., 2003; Bogacz et al., 2006).

Specifically, fractional activations ${\displaystyle r(t)}$ and ${\displaystyle r^{\mathrm{\prime }}(t)}$ develop stochastically according to Equation 5 (Materials and methods). Transition rates ${\displaystyle {\nu }_s^{\pm }}$ and ${\displaystyle {\nu }_{s^{\mathrm{\prime }}}^{\pm }}$ vary exponentially with activation energy (Equation 1), with baseline difference ${\displaystyle u_r^0}$ and baseline rate ${\displaystyle {\nu }_r}$. The variable components of activation energy, ${\displaystyle u_r}$ and ${\displaystyle u_{r^{\mathrm{\prime }}}}$, are synaptically modulated by evidence and decision activities:

where coupling constants ${\displaystyle w_{\mathit{e}\mathit{x}\mathit{c}}}$, ${\displaystyle w_{\mathit{i}\mathit{n}\mathit{h}}}$, ${\displaystyle w_{\mathit{c}\mathit{o}\mathit{o}\mathit{p}}}$, ${\displaystyle w_{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}}$ reflect feedforward excitation, feedforward inhibition, lateral cooperation within decision pools, and lateral competition between decision pools, respectively.

This biased competition circuit expresses a categorical decision by either raising ${\displaystyle r}$ towards unity (and lowering ${\displaystyle r^{\mathrm{\prime }}}$ towards zero) or vice versa. The choice is random when visual input is ambiguous, ${\displaystyle I\simeq I^{\prime }}$, but becomes deterministic with growing input bias ${\displaystyle |I-I^{\mathrm{\prime }}|\S gt;0}$ . This probabilistic sensitivity to input bias is reliable and robust under arbitrary initial conditions of ${\displaystyle e}$, ${\displaystyle e^{\mathrm{\prime }}}$, ${\displaystyle r}$ and ${\displaystyle r^{\mathrm{\prime }}}$ (see Appendix 1, section Categorical choice with Appendix 1—figure 3).

Finally, we assume feedback suppression, with each decision pool selectively targeting the corresponding evidence pool. A functional motivation for this systematic bias against the currently dominant appearance is given momentarily. Its effects include curtailment of dominance durations and ensuring that reversals occur from time to time. Specifically, we modify Equation 3 to

where ${\displaystyle w_{\mathit{s}\mathit{u}\mathit{p}\mathit{p}}}$ is a coupling constant.

Previous models of BR (Dayan, 1998; Hohwy et al., 2008) have justified selective feedback suppression of the evidence supporting a winning hypothesis in terms of ‘predictive coding’ and ‘hierarchical Bayesian inference’ (Rao and Ballard, 1999; Lee and Mumford, 2003). An alternative normative justification is that, in volatile environments, where the sensory situation changes frequently (‘volatility prior’), optimal inference requires an exponentially growing bias against evidence for the most likely hypothesis (Veliz-Cuba et al., 2016). Note that feedback suppression applies selectively to evidence for a winning hypothesis and is thus materially different from visual adaptation (Wark et al., 2009), which applies indiscriminately to all evidence present.

A representative example of the joint dynamics of evidence and decision pools is illustrated in Figure 1c,d, both at the level of pool activities ${\displaystyle e(t)}$, ${\displaystyle e^{\mathrm{\prime }}(t)}$, ${\displaystyle r(t)}$, ${\displaystyle r^{\mathrm{\prime }}(t)}$, and at the level of individual bistable variables ${\displaystyle x(t)}$. The top row shows decision pools ${\displaystyle R}$ and ${\displaystyle R^{\mathrm{\prime }}}$, with instantaneous active counts, ${\displaystyle Nr(t)}$ and ${\displaystyle N{r}^{\mathrm{\prime }}(t)}$ and active/inactive states of individual variables ${\displaystyle x(t)}$. The bottom row shows evidence pools ${\displaystyle E}$ and ${\displaystyle E^{\mathrm{\prime }}}$, with instantaneous active counts, ${\displaystyle Ne(t)}$ and ${\displaystyle N{e}^{\mathrm{\prime }}(t)}$ and active/inactive states of individual variables ${\displaystyle x(t)}$. Only a small fraction of evidence variables is active at any one time.

Phenomenal appearance reverses when the differential activity ${\displaystyle \mathrm{\Delta }e=e-e^{\prime }}$ of evidence pools, ${\displaystyle E}$ and ${\displaystyle E^{\mathrm{\prime }}}$, contradicts sufficiently strongly the differential activity ${\displaystyle \mathrm{\Delta }r=r-r^{\prime }}$ of decision pools, ${\displaystyle R}$ and ${\displaystyle R^{\mathrm{\prime }}}$, such that the steady state of decision pools is destabilized (see further below and Figure 4). As soon as the reversal has been effected at the decision level, feedback suppression lifts from the newly non-dominant evidence and descends upon the newly dominant evidence. Due to this asymmetric suppression, the newly non-dominant evidence recovers, whereas the newly dominant evidence habituates. This opponent dynamics progresses, past the point of equality ${\displaystyle s\simeq s^{\prime }}$, until differential evidence activity ${\displaystyle \mathrm{\Delta }e}$ once again contradicts differential decision activity ${\displaystyle \mathrm{\Delta }r}$. Whereas the activity of decision pools varies in phase (or counterphase) with perceptual appearance, the activity of evidence pools changes in quarterphase (or negative quarterphase) with perceptual appearance (e.g., Figures 1c,d,2a), consistent with some previous models (Gigante et al., 2009; Albert et al., 2017; Weilnhammer et al., 2017).

Figure 2

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Binocular rivalry

To compare predictions of the model described above to experimental observations, we measured spontaneous reversals of BR for different combinations of image contrast. BR is a particularly well-studied instance of multistable perception (Wheatstone, 1838; Diaz-Caneja, 1928; Levelt, 1965; Leopold and Logothetis, 1999; Brascamp et al., 2015). When conflicting images are presented to each eye (e.g., by means of a mirror stereoscope or of colored glasses, see Materials and methods), the phenomenal appearance reverses from time to time between the two images (Figure 1a). Importantly, the perceptual conflict involves also representations of coherent (binocular) patterns and is not restricted to eye-specific (monocular) representations (Logothetis et al., 1996; Kovács et al., 1996; Bonneh et al., 2001; Blake and Logothetis, 2002).

Specifically, our experimental observations established reversal sequences for ${\displaystyle 5\times 5}$ combinations of image contrast, ${\displaystyle c_{dom},c_{sup}\in \{\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},1\}}$. During any given dominance period, ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}}$ is the contrast of the phenomenally dominant image and ${\displaystyle c_{\mathit{s}\mathit{u}\mathit{p}}}$ the contrast of the other, phenomenally suppressed image (see Materials and methods). We analyzed these observations in terms of mean dominance durations ${\displaystyle ⟨T⟩}$, higher moments ${\displaystyle c_V}$ and ${\displaystyle {\gamma }_1/c_V}$ of the distribution of dominance durations, and sequential correlation ${\displaystyle c{c}_1}$ of successive dominance durations.

Additional aspects of serial dependence are discussed further below.

As described in Materials and methods, we fitted 11 model parameters to reproduce observations with more than 50 degrees of freedom: ${\displaystyle 5\times 5}$ mean dominance durations ${\displaystyle ⟨T⟩}$, ${\displaystyle 5\times 5}$ coefficients of variation ${\displaystyle c_V}$, one value of skewness ${\displaystyle {\gamma }_1/c_V=2}$, and one correlation coefficient ${\displaystyle c{c}_1=0.06}$. The latter two values were obtained by averaging over ${\displaystyle 5\times 5}$ contrast combinations and rounding. Importantly, minimization of the fit error, by random sampling of parameter space with a stochastic gradient descent, resulted in a three-dimensional manifold of suboptimal solutions. This revealed a high degree of redundancy among the 11 model parameters (see Materials and methods). Accordingly, we estimate that the effective number of degrees of freedom needed to reproduce the desired out-of-equilibrium dynamics was between 3 and 4. Model predictions and experimental observations are juxtaposed in Figures 3 and 4.

Figure 3

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Figure 4

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The complex and asymmetric dependence of mean dominance durations on image contrast — aptly summarized by Levelt’s ‘propositions’ I to IV (Levelt, 1965; Brascamp et al., 2015) — is fully reproduced by the model (Figure 3). Here, we use the updated definition of Brascamp et al., 2015: increasing the contrast of one image increases the fraction of time during which this image dominates appearance (‘predominance,’ Levelt I). Counterintuitively, this is due more to shortening dominance of the unchanged image than to lengthening dominance of the changed image (Levelt II, Figure 3b, left panel). Mean dominance durations grow (and alternation rates decline) symmetrically around equal predominance as contrast difference ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}-c_{sup}}$ increases (Levelt III, Figure 3b, right panel). Mean dominance durations shorten when both image contrasts ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}=c_{\mathit{s}\mathit{u}\mathit{p}}}$ increase (Levelt IV, Figure 3c).

Successive dominance durations are typically correlated positively (Fox and Herrmann, 1967; Walker, 1975; Pastukhov et al., 2013c). Averaging over all contrast combinations, observed and fitted correlation coefficients were comparable with ${\displaystyle c{c}_1=0.06\pm 0.06}$ (mean and standard deviation). Unexpectedly, both observed and fitted correlations coefficients increased systematically with image contrast ($\rho =0.9$, $p\S lt;.01$), growing from $c{c}_1=0.02\pm 0.05$ at ${\displaystyle c_{dom}=c_{sup}=\frac{1}{16}}$ to ${\displaystyle 0.21\pm 0.06}$ at ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}=c_{\mathit{d}\mathit{o}\mathit{m}}=1}$ (Figure 3e, blue symbols). It is important to that this dependence was not fitted. Rather, this previously unreported dependence constitutes a model prediction that is confirmed by observation.

The distribution of dominance durations typically takes a characteristic shape (Cao et al., 2016; Fox and Herrmann, 1967; Blake et al., 1971; Borsellino et al., 1972; Walker, 1975; De Marco et al., 1977; Murata et al., 2003; Brascamp et al., 2005; Pastukhov and Braun, 2007; Denham et al., 2018), approximating a gamma distribution with shape parameter ${\displaystyle r\simeq 3-4}$, or coefficient of variation ${\displaystyle c_V=1/\sqrt{r}\simeq 0.5-0.6}$. The fitted model fully reproduces this ‘scaling property’ (Figure 4). The observed coefficient of variation remained in the range ${\displaystyle c_V\simeq 0.05-0.06}$ for nearly all contrast combinations (Figure 4b). Unexpectedly, both observed and fitted values increased above, or decreased below, this range at extreme contrast combinations (Figure 4b, left panel). Along the main diagonal ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}=c_{\mathit{s}\mathit{u}\mathit{p}}}$ , where observed values had smaller error bars, both observed and fitted values of skewness were ${\displaystyle {\gamma }_1/c_V\simeq 2}$ and thus approximated a gamma distribution (Figure 4d, blue symbols).

The characteristic input dependence of average dominance durations emerges in two steps (as in Gigante et al., 2009). First, inputs and feedback suppression shape the birth-death dynamics of evidence pools ${\displaystyle E}$ and ${\displaystyle E^{\mathrm{\prime }}}$ (by setting disparate transition rates ${\displaystyle {\nu }^{\pm }}$, following Equation 3’ and Equation 1). Second, this sets in motion two opponent developments (habituation of dominant evidence activity and recovery of non-dominant evidence activity, both following Equation 2) that jointly determine dominance duration.

To elucidate this mechanism, it is helpful to consider the limit of large pools (${\displaystyle N\to \mathrm{\infty }}$) and its deterministic dynamics (Figure 2), which corresponds to the average stochastic dynamics. In this limit, periods of dominant evidence ${\displaystyle E}$ or ${\displaystyle E^{\mathrm{\prime }}}$ start and end at the same levels (${\displaystyle e_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}=e_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\prime }}$ and ${\displaystyle e_{\mathit{e}\mathit{n}\mathit{d}}={e^{\prime }}_{\mathit{e}\mathit{n}\mathit{d}}}$), because reversal thresholds ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ are the same for evidence difference ${\displaystyle e-e^{\prime }}$ and ${\displaystyle e^{\mathrm{\prime }}-e}$ (see section Levelt IV below).

The rates at which evidence habituates or recovers depend, in the first instance, on asymptotic levels ${\displaystyle e_{\mathrm{\infty }}}$ and ${\displaystyle {e^{\mathrm{\prime }}}_{\mathrm{\infty }}}$ (Equation 1 and 2, Figure 2b and Appendix 1—figure 4). In general, dominance durations depend on distance ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ between asymptotic levels: the further apart these are, the faster the development and the shorter the duration. As feedback suppression inverts the sign of the opponent developments, dominant evidence decreases (habituates) while non-dominant evidence increases (recovers). Due to this inversion, ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ is roughly proportional to ${\displaystyle e_{\mathrm{\infty }}^{\mathit{n}\mathit{o}\mathit{n}\mathit{-}\mathit{d}\mathit{o}\mathit{m}}-e_{\mathrm{\infty }}^{\mathit{d}\mathit{o}\mathit{m}}+w_{\mathit{s}\mathit{u}\mathit{p}\mathit{p}}}$. It follows that the distance ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ is smaller and the reversal dynamics slower when dominant input is stronger, and vice versa. It further follows that incrementing one input (and raising the corresponding asymptotic level) speeds up recovery or slows down habituation, shortening or lengthening periods of non-dominance and dominance, respectively (Levelt I).

In the second instance, rates of habituation or recovery depend on characteristic times ${\displaystyle {\tau }_e}$ and ${\displaystyle {\tau }_{e^{\mathrm{\prime }}}}$ (Equation 1 and 2). When these rates are unequal, dominance durations depend more sensitively on the slower process. This is why dominance durations depend more sensitively on non-dominant input (Levelt II): recovery of non-dominant evidence is generally slower than habituation of dominant evidence, independently of which input is weaker or stronger. The reason is that the respective effects of characteristic times ${\displaystyle {\tau }_e}$ and ${\displaystyle {\tau }_{e^{\mathrm{\prime }}}}$ and asymptotic levels ${\displaystyle e_{\mathrm{\infty }}}$ and ${\displaystyle {e^{\mathrm{\prime }}}_{\mathrm{\infty }}}$ are synergistic for weaker-input evidence (in both directions), whereas they are antagonistic for stronger-input evidence (see Appendix 1, section Deterministic dynamics: Evidence pools and Appendix 1—figure 4).

In general, dominance durations depend hyperbolically on ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ (Figure 2c and Equation 7 in Appendix 1). Dominance durations become infinite (and reversals cease) when ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ falls below the reversal threshold ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$. This hyperbolic dependence is also why alternation rate peaks at equidominance (Levelt III): increasing the difference between inputs always lengthens longer durations more than it shortens shorter durations, thus lowering alternation rate.

For all combinations of image contrast, the mechanism accurately predicts the experimentally observed distributions of dominance durations. This is owed to the stochastic activity of pools of bistable variables.

Firstly, dominance distributions retain nearly the same shape, even though average durations vary more than threefold with image contrast (see also Appendix 1—figure 6a,b). This ‘scaling property’ is due to the Poisson-like variability of birth-death processes (see Appendix 1, section Stochastic dynamics). Generally, when a stochastic accumulation approaches threshold, the rates of both accumulation and dispersion of activity affect the distribution of first-passage-times (Cao et al., 2014; Cao et al., 2016). In the special case of Poisson-like variability, the two rates vary proportionally and preserve distribution shape (see also Appendix 1—figure 6c,d).

Secondly, predicted distributions approximate gamma distributions with scale factor ${\displaystyle r\simeq 3-4}$. As shown previously (Cao et al., 2014; Cao et al., 2016), this is due to birth-death processes accumulating activity within a narrow range (i.e., evidence difference ${\displaystyle \mathrm{\Delta }e\le 0.2}$). In this low-threshold regime, the first-passage-times of birth-death processes are both highly variable and gamma distributed, consistent with experimental observations.

Thirdly, the predicted variability (coefficients of variation) of dominance periods varies along the ${\displaystyle c+c^{\prime }=1}$ axis, being larger for longer than for shorter dominance durations (Figure 4a,b). The reason is that stochastic development becomes noise-dominated. For longer durations, stronger-input evidence habituates rapidly into a regime where random fluctuations gain importance (see also Appendix 1—figure 4a,b).

The model accurately predicts how dominance durations shorten with higher image contrast ${\displaystyle c=c^{\mathrm{\prime }}}$ (Levelt IV). Surprisingly, this reflects the dynamics of decision pools ${\displaystyle R}$ and ${\displaystyle R^{\mathrm{\prime }}}$ (Figure 5).

Figure 5

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Here again it is helpful to consider the deterministic limit of large pools (${\displaystyle N\to \mathrm{\infty }}$). In this limit, a dominant decision state ${\displaystyle r^{\mathrm{\prime }}\simeq 1}$ is destabilized when a contradictory evidence difference ${\displaystyle \mathrm{\Delta }e=e-e^{\prime }}$ exceeds a certain threshold value ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ (Figure 5b and Appendix 1, section Deterministic dynamics: Decision pools). Due to the combined effect of excitatory and inhibitory feedforward projections, ${\displaystyle w_{\mathit{e}\mathit{x}\mathit{c}}}$ and ${\displaystyle w_{\mathit{i}\mathit{n}\mathit{h}}}$ (Equation 4 and Figure 5a), this average reversal threshold decreases with mean evidence activity ${\displaystyle \overline{e}=(e+e^{\prime })/2}$. Simulations of the fully stochastic model (${\displaystyle N=25}$) confirm this analysis (Figure 5c). As average evidence activity ${\displaystyle ⟨\overline{e}⟩}$ increases with image contrast, the average evidence bias ${\displaystyle ⟨\mathrm{\Delta }e⟩}$ at the time of reversals decreases, resulting in shorter dominance periods (Figure 5d).

Serial dependence

The proposed mechanism predicts positive correlations between successive dominance durations, a well-known characteristic of multistable phenomena (Fox and Herrmann, 1967; Walker, 1975; Van Ee, 2005; Denham et al., 2018). In addition, it predicts further aspects of serial dependence not reported previously.

In both model and experimental observations, a long dominance period tends to be followed by another long period, and a short dominance period by another short period (Figure 6). In the model, this is due to mean evidence activity ${\displaystyle \overline{e}=(e+e^{\mathrm{\prime }})/2}$ fluctuating stochastically above and below its long-term average. The autocorrelation time of these fluctuations increases monotonically with image contrast and, for high contrast, spans multiple dominance periods (see Appendix 1, section Characteristic times and Appendix 1—figure 7). Note that fluctuations of ${\displaystyle \overline{e}}$ diminish as the number of bistable variables increases and vanishe in the deterministic limit ${\displaystyle N\to \mathrm{\infty }}$.

Figure 6

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Crucially, fluctuations of mean evidence ${\displaystyle \overline{e}}$ modulate both reversal threshold ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ and dominance durations ${\displaystyle T}$, as illustrated in Figure 6a,b. To obtain Figure 6a, dominance durations were grouped into quantiles and the average duration ${\displaystyle ⟨T_0⟩}$ of each quantile was compared to the conditional expectation of preceding and following durations ${\displaystyle ⟨T_{\pm n}⟩}$ (upper graph). For the same quantiles (compare color coding), average evidence activity ${\displaystyle ⟨{\overline{e}}_0⟩}$ was compared to the conditional expectation ${\displaystyle ⟨{\overline{e}}_{\pm n}⟩}$ at the end of preceding and following periods (lower graph). Both the inverse relation between ${\displaystyle ⟨T_{\pm n}⟩}$ and ${\displaystyle ⟨{\overline{e}}_{\pm n}⟩}$ and the autocorrelation over multiple dominance periods are evident.

This source of serial dependency – comparatively slow fluctuations of ${\displaystyle \overline{e}}$ and ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ – predicts several qualitative characteristics not reported previously and now confirmed by experimental observations. First, sequential correlations are predicted (and observed) to be strictly positive at all lags (next period, one-after-next period, and so on) (Figure 6d). In other words, it predicts that several successive dominance periods are shorter (or longer) than average.

Second, due to the contrast dependence of autocorrelation time, sequential correlations are predicted (and observed) to increase with image contrast (Figure 6d). The experimentally observed degree of contrast dependence is broadly consistent with pool sizes between ${\displaystyle N=25}$ and ${\displaystyle N=40}$ (black and red curves in Figure 3e). Larger pools with hundreds of bistable variables do not express the observed dependence on contrast (not shown).

Third, for high image contrast, reversal sequences are predicted (and observed) to contain extended episodes with dominance periods that are short or extended episodes with periods that are long (Figure 6c). When quantified in terms of a ‘burstiness index,’ the degree of inhomogeneity in predicted and observed reversal sequences is comparable (see Appendix 1, section Burstiness and Appendix 1—figure 8).

Many previous models of BR (e.g., Laing and Chow, 2002) postulated selective adaptation of competing representations to account for serial dependency. However, selective adaptation is an opponent process that favors positive correlations between different dominance periods, but negative correlations between same dominance periods. To demonstrate this point, we fitted such a model to reproduce our experimental observations (${\displaystyle T}$, ${\displaystyle c_V}$, ${\displaystyle {\gamma }_1}$, and ${\displaystyle c{c}_1}$) for five image contrasts ${\displaystyle c=c^{\mathrm{\prime }}}$. As expected, the alternative model predicts negative correlations ${\displaystyle c{c}_2}$ for same dominance periods (Figure 6d, right panel), contrary to what is observed.

Two principal alternatives have been considered for the dynamical mechanism of perceptual decision-making: drift-diffusion models (Luce, 1986; Ratcliff and Smith, 2004) and recurrent network models (Wang, 2008; Wang, 2012). The mechanism proposed here combines both alternatives: at its evidence level, sensory information is integrated, over both space and time, by ‘local attractors’ in a discrete version of a drift-diffusion process. At its decision level, the population dynamics of a recurrent network implements a winner-take-all competition between ‘non-local attractors.’ Together, the two levels form a ‘nested attractor’ system (Braun and Mattia, 2010) operating perpetually out of equilibrium.

A recurrent network with strong competition typically ‘normalizes’ individual responses relative to the total response (Miller, 2016). Divisive normalization is considered a canonical cortical computation (Carandini and Heeger, 2011), for which multiple rationales can be found. Here, divisive normalization is augmented by indiscriminate feedforward inhibition. This combination ensures that decision activity rapidly and reliably categorizes differential input strength, largely independently of total input strength.

Another key feature of the proposed mechanism is that a ‘dominant’ decision pool applies feedback suppression to the associated evidence pool. Selective suppression of evidence for a winning hypothesis features in computational theories of ‘hierarchical inference’ (Rao and Ballard, 1999; Lee and Mumford, 2003; Parr and Friston, 2017b; Pezzulo et al., 2018), as well as in accounts of multistable perception inspired by such theories (Dayan, 1998; Hohwy et al., 2008; Weilnhammer et al., 2017). A normative reason for feedback suppression arises during continuous inference in uncertain and volatile environments, where the accumulation of sensory information is ongoing and cannot be restricted to appropriate intervals (Veliz-Cuba et al., 2016). Here, optimal change detection requires an exponentially rising bias against evidence for the most likely state, ensuring that even weak changes are detected, albeit with some delay.

The pivotal feature of the proposed mechanism are pools of bistable variables or ‘local attractors.’ Encoding sensory inputs in terms of persistent ‘activations’ of local attractors assemblies (rather than in terms of transient neuronal spikes) creates an intrinsically retentive representation: sites that respond are also sites that retain information (for a limited time). Our results are consistent with a few tens of bistable variables in each pool. In the proposed mechanism, differential activity of two pools accumulates evidence against the dominant appearance until a threshold is reached and a reversal ensues (see also Barniv and Nelken, 2015; Nguyen et al., 2020). Conceivably, this discrete non-equilibrium dynamics might instantiate a variational principle of inference such as ‘maximum caliber’ (Pressé et al., 2013; Dixit et al., 2018).

The components of the proposed mechanism interact to guarantee the statistical features that characterize BR and other multistable phenomena. Discretely stochastic accumulation of differential evidence against the dominant appearance ensures sensitivity of dominance durations to non-dominant input. It also ensures the invariance of relative variability (‘scaling property’) and gamma-like distribution shape of dominance durations. Due to a non-trivial interaction with the competitive decision, discretely stochastic fluctuations of evidence-level activity express themselves in a serial dependency of dominance durations. Several features of this dependency were unexpected and not reported previously, for example, the sensitivity to image contrast and the ‘burstiness’ of dominance reversals (i.e., extended episodes in which dominance periods are consistently longer or shorter than average). The fact that these predictions are confirmed by our experimental observations provides further support for the proposed mechanism.

How does the proposed mechanism compare to previous ‘dynamical’ models of multistable phenomena? It is of similar complexity as previous minimal models (Laing and Chow, 2002; Wilson, 2007; Moreno-Bote et al., 2010) in that it assumes four state variables at two dynamical levels, one slow (accumulation) and one fast (winner-take-all competition). It differs in reversing their ordering: visual input impinges first on the slow level, which then drives the fast level. It also differs in that stochasticity dominates the slow dynamics (as suggested by van Ee, 2009), not the fast dynamics. However, the most fundamental difference is discreteness (pools of bistable variables), which shapes all key dynamical properties.

Unlike many previous models (e.g., Laing and Chow, 2002; Wilson, 2007; Moreno-Bote et al., 2007; Moreno-Bote et al., 2010; Cohen et al., 2019), the proposed mechanism does not include adaptation (stimulation-driven weakening of evidence), but a phenomenologically similar feedback suppression (perception-driven weakening of evidence). Evidence from perceptual aftereffects supports the existence of both stimulation- and perception-driven adaptation, albeit at different levels of representation. Aftereffects in the perception of simple visual features – such as orientation, spatial frequency, or direction of motion (Blake and Fox, 1974; Lehmkuhle and Fox, 1975; Wade and Wenderoth, 1978) – are driven by stimulation rather than by perceived dominance, whereas aftereffects in complex features – such as spiral motion, subjective contours, rotation in depth (Wiesenfelder and Blake, 1990; Van der Zwan and Wenderoth, 1994; Pastukhov et al., 2014a) – typically depend on perceived dominance. Several experimental observations related to BR have been attributed to stimulation-driven adaptation (e.g., negative priming, flash suppression, generalized flash suppression; Tsuchiya et al., 2006). The extent to which a perception-driven adaptation could also explain these observations remains an open question for future work.

Multistable perception induces a positive priming or ‘sensory memory’ (Pearson and Clifford, 2005; Pastukhov and Braun, 2008; Pastukhov et al., 2013a), which can stabilize a dominant appearance during intermittent presentation (Leopold et al., 2003; Maier et al., 2003; Sandberg et al., 2014). This positive priming exhibits rather different characteristics (e.g., shape-, size- and motion-specificity, inducement period, persistence period) than the negative priming/adaptation of rivaling representations (de Jong et al., 2012; Pastukhov et al., 2013a; Pastukhov and Braun, 2013b; Pastukhov et al., 2014a; Pastukhov et al., 2014b; Pastukhov, 2016). To our mind, this evidence suggest that sensory memory is mediated by additional levels of representation and not by self-stabilization of rivaling representations, as has been suggested (Noest et al., 2007; Leptourgos, 2020). To incorporate sensory memory, the present model would have to be extended to include three hierarchical levels (evidence, decision, and memory), as previously proposed by Gigante et al., 2009.

BR arises within local regions of the visual field, measuring approximately ${0.25}^{\circ }$ to ${0.5}^{\circ }$ in the fovea (Leopold, 1997; Logothetis, 1998). No rivalry ensues when the stimulated locations in the left and right eye are more distant from each other. The computational model presented here encompasses only one such local region, and therefore cannot reproduce spatially extended phenomena such as piecemeal rivalry (Blake et al., 1992) or traveling waves (Wilson et al., 2001). To account for these phenomena, the visual field would have to be tiled with replicant models linked by grouping interactions (Knapen et al., 2007; Bressloff and Webber, 2012).

A particularly intriguing previous model (Wilson, 2003) postulated a hierarchy with competing and adapting representations in eight state variables at two separate levels, one lower (monocular) and another higher (binocular) level. This ‘stacked’ architecture could explain the fascinating experimental observation that one image can continue to dominate (dominance durations ${\displaystyle \sim 2\mathit{s}}$) even when images are rapidly swapped between eyes (period ${\displaystyle 1/3\mathit{s}}$) (Kovács et al., 1996; Logothetis et al., 1996). We expect that our hierarchical model could also account for this phenomenon if it were to be replicated at two successive levels. It is tempting to speculate that such ‘stacking’ might have a normative justification in that it might subserve hierarchical inference (Yuille and Kersten, 2006; Hohwy et al., 2008; Friston, 2010).

Another previous model (Li et al., 2017) used a hierarchy with 24 state variables at three separate levels to show that a stabilizing influence of selective visual attention could also explain slow rivalry when images are swapped rapidly. Additionally, this rather complex model reproduced the main features of Levelt’s propositions, but did not consider scaling property and sequential dependency. The model shared some of the key features of the present model (divisive inhibition, differential excitation-inhibition), but added a multiplicative attentional modulation. As the present model already incorporates the ‘biased competition’ that is widely thought to underlie selective attention (Sabine and Ungerleider, 2000; Reynolds and Heeger, 2009), we expect that it could reproduce attentional effects by means of additive modulations.

The notion that multistable phenomena such as BR reflect active exploration of explanatory hypotheses for sensory evidence has a venerable history (von Helmholtz, 1867; Barlow et al., 1972; Gregory, 1980; Leopold and Logothetis, 1999). The mechanism proposed here is in keeping with that notion: higher-level ‘explanations’ compete for control (‘dominance’) of phenomenal appearance in terms of their correspondence to lower-level ‘evidence.’ An ‘explanation’ takes control if its correspondence is sufficiently superior to that of rival ‘explanations.’ The greater the superiority, the longer control is retained. Eventually, alternative ‘explanations’ seize control, if only briefly. This manner of operation is also consistent with computational theories of ‘analysis by synthesis’ or ‘hierarchical inference,’ although there are many differences in detail (Rao and Ballard, 1999; Parr and Friston, 2017b; Pezzulo et al., 2018).

Interacting with an uncertain and volatile world necessitates continuous and concurrent evaluation of sensory evidence and selection of motor action (Cisek and Kalaska, 2010; Gold and Stocker, 2017). Multistable phenomena exemplify continuous decision-making without external prompting (Braun and Mattia, 2010). Sensory decision-making has been studied extensively, mostly in episodic choice-task, and the neural circuits and activity dynamics underlying episodic decision-making – including representations of potential choices, sensory evidence, and behavioral goals – have been traced in detail (Cisek and Kalaska, 2010; Gold and Shadlen, 2007; Wang, 2012; Krug, 2020). Interestingly, there seems to be substantial overlap between choice representations in decision-making and in multistable situations (Braun and Mattia, 2010).

Continuous inference has been studied extensively in auditory streaming paradigms (Winkler et al., 2012; Denham et al., 2014). The auditory system seems to continually update expectations for sound patterns on the basis of recent experience. Compatible patterns are grouped together in auditory awareness, and incompatible patterns result in spontaneous reversals between alternatives. Many aspects of this rich phenomenology are reproduced by computational models driven by some kind of ‘prediction error’ (Mill et al., 2013). The dynamics of two recent auditory models (Barniv and Nelken, 2015; Nguyen et al., 2020) are rather similar to the model presented here: while one sound pattern dominates awareness, evidence against this pattern is accumulated at a subliminal level.

What might be the neural basis of the bistable variables/‘local attractors’ proposed here? Ongoing activity in sensory cortex appears to be low-dimensional, in the sense that the activity of neurons with similar response properties varies concomitantly (‘shared variability,’ ‘noise correlations,’ Ponce-Alvarez et al., 2012, Mazzucato et al., 2015, Engel et al., 2016, Rich and Wallis, 2016, Mazzucato et al., 2019). This shared variability reflects the spatial clustering of intracortical connectivity (Muir and Douglas, 2011; Okun et al., 2015; Cossell et al., 2015; Lee et al., 2016; Rosenbaum et al., 2017) and unfolds over moderately slow time scales (in the range of ${\displaystyle 100\mathit{m}\mathit{s}}$ to ${\displaystyle 500\mathit{m}\mathit{s})}$ both in primates and rodents (Ponce-Alvarez et al., 2012; Mazzucato et al., 2015; Cui et al., 2016; Engel et al., 2016; Rich and Wallis, 2016; Mazzucato et al., 2019).

Possible dynamical origins of shared and moderately slow variability have been studied extensively in theory and simulation (for reviews, see Miller, 2016; Huang and Doiron, 2017; La Camera et al., 2019). Networks with weakly clustered connectivity (e.g., 3% rewiring) can express a metastable attractor dynamics with moderately long time scales (Litwin-Kumar and Doiron, 2012; Doiron and Litwin-Kumar, 2014; Schaub et al., 2015; Rosenbaum et al., 2017). In a metastable dynamics, individual (connectivity-defined) clusters transition spontaneously between distinct and quasi-stationary activity levels (‘attractor states’) (Tsuda, 2001; Stern et al., 2014).

Evidence for metastable attractor dynamics in cortical activity is accumulating steadily (Mattia et al., 2013; Mazzucato et al., 2015; Rich and Wallis, 2016; Engel et al., 2016; Marcos et al., 2019; Mazzucato et al., 2019). Distinct activity states with exponentially distributed durations have been reported in sensory cortex (Mazzucato et al., 2015; Engel et al., 2016), consistent with noise-driven escape transitions (Doiron and Litwin-Kumar, 2014; Huang and Doiron, 2017). And several reports are consistent with external input modulating cortical activity mostly indirectly, via the rate of state transitions (Fiser et al., 2004; Churchland et al., 2010; Mazzucato et al., 2015; Engel et al., 2016; Mazzucato et al., 2019).

The proposed mechanism assumes bistable variables with noise-driven escape transitions, with transition rates modulated exponentially by external synaptic drive. Following previous work (Cao et al., 2016), we show this to be an accurate reduction of the population dynamics of metastable networks of spiking neurons.

Unfortunately, the spatial structure of the ‘shared variability’ or ‘noise correlations’ in cortical activity described above is poorly understood. However, we estimate that the cortical representation of our rivaling display involves approximately ${\displaystyle 400{\mathit{m}\mathit{m}}^2}$ and ${\displaystyle 200{\mathit{m}\mathit{m}}^2}$ of cortical surface in cortical areas V1 and V4, respectively (Winawer and Witthoft, 2015; Winawer and Benson, 2021). Accordingly, in each of these two cortical areas, the neural representation of rivaling stimulation can comfortably accommodate several thousand recurrent local assemblies, each capable of expressing independent collective dynamics (i.e., ‘classic columns’ comprising several ‘minicolumns’ with distinct stimulus selectivity Nieuwenhuys R, 1994, Kaas, 2012). Thus, our model assumes that the representation of two rivaling images engages approximately 1–2% of the available number of recurrent local assemblies.

Neurophysiological correlates of BR have been studied extensively, often by comparing reversals of phenomenal appearance during binocular stimulation with physical alternation (PA) of monocular stimulation (e.g., Leopold and Logothetis, 1996; Scheinberg and Logothetis, 1997; Logothetis, 1998; Wilke et al., 2006; Aura et al., 2008; Keliris et al., 2010; Panagiotaropoulos et al., 2012; Bahmani et al., 2014; Xu et al., 2016; Kapoor et al., 2020; Dwarakanath et al., 2020). At higher cortical levels, such as inferior temporal cortex (Scheinberg and Logothetis, 1997) or prefrontal cortex (Panagiotaropoulos et al., 2012; Kapoor et al., 2020; Dwarakanath et al., 2020), BR and PA elicit broadly comparable neurophysiological responses that mirror perceptual appearance. Specifically, activity crosses its average level at the time of each reversal, roughly in phase with perceptual appearance (Scheinberg and Logothetis, 1997; Kapoor et al., 2020). In primary visual cortex (area V1), where many neurons are dominated by input from one eye, neurophysiological correlates of BR and PA diverge in an interesting way: whereas modulation of spiking activity is weaker during BR than PA (Leopold and Logothetis, 1996; Logothetis, 1998; Wilke et al., 2006; Aura et al., 2008; Keliris et al., 2010), measures thought to record dendritic inputs are modulated comparably under both conditions (Aura et al., 2008; Keliris et al., 2010; Bahmani et al., 2014; Yang et al., 2015; Xu et al., 2016). A stronger divergence is observed at an intermediate cortical level (visual area V4), where neurons respond to both eyes. Whereas some units modulate their spiking activity comparably during BR and PA (i.e., increased activity when preferred stimulus becomes dominant), other units exhibit the opposite modulation during BR (i.e., reduced activity when preferred stimulus gains dominance) (Leopold and Logothetis, 1996; Logothetis, 1998; Wilke et al., 2006). Importantly, at this intermediate cortical level, activity crosses its average level well before and after each reversal (Leopold and Logothetis, 1996; Logothetis, 1998), roughly in quarter phase with perceptual appearance.

Some of these neurophysiological observations are directly interpretable in terms of the model proposed here. Specifically, activity modulation at higher cortical levels (inferotemporal cortex, prefrontal cortex) could correspond to ‘decision activity,’ predicted to vary in phase with perceptual appearance. Similarly, activity modulation at intermediate cortical levels (area V4) could correspond to ‘evidence activity,’ which is predicted to vary in quarter phase with perceptual appearance. This identification would also be consistent with the neurophysiological evidence for attractor dynamics in columns of area V4 (Engel et al., 2016). The subpopulation of area V4 with opposite modulation could mediate feedback suppression from decision levels. If so, our model would predict this subpopulation to vary in counterphase with perceptual appearance. Finally, the fascinating interactions observed within primary visual cortex (area V1) are well beyond the scope of our simple model. Presumably, a ‘stacked’ model with two successive levels of competitive interactions at monocular and binocular levels or representation (Wilson, 2003; Li et al., 2017) would be required to account for these phenomena.

As multistable phenomena and their characteristics are ubiquitous in visual, auditory, and tactile perception, the mechanism we propose may form a general part of sensory processing. It bridges neural, perceptual, and normative levels of description and potentially offers a ‘comprehensive task-performing model’ (Kriegeskorte and Douglas, 2018) for sensory decision-making.

Six practiced observers participated in the experiment (four males, two females). Informed consent, and consent to publish, was obtained from all observers, and ethical approval Z22/16 was obtained from the Ethics Commission of the Faculty of Medicine of the Otto-von-Guericke University, Magdeburg. Stimuli were displayed on an LCD screen (EIZO ColorEdge CG303W, resolution ${\displaystyle 2560\times 1600}$ pixels, viewing distance was 104 cm, single pixel subtended ${\displaystyle {0.014}^{\circ }}$, refresh rate 60 Hz) and were viewed through a mirror stereoscope, with viewing position being stabilized by chin and head rests. Display luminance was gamma-corrected and average luminance was ${\displaystyle 50\mathit{c}\mathit{d}/m^2}$.

Two grayscale circular orthogonally oriented gratings (${\displaystyle +{45}^{\circ }}$ and ${\displaystyle -{45}^{\circ }}$) were presented foveally to each eye. Gratings had diameter of ${\displaystyle {1.6}^{\circ }}$, spatial period ${\displaystyle 2\mathit{c}\mathit{y}\mathit{c}/\mathit{d}\mathit{e}\mathit{g}}$. To avoid a sharp outer edge, grating contrast was modulated with Gaussian envelope (inner radius ${\displaystyle {0.6}^{\circ }}$, ${\displaystyle \sigma ={0.2}^{\circ }}$). Tilt and phase of gratings was randomized for each block. Five contrast levels were used: 6.25, 12.5, 25, 50, and 100%. Contrast of each grating was systematically manipulated, so that each contrast pair was presented in two blocks (50 blocks in total). Blocks were ${\displaystyle 120\mathit{s}}$ long and separated by a compulsory 1 min break. Observers reported on the tilt of the visible grating by continuously pressing one of two arrow keys. They were instructed to press only during exclusive visibility of one of the gratings, so that mixed percepts were indicated by neither key being pressed (25% of total presentation time). To facilitate binocular fusion, gratings were surrounded by a dichoptically presented square frame (outer size 9.8°, inner size 2.8°).

Dominance periods of ‘clear visibility’ were extracted in sequence from the final ${\displaystyle 90\mathit{s}}$ of each block and the mean linear trend was subtracted from all values. Values from the initial ${\displaystyle 30\mathit{s}}$ were discarded. To make comparable the dominance periods of different observers, values were rescaled by the ratio of the all-condition-all-observer average (${\displaystyle 2.5\mathit{s}}$) and the all-condition average of each observer (${\displaystyle 2.5\pm 1.3\mathit{s}}$). Finally, dominance periods from symmetric conditions ${\displaystyle (c_{\mathit{l}\mathit{e}\mathit{f}\mathit{t}},c_{\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}})}$ with ${\displaystyle c_{\mathit{l}\mathit{e}\mathit{f}\mathit{t}}=c_{\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}}}$ were combined into a single category ${\displaystyle (c_{\mathit{d}\mathit{o}\mathit{m}},c_{\mathit{s}\mathit{u}\mathit{p}})}$, where ${\displaystyle c_{\mathit{d}\mathit{o}\mathit{m}}}$ (${\displaystyle c_{\mathit{s}\mathit{u}\mathit{p}}}$) was the contrast viewed by the dominant (suppressed) eye. The number of observed dominance periods ranged from 900 to 1700 per contrast combination (${\displaystyle 1300\pm 240}$).

For the dominance periods ${\displaystyle T}$ observed in each condition, first, second, and third central moments were computed, as well as coefficient of variation ${\displaystyle c_V}$ and skewness ${\displaystyle {\gamma }_1}$ relative to coefficient of variation:

The expected standard error of the mean for distribution moments is 2% for the mean, 3% for the coefficient of variation, and 12% for skewness relative to coefficient of variation, assuming 1000 gamma-distributed samples.

Coefficients of sequential correlations were computed from pairs of periods ${\displaystyle (T_i,T_j)}$ with opposite dominance (first and next: ‘lag’ ${\displaystyle j-i=1}$), pairs of periods with same dominance (first and next but one: ‘lag’ ${\displaystyle j-i=2}$), and so on,

where ${\displaystyle ⟨T⟩}$ and ${\displaystyle ⟨T^2⟩}$ are mean duration and mean square duration, respectively. The expected standard deviation of the coefficient of correlation is 0.03, assuming 1000 gamma-distributed samples.

To analyze ‘burstiness,’ we adapted a statistical measure used in neurophysiology (Compte et al., 2003). First, sequences of dominance periods were divided into all possible subsets of ${\displaystyle k\in \{2,3,\dots ,16\}}$ successive periods and mean durations computed for each subset. Second, heterogeneity was assessed by computing, for each size ${\displaystyle k}$, the coefficient of variation c_V over mean durations, compared to the mean and variance of the corresponding coefficient of variation for randomly shuffled sequences of dominance periods. Specifically, a ‘burstiness index’ was defined for each subset size ${\displaystyle k}$ as.

where ${\displaystyle c_V}$ is the coefficient of variation over subsets of size ${\displaystyle k}$ and where ${\displaystyle ⟨c_V{⟩}_{\mathit{s}\mathit{h}\mathit{u}\mathit{f}\mathit{f}\mathit{l}\mathit{e}}}$ and ${\displaystyle ⟨c_V^2{⟩}_{\mathit{s}\mathit{h}\mathit{u}\mathit{f}\mathit{f}\mathit{l}\mathit{e}}}$ are, respectively, mean and mean square of the coefficients of variation from shuffled sequences.

The proposed mechanism for BR dynamics relies on discretely stochastic processes (‘birth-death’ or generalized Ehrenfest processes). Bistable variables ${\displaystyle x\in \{0,1\}}$ transition between active and inactive states with time-varying Poisson rates ${\displaystyle {\nu }^{+}(t)}$ (activation) and ${\displaystyle {\nu }^{-}(t)}$ (inactivation). Two ‘evidence pools’ of ${\displaystyle N}$ such variables, ${\displaystyle E}$ and ${\displaystyle E^{\mathrm{\prime }}}$, represent two kinds visual evidence (e.g., for two visual orientations), whereas two ‘decision pools,’ ${\displaystyle R}$ and ${\displaystyle R^{\mathrm{\prime }}}$, represent alternative perceptual hypotheses (e.g., two grating patterns) (see also Appendix 1—figure 1). Thus, instantaneous dynamical state is represented by four active counts ${\displaystyle n_e,n_{e^{\mathrm{\prime }}},n_r,n_{r^{\mathrm{\prime }}}\in [0,N]}$ or, equivalently, by four active fractions ${\displaystyle e,e^{\mathrm{\prime }},r,r^{\mathrm{\prime }}\in [0,1]}$.

The development of pool activity over time is described by a master equation for probability ${\displaystyle P_n(t)}$ of the number ${\displaystyle n(t)\in [0,N]}$ active variables.

For constant ${\displaystyle {\nu }^{\pm }}$, the distribution ${\displaystyle P_n(t)}$ is binomial at all times Karlin and McGregor, 1965, van Kampen, 1981. The time development of the number of active units ${\displaystyle n_X(t)}$ in pool ${\displaystyle X}$ is an inhomogeneous Ehrenfest process and corresponds to the count of activations, minus the count of deactivations,

where ${\displaystyle \mathcal{B}\left(n,\nu \mathrm{\Delta }t\right)}$ is a discrete random variable drawn from a binomial distribution with trial number ${\displaystyle n}$ and success probability ${\displaystyle \nu \mathrm{\Delta }t}$.

All variables of a pool have identical transition rates, which depend exponentially on the ‘potential difference’ ${\displaystyle \mathrm{\Delta }u=u+u^0}$ between states, with a input-dependent component ${\displaystyle u}$ and a baseline component ${\displaystyle u^0}$:

where ${\displaystyle {\nu }_e}$ and ${\displaystyle {\nu }_r}$ are baseline rates and ${\displaystyle u_e^0}$ and ${\displaystyle u_r^0}$ baseline components. The input-dependent components of effective potentials are modulated linearly by synaptic couplings

Visual inputs are ${\displaystyle I=f(c)}$ and ${\displaystyle I^{\mathrm{\prime }}=f(c^{\mathrm{\prime }})}$, respectively, where

is a monotonically increasing, logarithmic function of image contrast, with parameter γ.

The proposed mechanism has 11 independent parameters – 6 synaptic couplings, 2 baseline rates, 2 baseline potentials, 1 contrast nonlinearity – which were fitted to experimental observations. A 12th parameter – pool size – remained fixed.

The experimental dataset consisted of two 5 × 5 arrays ${\displaystyle X_i^{\mathit{e}\mathit{x}\mathit{p}}}$ for mean ${\displaystyle ⟨T⟩}$ and coefficient of variation ${\displaystyle c_V}$, plus two scalar values for skewness ${\displaystyle {\gamma }_1=2}$ and correlation coefficient ${\displaystyle c{c}_1=0.06}$. The two scalar values corresponded to the (rounded) average values observed over the 5 × 5 combinations of image contrast. In other words, the fitting procedure prescribed contrast dependencies for the first two distribution moments, but not for correlation coefficients.

The fit error ${\displaystyle E_{\mathit{f}\mathit{i}\mathit{t}}}$ was computed as a weighted sum of relative errors

with weighting ${\displaystyle w=[1,1,1,{\displaystyle {\displaystyle 1}}/{\displaystyle {\displaystyle 4}}]}$ emphasizing distribution moments.

Approximately 400 minimization runs were performed, starting from random initial configurations of model parameters. For the optimal parameter set, the resulting fit error for the mean observer dataset was approximately 13%. More specifically, the fit errors for mean dominance ${\displaystyle ⟨T⟩}$, coefficient of variation ${\displaystyle c_V}$, relative skewness ${\displaystyle {\gamma }_1/c_V}$, and correlation coefficients ${\displaystyle c{c}_1}$ and $c{c}_2$ were 9.8, 7.9, 8.7, 70, and 46%, respectively. Here, fit errors for relative skewness and correlation coefficients were computed for the isocontrast conditions, where experimental observations were least noisy.

To confirm that resulting fit was indeed optimal and could not be further improved, we studied the behavior of the fit error in the vicinity of the optimal parameter set. For each parameter ${\displaystyle {\alpha }_i}$, 30 values ${\displaystyle {\alpha }_i^{(j)}}$ were picked in the direct vicinity of the optimal parameter ${\displaystyle {\alpha }_i^{\mathit{o}\mathit{p}\mathit{t}}}$ (Appendix 1—figure 9). The resulting scatter plot of value pairs ${\displaystyle {\alpha }_i^{(j)}}$ and fit error ${\displaystyle E_{\mathit{f}\mathit{i}\mathit{t}}^{(j)}}$ was approximated by a quadratic function, which provided 95% confidence intervals for ${\displaystyle {\alpha }_i^{(j)}}$. For all parameters except ${\displaystyle {\nu }_r}$, the estimated quadratic function was convex and the coefficient of the Hessian matrix associated with the fit error was positive. Additionally, the estimated extremum of each parabola was close to the corresponding optimal parameter, confirming that the parameter set was indeed optimal (Appendix 1—figure 9).

To minimize fit error, we repeated a stochastic gradient descent from randomly chosen initial parameter. Interestingly, the ensemble of suboptimal solutions found by this procedure populated a low-dimensional manifold of the parameter space in three principal components accounted for 95% of the positional variance. Thus, models that reproduce experimental observations with varying degrees of freedom exhibit only 3–4 effective degrees of freedom. We surmise that this is due, on the one hand, to the severe constraints imposed by our model architecture (e.g., discrete elements, exponential input dependence of transition rates) and, on the other hand, by the requirement that the dynamical operating regime behaves as a relaxation oscillator.

In support of this interpretation, we note that our 5 × 5 experimental measurements of ${\displaystyle ⟨T⟩}$ and ${\displaystyle c_V}$ were accurately described by ‘quadric surfaces’ (${\displaystyle z=a_1+a_{2}x+a_{3}y+a_4{x}^2+a_{5}xy+a_6{y}^2}$) with six coefficients each. Together with the two further measurements of ${\displaystyle {\gamma }_1/c_V}$ and ${\displaystyle c{c}_1}$, our experimental observations accordingly exhibited approximately ${\displaystyle 6\times 2+2=14}$ effective degrees of freedom. This number was sufficient to constrain the 3–4 dimensional manifold of parameters, where the model operated as a relaxation oscillator with a particular dynamics, specifically, a slow-fast dynamics associated, respectively, with the accumulation and reversal phases of BR.

As an alternative model (Laing and Chow, 2002), a combination of competition, adaptation, and image-contrast-dependent noise was fitted to reproduce four 5 × 5 arrays ${\displaystyle X_i^{\mathit{e}\mathit{x}\mathit{p}}}$ for mean ${\displaystyle ⟨T⟩}$, coefficient of variation ${\displaystyle c_V}$, skewness ${\displaystyle {\gamma }_1}$, and correlation coefficient ${\displaystyle c{c}_1}$. Fit error ${\displaystyle E_{\mathit{f}\mathit{i}\mathit{t}}}$ was computed as the average of relative errors

For purposes of comparison, a weighted fit error with weighting ${\displaystyle w=[1,1,1,{\displaystyle {\displaystyle 1}}/{\displaystyle {\displaystyle 4}}]}$ was computed, as well.

The model comprised four state variables and independent colored noise:

where ${\displaystyle F(x)=[1+\exp (-x/\kappa ){]}^{-1}}$ is a nonlinear activation function and ${\displaystyle \xi (t)}$ is white noise.

Additionally, both input ${\displaystyle I_{1,2}}$ and noise amplitude ${\displaystyle {\sigma }_{1,2}}$ were assumed to depend nonlinearly on image contrast ${\displaystyle c_{1,2}}$:

This coupling between input and noise amplitude served stabilizes the shape of dominance distributions over different image contrasts (‘scaling property’).

Parameters for competition β = 10, activity time constant ${\displaystyle {\tau }_r=50\mathit{m}\mathit{s}}$, noise time constant ${\displaystyle {\tau }_n=500\mathit{m}\mathit{s}}$, and activation function ${\displaystyle k=0.1}$ were fixed. Parameters for adaptation strength ${\displaystyle {\varphi }_a\in [1,100]}$, adaptation time constant ${\displaystyle {\tau }_a\in [1,00]}$, contrast dependence of input ${\displaystyle b_I\in [1,5]}$, ${\displaystyle k_I\in [0.1,5]}$, and contrast dependence of noise amplitude ${\displaystyle b_{\sigma }\in [0.1,1]}$, ${\displaystyle k_{\sigma }\in [0.1,1]}$ were explored within the ranges indicated.

The best fit (determined with a genetic algorithm) was as follows: ${\displaystyle {\varphi }_a=18.39}$, ${\displaystyle {\tau }_a=22.78}$, ${\displaystyle k_I=1.52}$, ${\displaystyle b_I=2.92}$, ${\displaystyle k_{\sigma }=0.57}$, ${\displaystyle b_{\sigma }=0.19}$. The fit errors for mean dominance ${\displaystyle ⟨T⟩}$, coefficient of variation ${\displaystyle c_V}$, skewness ${\displaystyle {\gamma }_1}$, and correlation coefficient ${\displaystyle c{c}_1}$ were, respectively, 11.3, 8.3, 20, and 55%. The fit error for correlation coefficient ${\displaystyle c{c}_2}$ was 180% (because the model predicted negative values). The combined average for ${\displaystyle ⟨T⟩}$, ${\displaystyle c_V}$, and ${\displaystyle {\gamma }_1}$ was 13.2%. The fit error obtained with weighting ${\displaystyle w=(1,1,1,{\displaystyle {\displaystyle 1}}/{\displaystyle {\displaystyle 4}})}$ was 16.4%.

For Figure 6d, the alternative model was fitted only to observations at equal image contrast, ${\displaystyle c=c^{\mathrm{\prime }}}$: mean dominance ${\displaystyle ⟨T⟩}$, coefficient of variation ${\displaystyle c_V}$, skewness ${\displaystyle {\gamma }_1}$, and correlation coefficient ${\displaystyle c{c}_1}$. The combined average fit error for ${\displaystyle ⟨T⟩}$, ${\displaystyle c_V}$, and ${\displaystyle {\gamma }_1}$ was 11.2%. The combined average for all four observables was 22%.

To illustrate a possible neural realization of ‘local attractors,’ we simulated a competitive network with eight identical assemblies of excitatory and inhibitory neurons, which collectively expresses a spontaneous and metastable dynamics (Mattia et al., 2013). One assembly (denoted as ‘foreground’) comprised 150 excitatory leaky-integrate-and-fire neurons, which were weakly coupled to the 1050 excitatory neurons of the other assemblies (denoted as ‘background’), as well as 300 inhibitory neurons. Note that background assemblies are not strictly necessary and are included only for the sake of verisimilitude. The connection probability between any two neurons was ${\displaystyle c=2/3}$. Excitatory synaptic efficacy between neurons in the same assembly and in two different assemblies was ${\displaystyle J_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}=0.612\mathit{m}\mathit{V}}$ and ${\displaystyle J_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}=0.403\mathit{m}\mathit{V}}$, respectively. Inhibitory synaptic efficacy was ${\displaystyle J_I=-1.50\mathit{m}\mathit{V}}$, and the efficacy of excitatory synapses onto inhibitory neurons was ${\displaystyle J_{IE}=0.560\mathit{m}\mathit{V}}$. Finally, ‘foreground’ neurons, ‘background neurons,’ and ‘inhibitory neurons’ each received independent Poisson spike trains of ${\displaystyle 2400\mathit{H}\mathit{z}}$, ${\displaystyle 2280\mathit{H}\mathit{z}}$ and ${\displaystyle 2400\mathit{H}\mathit{z}}$, respectively. Other settings were as in Mattia et al., 2013. As a result of these settings, ‘foreground’ activity transitioned spontaneously between an ‘off’ state of approximately ${\displaystyle 4\mathit{H}\mathit{z}}$ and an ‘on’ state of approximately ${\displaystyle 40\mathit{H}\mathit{z}}$.

The ‘biased competition’ circuit proposed here expresses a categorical decision by either raising ${\displaystyle r}$ towards unity (and lowering ${\displaystyle r^{\mathrm{\prime }}}$ towards zero) or vice versa. Here, we describe its stochastic steady-state response to constant visual inputs ${\displaystyle I=f(c)}$ and ${\displaystyle I^{\mathrm{\prime }}=f(c^{\mathrm{\prime }})}$ and for arbitrary initial conditions of ${\displaystyle e}$, ${\displaystyle e^{\mathrm{\prime }}}$, ${\displaystyle r}$ and ${\displaystyle r^{\mathrm{\prime }}}$ (Appendix 1—figure 3). Note that, for purposes of this analysis, evidence activity ${\displaystyle e}$, ${\displaystyle e^{\mathrm{\prime }}}$ was not subject to feedback suppression.

The choice is random when the input is ambiguous, ${\displaystyle I\simeq I^{\prime }}$, but quickly becomes deterministic with growing input bias ${\displaystyle |I-I^{\mathrm{\prime }}|\S gt;0}$. Importantly, the choice is consistently determined by visual input for all initial conditions. The 75% performance level is reached for biases ${\displaystyle |I-I^{\mathrm{\prime }}|\approx 0.04\mathrm{t}\mathrm{o}0.06}$.

Mutual inhibition ${\displaystyle w_{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}}$ controls the width of the ambiguous region around ${\displaystyle I=I^{\mathrm{\prime }}}$, and self-excitation ${\displaystyle w_{\mathit{c}\mathit{o}\mathit{o}\mathit{p}}}$ ensures a categorical decision even for small ${\displaystyle I,I^{\mathrm{\prime }}\simeq 0}$. The balance between feedforward excitation ${\displaystyle w_{\mathit{e}\mathit{x}\mathit{c}}}$ and inhibition ${\displaystyle w_{\mathit{i}\mathit{n}\mathit{h}}}$ eliminates decision failures for all but the largest values of ${\displaystyle I,I^{\mathrm{\prime }}\S gt;0.7}$ and reduces the degree to which sensitivity to differential input ${\displaystyle |I-I^{\prime }|}$ varies with total input ${\displaystyle I+I^{\prime }}$.

For particularly high values of input ${\displaystyle I,I^{\mathrm{\prime }}\S gt;0.7}$, no categorical decision is reached and activities of both ${\displaystyle r}$ and ${\displaystyle r^{\mathrm{\prime }}}$ grow above 0.5. In the full model, such inconclusive outcomes are eliminated by feedback suppression.

In the deterministic limit of ${\displaystyle N\to \mathrm{\infty }}$, fractional pool activity ${\displaystyle x}$ equals its expectation $⟨x⟩$ and the relaxation dynamics of Equation 2 becomes

with characteristic time ${\displaystyle {\tau }_x=\frac{1}{{\nu }^{+}+{\nu }_{-}}=\mathrm{{\rm Y}}(\mathrm{\Delta }u)}$ and asymptotic values ${\displaystyle x_{\mathrm{\infty }}=\frac{{\nu }^{+}}{{\nu }^{+}+{\nu }^{-}}=\mathrm{\Phi }(\mathrm{\Delta }u)}$, where ${\displaystyle \mathrm{\Delta }u}$ is the potential difference. Input dependencies of characteristic time and of asymptotic value follow from Equation 1:

Evidence pools

The relaxation dynamics of evidence pools is given by Equation 2 and Equation 3′. As shown in the next section, reversals occur when evidence difference ${\displaystyle |e-e^{\mathrm{\prime }}|}$ reaches a reversal threshold ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$. For example, a dominance period of evidence ${\displaystyle e^{\mathrm{\prime }}}$ begins with ${\displaystyle {e^{\mathrm{\prime }}}_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}=e_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}+{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ and ends when the concurrent habituation of ${\displaystyle e^{\mathrm{\prime }}}$ and recovery of ${\displaystyle e}$ have inverted the situation to ${\displaystyle e_{\mathit{e}\mathit{n}\mathit{d}}={e^{\mathrm{\prime }}}_{\mathit{e}\mathit{n}\mathit{d}}+{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$ (Appendix 1—figures 4). Once the deterministic limit has settled into a limit cycle, all dominance periods start from, and end at, the same evidence levels.

Appendix 1—figure 4

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If pool ${\displaystyle R^{\mathrm{\prime }}}$ has just become dominant, so that ${\displaystyle r^{\mathrm{\prime }}\simeq 1}$ and ${\displaystyle r\simeq 0}$, the state-dependent potential differences are

$\begin{array}{l}{\displaystyle u_e\simeq w_{\mathit{v}\mathit{i}\mathit{s}}f(c)}\\ {\displaystyle u_{e^{\mathrm{\prime }}}\simeq w_{\mathit{v}\mathit{i}\mathit{s}}f(c^{\mathrm{\prime }})-w_{\mathit{c}\mathit{o}\mathit{o}\mathit{p}}}\end{array}$

and the deterministic development is

${\displaystyle {\tau }_e\frac{de}{dt}=-e+e_{\mathrm{\infty }},{\tau }_{e^{\mathrm{\prime }}}\frac{d{e}^{\mathrm{\prime }}}{dt}=-e^{\mathrm{\prime }}+{e^{\mathrm{\prime }}}_{\mathrm{\infty }}}$

with asymptotic values

${\displaystyle e_{\mathrm{\infty }}=\mathrm{\Phi }(u_e+u_e^0),{e^{\mathrm{\prime }}}_{\mathrm{\infty }}=\mathrm{\Phi }(u_{e^{\mathrm{\prime }}}+u_e^0)}$

and characteristic times

${\displaystyle {\tau }_e=\mathrm{{\rm Y}}\left(u_e+u_e^0\right),{\tau }_{e^{\prime }}=\mathrm{{\rm Y}}\left(u_{e^{\prime }}+u_e^0\right),}$

Appendix 1—figure 5

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The starting points of the development, ${\displaystyle e_{\mathit{r}\mathit{e}\mathit{v}}}$ and ${\displaystyle {e^{\mathrm{\prime }}}_{\mathit{r}\mathit{e}\mathit{v}}}$ (dashed lines in Appendix 1—figure 4a,b), depend mostly on total input ${\displaystyle c+c^{\prime }}$ and only little on input difference ${\displaystyle c-c^{\prime }}$. Accordingly, for a given level of total input ${\displaystyle c+c^{\mathrm{\prime }}}$, the situation is governed by the distance between asymptotic evidence levels ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}={e^{\prime }}_{\mathrm{\infty }}-e_{\mathrm{\infty }}}$ and by characteristic times ${\displaystyle {\tau }_e}$, ${\displaystyle {\tau }_{e^{\mathrm{\prime }}}}$.

The dependence on input bias ${\displaystyle c-c^{\prime }}$ of effective potential ${\displaystyle \mathrm{\Delta }{u}_e+u_e}$, characteristic time ${\displaystyle {\tau }_e}$, and asymptotic value ${\displaystyle e_{\mathrm{\infty }}}$ is illustrated in Appendix 1—figure 4c–e. The potential range of relaxation is ${\displaystyle e_{\mathit{r}\mathit{e}\mathit{v}}\to e_{\mathrm{\infty }}}$ and ${\displaystyle {e^{\mathrm{\prime }}}_{\mathit{r}\mathit{e}\mathit{v}}\to {e^{\prime }}_{\mathrm{\infty }}}$, where reversal levels ${\displaystyle e_{\mathit{r}\mathit{e}\mathit{v}}}$ and ${\displaystyle {e^{\mathrm{\prime }}}_{\mathit{r}\mathit{e}\mathit{v}}}$ can be obtained numerically.

Dominance durations depend more sensitively on the slower of the two concurrent processes as it sets the pace of the combined development. The initial rates ${\displaystyle {\rho }_e}$ and ${\displaystyle {\rho }_{e^{\mathrm{\prime }}}}$ after a reversal of the two opponent relaxations

${\displaystyle {\rho }_e=\frac{d|e|}{dt}=\frac{|e_{\mathit{r}\mathit{e}\mathit{v}}-e_{\mathrm{\infty }}|}{{\tau }_e},{\rho }_{e^{\prime }}=\frac{d|e^{\prime }|}{dt}=\frac{|e_{\mathit{r}\mathit{e}\mathit{v}}^{\prime }-e_{\mathrm{\infty }}^{\prime }|}{{\tau }_{e^{\prime }}}}$

provide a convenient proxy for relative rate. As shown in app. Figure 4f, when stronger-input evidence ${\displaystyle e}$ dominates, recovery of weaker-input evidence (red up arrow on blue background) is slower than habituation of stronger-input evidence (blue down arrow on blue background). Conversely, when weaker-input evidence ${\displaystyle e^{\mathrm{\prime }}}$ dominates, recovery of stronger-input evidence (blue up arrow on red background) is slower than habituation of weaker-input evidence (red down arrow on red background). In short, dominance durations always depend more sensitively on the recovery of the currently non-dominant evidence than on the habituation of the currently dominant evidence.

If the two evidence populations ${\displaystyle E}$, ${\displaystyle E^{\mathrm{\prime }}}$ have equal and opposite potential differences, ${\displaystyle \mathrm{\Delta }{u}_e=-\mathrm{\Delta }{u}_{e^{\mathrm{\prime }}}}$, then they also have equal and opposite activation and inactivation rates (Equation 1)

${\displaystyle {\nu }_e^{+}={\nu }_{e^{\mathrm{\prime }}}^{-}={\nu }^{+},{\nu }_e^{-}={\nu }_{e^{\mathrm{\prime }}}^{+}={\nu }^{-}}$

and identical characteristic times ${\displaystyle {\tau }_e}$ (recovery of ${\displaystyle E}$) and ${\displaystyle {\tau }_{e^{\mathrm{\prime }}}}$ (habituation of ${\displaystyle E^{\mathrm{\prime }}}$). In this special case, the two processes may be combined and the development of evidence difference ${\displaystyle \mathrm{\Delta }e=e-e^{\prime }}$ is

${\displaystyle {\tau }_{\mathrm{\Delta }}\frac{d\mathrm{\Delta }e(t)}{dt}=-\mathrm{\Delta }e(t)+{\mathrm{\Delta }}_{\mathrm{\infty }}}$

${\displaystyle {\tau }_{\mathrm{\Delta }}=\frac{1}{{\nu }^{+}+{\nu }^{-}},{\mathrm{\Delta }}_{\mathrm{\infty }}=\frac{{\nu }^{+}-{\nu }^{-}}{{\nu }_{+}+{\nu }^{-}}}$

Starting from ${\displaystyle \mathrm{\Delta }e(0)=-{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$, we consider the first-passage-time of ${\displaystyle \mathrm{\Delta }e(t)}$ through ${\displaystyle +{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$. If a crossing is certain (i.e. when ${\displaystyle \frac{{\nu }^{+}}{{\nu }^{+}+{\nu }^{-}}>{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$), the first-passage-time ${\displaystyle T}$ writes

(2) ${\displaystyle T={\tau }_{\mathrm{\Delta }}\ln \left(\frac{{\mathrm{\Delta }}_{\mathrm{\infty }}+{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}{{\mathrm{\Delta }}_{\mathrm{\infty }}-{\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}\right)}$

A similar hyperbolic dependence obtains also in all other cases. When the distance between asymptotic levels ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ falls below the reversal threshold ${\displaystyle {\mathrm{\Delta }}_{\mathit{r}\mathit{e}\mathit{v}}}$, dominance durations become infinite and reversals cease.

The hyperbolic dependence of dominance durations, illustrated in Appendix 1—figure 4d, has an interesting implication. Consider the point of equidominance, at which both dominance durations are equal and of moderate duration. Increasing the difference between image contrasts (e.g., increasing ${\displaystyle c\to c+\mathrm{\Delta }c}$ and decreasing ${\displaystyle c^{\prime }\to c^{\prime }-\mathrm{\Delta }c}$) increases ${\displaystyle {\mathrm{\Delta }}_{\mathrm{\infty }}}$ during the dominance of ${\displaystyle e}$ and decreases it during the dominance of ${\displaystyle e^{\mathrm{\prime }}}$. Due to the hyperbolic dependence, longer dominance periods lengthen more (${\displaystyle T\to T+\mathrm{\Delta }T}$) than shorter dominance periods shorten (${\displaystyle T^{\prime }\to T^{\prime }-\mathrm{\Delta }{T}^{\prime }}$), consistent with the contemporary formulation of Levelt III [Brascamp et al., 2015].