Existing neural explanations of spontaneous percept switching under steady viewing of an ambiguous stimulus do not fit the fact that stimulus interruptions cause the *same* percept to reappear across many ON/OFF cycles. We present a simple neural model that explains the observed behavior and predicts several more complicated percept sequences, without invoking any “high-level” decision making or memory. Percept choice at stimulus onset, which differs fundamentally from standard percept switching, depends crucially on a hitherto neglected interaction between local “shunting” adaptation and a near-threshold neural baseline. Stimulus ON/OFF timing then controls the generation of repeating, alternating, or more complex choice sequences. Our model also explains “priming” versus “habituation” effects on percept choice, reinterprets recent neurophysiological data, and predicts the emergence of hysteresis at the level of percept sequences, with occasional noise-induced sequence “hopping.”

*outputs*as the primary dynamical variables. Given the thresholded sigmoidal shape of neural firing-rate functions, this makes such models blind to a subthreshold side effect of unequal adaptation that we will identify as sufficient for producing all hitherto unexplained percept-choice phenomena. Without this, one would be led to assume much more complicated mechanisms (see the 1 for details). In our approach, neural outputs occur only as sigmoidal transformations

*S*(

*H*

_{i}) of the primary dynamical variables

*H*

_{i}, called “local fields”, which correspond to the percept (

*i*)-related component of the membrane potentials of the neurons that encode the two rivaling percepts, indexed by

*i*∈ {1, 2}. Reducing the fast (timescale

*τ*≪ 1) neural dynamics to one of the simplest forms that captures all the phenomena of interest here, we employ the pair of differential equations

*H*

_{i}integrates its (preprocessed) visual input

*X*

_{i}with the “shunting”-type gain control (1 +

*A*

_{i})

*H*

_{i}that implements adaptation and the subtractive cross-inhibition

*γS*(

*H*

_{j}). The crucial role and meaning of the term

*βA*

_{i}(or its equivalent alternatives) will become clear in the course of our analysis. As adaptation dynamics, we use the simplest possibility, a standard “leaky integrator”

*A*

_{ i}dynamics timescale (taken as 1) is much slower than the

*H*

_{ i}dynamics timescale (

*τ*≪ 1), it does not affect the pattern of our results. Unless stated otherwise, we assume strictly ambiguous inputs

*X*

_{ i}=

*X,*because we focus on how the system resolves ambiguities based purely on its slowly evolving adaptation levels

*A*

_{ i}. Visit Supplementary Material for Simulink code of a neural network directly matching this form of the model.

*H*

_{ i}trajectories that start near the origin (reached during the stimulus interruption) suddenly diverge when approaching the saddle point (aptly named for its combination of stable and unstable directions) that lies between the two coexisting attractors that encode the two potential percepts. Hence, ambiguity resolution is best probed by responses to the onset of ambiguous stimuli. By contrast, a percept switch ( Figure 1b) occurs after prolonged stimulus presentation, when neural adaptation of the current percept representation (here,

*A*

_{2}≫

*A*

_{1}) has slowly destabilized its own attractor. Shortly before the attractor disappears by fusion with the saddle point, neural noise will make the system escape from the remnant of the attractor, followed by motion toward the opposite percept attractor. As expected, moderate stimulus perturbations (with no correlation or with anticorrelation between the two

*X*

_{ i}values) modulate the timing of each switch event (Kim, Grabowecky, & Suzuki, 2006; Lankheet, 2006), and this can be a rich source of information about how adaptation and noise interact in triggering a percept-switch event. However, it is important to realize that perturbing the percept-switching process probes how bistability is

*lost,*not how the neural network can choose systematically between percepts with concurrently stable representations.

*X*

_{1}=

*X*

_{2}). With symmetric adaptation (

*A*

_{1}=

*A*

_{2}, as in Figure 1a), the fate depends only on whether the (

*H*

_{1},

*H*

_{2}) state at onset of the stimulus is below or above the diagonal

*H*

_{1}=

*H*

_{2}. Independently of

*β,*the initial (

*H*

_{1},

*H*

_{2}) is ideally exactly on the diagonal and close to (0, 0), but it is instructive to consider small perturbations of this, which then determine the fate of their trajectories as shown in Figure 1a. One also notices the existence of one special (unstable) trajectory that ends at the saddle point; this special trajectory is key to understanding the choice process in any situation because it forms part of the “separatrix”, the locus that separates all trajectories leading to Percept 1 from those leading to Percept 2. An immediate consequence is that systems without such a separatrix (e.g., Freeman, 2005) do not cover the well-defined percept-choice phenomena we address here.

*A*

_{2}>

*A*

_{1}and vice versa. Such adaptation asymmetry occurs, for example, when the system was predominantly in one of its attractors before the most recent stimulus interruption. All models that (like our case

*β*= 0) cannot shift the initial

*H*

_{ i}point relative to the separatrix thus predict that the next percept choice is always

*opposite*to the previous one, contrary to the observations (Orbach et al., 1963; say, for off-times of more than 1 s). Reduction of adaptation asymmetry has been suggested (Blake, Sobel, & Gilroy, 2003; Chen & He, 2004; Orbach et al., 1963) as a possible cause of percept repetition, but our analysis makes clear that this does not suffice—even small

*A*

_{i}asymmetry in the usual (

*β*= 0) types of models still predicts choice alternation, and increasing the off-time until the

*A*

_{i}asymmetry decays to the noise level only produces random percept choices. The initial

*H*

_{i}point must be shifted off the diagonal in the same direction as the separatrix is shifted by adaptation asymmetry to allow the previous percept to be chosen—in fact, the initial

*H*

_{i}shift must

*over*compensate for the separatrix shift. Reconsidering the

*H*

_{i}dynamics (Equation 1), one concludes that the model must have

*β*> 0.

*βA*

_{ i}must be in the model simply because it is the lowest order symmetry-allowed term that, even when small, can drive a bifurcation, that is, an addition or deletion of qualitatively different types of system behavior. This standard demand on models is prior to any interpretation of its equations or solutions. In our case, all other formally allowed small changes to the model are found to be either equivalent to

*β*or incapable of driving a bifurcation; thus, leaving all these irrelevant but often “realistic” terms out of the model is safe, and it reduces the analysis to the simplest possible level. Psychophysically, capturing the hitherto unexplained “repetition” choice sequences within the model requires

*β*> 0, as loosely argued above. The precise dynamical explanation is given in the rest of this article, together with several new predictions. The switching process is not substantially affected by the small

*β*values that explain observed choice sequences.

*β*> 0 term in the model, which often explain other surprising findings as well. For example, literal translation of both

*A*

_{ i}and

*H*

_{ i}equations into separate neuron pools yields

*A*

_{ i}signaling neurons coupled (with strength

*β*) to the

*H*

_{ i}neurons whose gain they down-regulate via shunting inhibition. Such

*A*

_{ i}signals explain the hitherto perplexing observation (Leopold & Logothetis, 1996) that many of the neurons that carry percept

*i*signals when driven by unambiguous stimuli, carry a signal with large phase shift relative to percept

*i*dominance when driven by steady ambiguous stimuli (i.e., under spontaneous percept switching, as in Figure 1b): Switching from, say, Percept 2 to Percept 1 corresponds to

*A*

_{2}reaching its peak, and this implies that

*H*

_{2}peaked nearly half a cycle earlier, shortly after Percept 2 started.

*H*

_{ i}by shifted

*h*

_{ i}≡

*H*

_{ i}−

*β*puts our model into the exactly equivalent form

*β*can also be interpreted simply as a fixed signal “baseline”. It now performs the same function purely intraneurally (note that this role is now less obvious from the equations), but the existence of phase-shifted neural signals (Leopold & Logothetis, 1996) could still be attributed to separate

*A*

_{i}-carrying neurons. Visit Supplementary Material for Simulink code of a neural network directly matching this form of the model. More generally, any mixture of the two equivalent model forms and their neural interpretations is equivalent, and this extends to a much wider but precisely delimited range of model variants (see the 1).

*β*interacts with the shunting-type adaptation term −(1 +

*A*

_{ i})

*H*

_{ i}to yield offsets ≈

*βA*

_{ i}/(1 +

*A*

_{ i}) in the

*H*

_{ i}value at each stimulus onset, relative to their nonadapted values (for details, see the 1). As explained below, it is the size of these

*A*

_{ i}-dependent offsets that determines each choice of percept. The fact that their values depend on percept history (via the

*A*

_{ i}) will then be shown to explain all known percept-repetition sequences and to predict a host of others. Before we analyze these two temporal scales of choice dynamics, it is worth noting that the crucial

*H*

_{ i}shift signals ≈

*βA*

_{ i}/(1 +

*A*

_{ i}) arise from the shunting (or “gain control”) adaptation we use (but if this is present, the effect is not destroyed by common-mode subtractive adaptation, see the 1). Most rivalry models (e.g., Hock, Schöner, & Giese, 2003; Laing & Chow, 2002; Matsuoka, 1984; Wilson, 2003) use purely subtractive adaptation. This leads to the opposite of our predictions because it replaces our product term (1 +

*A*

_{i})

*H*

_{i}by a linear term of the same form as our

*βA*

_{i}but with a strongly

*negative β*value.

_{ i}and

*β*with the slowly varying adaptation state (

*A*

_{1},

*A*

_{2}) determine whether the less adapted or the more adapted attractor is chosen at each stimulus onset? Having identified that the interaction of these factors generates small offsets ≈

*βA*

_{ i}/(1 +

*A*

_{ i}) in the

*H*

_{ i}at stimulus onset, one could guess that any (above-noise) difference between these offsets simply gives one of the two

*H*

_{ i}a decisive “head start” in the “race” to dominant activation. Analyzing the actual dynamics of this race leads to less obvious but crucial insights and predictions. The first result, illustrated in Figure 2a, is that

*β*must exceed a finite (but moderate) value before the

*H*

_{ i}offset that it creates is sufficient to overcompensate for the shift of the separatrix caused by moderate

*A*

_{ i}asymmetry. In psychophysical terms, this “primes” the system to choose the more adapted percept (Pearson & Clifford, 2005; Figure A2 explains how this does not prevent normal aftereffects [van de Grind et al., 2004], that is, choice of the lesser adapted percept [Pearson & Clifford, 2005], when unambiguous stimuli are used to induce adaptation).

*β,*the choice of the more adapted percept persists in a large but finite regime of adaptation states

*A*

_{ i}, the two lower left sectors. However, for very large

*A*

_{ i}(upper right pair of sectors), the difference in

*H*

_{ i}offsets

*βA*

_{ i}/(1 +

*A*

_{ i}) can no longer surpass the separatrix shift; the system then chooses the less adapted percept.

*A*

_{ i}directly, especially because any induction of unbalanced adaptation states in the preceding stages destroys the ambiguity of the inputs

*X*

_{ i}to the percept-choice network, thereby obliterating the process of interest. The simplest suitable stimulus sequence is a periodic ON/OFF cycle of the same ambiguous stimulus, which allows indirect control over the

*A*

_{ i}state at each stimulus onset by means of the

*T*

_{ON}and

*T*

_{OFF}durations. This is precisely the type of stimulus by which interrupt-induced percept repetition and alternation were discovered (Orbach et al., 1963, 1966). The ensuing choice sequence can then be understood fully in terms of the

*A*

_{i}dynamics during the on and off phases of the cycle—the fast (

*H*

_{i}) dynamics of the percept-choice map shown in Figure 2b acts only at each stimulus onset. This also reduces the choice-relevant

*A*

_{i}dynamics to a discrete-time process,

*A*

_{i}(

*t*) →

*A*

_{i}(

*t*+

*T*

_{ON}+

*T*

_{OFF}). This (effectively discontinuous) Poincaré map (Guckenheimer & Holmes, 1983) thus contains all information about the existence and stability of all possible choice sequences.

*A*

_{ i}trajectories are plotted for the two most prominent types of percept-choice sequences that can be produced: For relatively long

*T*

_{OFF}(Panel a), the

*A*

_{ i}trajectories quickly settle into an attractor that corresponds to repeatedly choosing the same percept, either Percept 1 or Percept 2 depending on the initial condition. For much shorter

*T*

_{OFF}(Panel b), the system settles into a sequence of choosing Percept 1 or Percept 2 in alternation. Here, the initial condition merely determines the eventual phase of the percept sequence. For intermediate

*T*

_{ON},

*T*

_{OFF}values, the “repeat” and “alternate” attractors coexist, leading to dependence on the initial conditions, to hysteresis, and to noise-induced hopping between the three possible choice sequences (see Figure A4a and its discussion). Panel c shows the (

*T*

_{ON},

*T*

_{OFF}) regimes in which various choice-sequence types occur, when using low-adaptation initial conditions. In addition to the repeat and alternation sequences, one may note that, for large

*T*

_{ON}, the model also produces sequences with spontaneous percept switches inserted between, and nonlinearly interacting with, the choice events. Given the known highly stochastic timing of percept-switch events (as expected for noisy

*H*

_{ i}and/or

*A*

_{ i}), these regimes are not optimally probed by regular ON/OFF cycles; hence, we leave their detailed study to another occasion.

*β,*which are crucial to percept choice but hardly affect percept switching, may well differ between the neural stages that encode different visual modalities. So far, repeat and alternation sequences have been reported with certainty only for Necker cubes (Kornmeier & Bach, 2005; Orbach et al., 1966). More diverse experiments and systematic studies of the predicted stimulus timing dependence are clearly called for.

*T*

_{OFF}intervals tends to replace percept-choice repetition by choice alternation. The crossover

*T*

_{OFF}timescale we found is roughly one order of magnitude smaller than the (independently measured) timescale of spontaneous percept switches. The neural adaptation timescale is unknown, but it is probably of the same order of the switching timescale. Thus, our finding of a relatively short

*T*

_{OFF}timescale for the coexistence and/or transition zone between percept-repetition and alternation sequences suggests that

*β*is larger than the value used in plotting Figure 3c, although the position and shape of the crossover region depends on all other parameters, including the unknown effective shape of the nonlinearities. The crossover (

*T*

_{OFF},

*T*

_{ON}) regimes were roughly overlapping among our four subjects, but the predicted lack of reproducibility due to sequence hysteresis and noise-induced sequence hopping makes precise comparisons difficult. We avoided the long

*T*

_{ON}regime where spontaneous switching complicates the dynamics but sampled

*T*

_{OFF}down to very small values where temporal blurring of short interruptions begins to make our modeled processes break down (see the 1 for details).

*T*

_{OFF}used), whereas unambiguous pulses reduce percept repetition (the classical aftereffect) by unbalancing the inputs

*X*

_{i}to our stage through unbalanced adaptation of preceding stages (see Figure A2 for details). Furthermore, the disruption of percept repetition by retinally displacing the stimuli during each interruption (Chen & He, 2004) fits our result that the adaptation states of competing local feature representations are required to enable the dynamics of percept repetition.

*T*

_{OFF},

*T*

_{ON}) as a practical means of influencing the adaptation state of the system at each stimulus onset so that it favors percept repetition or alternation sequences (although this bias is also predicted to show hysteresis and noise effects). An early psychophysical study (Orbach et al., 1966) found some signs that percept repetition breaks down at short

*T*

_{OFF}, as we calculate, but studies across several decades since then have focused on the repetition regime (Leopold et al., 2002; Maier et al., 2003; Pearson & Clifford, 2004, 2005)—in the few cases where some percept alternation was reported, it appears to be due at least partly to the insertion of unambiguous stimuli or other perturbations (Chen & He, 2004; Kanai, Moradi, Shimojo, & Verstraten, 2005; Maloney et al., 2005; Pearson & Clifford, 2004, 2005; see also the 1). Very recently (Kornmeier & Bach, 2005), however, alternating choice sequences evoked by Necker-cube stimuli with short

*T*

_{OFF}were used to allow stimulus-locked EEG measurements intended to probe the

*switch*process. Our analysis indicates that this experiment, in fact, probes the very different process of percept choice, but this only increases the relevance of this experiment to understanding how the visual system handles actually bistable (rather than temporarily monostable) percept representations. The finding (Kornmeier & Bach, 2005) of occipital choice-related signals at around 100 ms after stimulus onset, well before any response in “higher” stages, strongly supports our theory and contradicts the attribution of the choice process to high-level factors (Leopold et al., 2002; Maier et al., 2003; Maloney et al., 2005).

*β*) that control the stimulus-timing regimes where various choice sequences occur, without deciding each individual percept choice.

*H*

_{ i}or

*h*

_{ i}), rather than neural outputs

*S*(

*H*

_{ i}), as the primary dynamic variables of the model is not only more realistic but also crucial to discovering that all known percept-choice behavior already emerges from extremely basic, one-stage rivalry models. This result contradicts the long-standing and widely held intuition that the production of percept-repetition choice sequences requires some form of intervention by high-level processes into the low-level neural competition and adaptation dynamics that we modeled. In terms of explicit mechanisms, it disproves the logical necessity of additional mechanisms or stages or cross-level feedback. The pair of sub-/near-threshold signals parametrized by

*β*(in any of their equivalent forms) play the central role: They enable this simple type of models to describe all the known relevant psychophysics, plus many detailed predictions. In models that take the neural outputs

*S*(

*H*

_{ i}) as the primary dynamical variables (e.g., as in Laing & Chow, 2002; Wilson, 2003), subthreshold signals are typically unable to determine the percept choice because, at stimulus onset, both

*S*(

*H*

_{i}) are at or near zero. As illustrated in Figure A1, the observation of adaptation-dependent percept choice then implies that multiple

*S*(

*H*

_{i}) trajectories emerge from the same (nonsingular) starting point, thus proving this type of model to be insufficient.

*S*(

*H*

_{ i})-based models, one would thus be forced to attribute this (and other) percept-choice type to other dynamical variables and interactions that, in effect, take control over the neural output dynamics in a manner that must depend on previous output states. This approach has been explored very recently (H. R. Wilson, personal communication), and it appears to require doubling the number of equations, with many more parameters and a specific choice of third-order nonlinearity, so as to implement a two-state “memory” at the synapse level in each of the two competing neural populations. This pair of memory subsystems then controls the

*S*(

*H*

_{ i}) dynamics at stimulus onset, producing a roughly similar behavior as emerges from our much simpler and generic

*H*

_{ i}-based model. Of course, the brain may actually use even larger complexity, but we favor the simpler explanation until it is refuted by experiment.

*β*. We now sketch the role of the fast-timescale null clines and fixed points in more detail.

*X*

_{1}=

*X*

_{2}=

*X,*the fixed points of the fast-timescale dynamics are determined by the solutions

*H*

_{ i}=

*H*

_{ i}* of the two “null cline” equations

*h*

_{ i}≡

*H*

_{ i}−

*β*as dynamical variables

*H*

_{1},

*H*

_{2}) at the end of the stimulus interruption. For all but the shortest

*T*

_{OFF}(i.e., at least a few times the

*τ*timescale of all stages up to and including the stage[s] modeled here), this (

*H*

_{1},

*H*

_{2}) has effectively decayed to the fixed point (

*H*

_{1}*,

*H*

_{2}*) for

*X*= 0. In this regime, the sigmoidal terms may be neglected; thus, we find

*H*

_{ i}* ≈

*βA*

_{ i}/(1 +

*A*

_{ i}) and

*h*

_{ i}* ≈ −

*β*/(1 +

*A*

_{ i}), respectively. In either form, one notices that (

*H*

_{1},

*H*

_{2}) acquires

*A*

_{ i}-dependent shift components ≈

*βA*

_{ i}/(1 +

*A*

_{ i}) that can compensate (or not, see Figure 2) the oppositely

*A*

_{ i}-dependent shift of the separatrix that determines the fate of the trajectory after stimulus onset, that is, the outcome of a percept choice.

*β*> 0 term (in any of its forms) with a shunting-type adaptation term such as (1 +

*A*

_{ i})

*H*

_{ i}, it also became clear that a purely subtractive adaptation mechanism would not allow percept repetition. However, this does not forbid the existence of any subtractive components of adaptation in addition to the required shunting-type component. As expected from the fact that our basic model was the product of keeping only terms with the relevant effect and

*i,*

*j*symmetry, we can freely add back even large

*common-mode*subtractive terms (e.g., −(

*A*

_{ i}+

*A*

_{ j})) without losing any of our original types of model behavior. For example, we find that adding twice as much common-mode subtractive adaptation as our original

*β*term, that is, replacing the fast dynamics by

*α*= 6,

*γ*= 3.5,

*X*= 1.3). Note that with this strong subtractive adaptation component, the effect of any single adaptation change, say, in

*A*

_{1}, on the local field

*H*

_{1}is now suppressive instead of stimulatory, that is,

*opposite*to what it was in the original model version. This suppressive effect makes the model much more realistic in terms of recordings from individual neurons. However, what is important is that it is of no relevance to how percept choice is produced within the antagonistic neural network and that neurophysiological testing of our choice mechanism is likely to require extracting the small but crucial signals

*H*

_{1}* −

*H*

_{2}* from the probably larger but irrelevant common-mode signals.

*S*(

*h*) = max(

*h,*0). Our reason for not using this is that its special feature (piecewise linearity) is not stable under our averaging (Cross & Hohenberg, 1993) over the many neurons that make up a single percept representation. This implies averaging over the neural noise (say,

*ξ,*with some distribution

*p*(

*ξ*)), which renormalizes the cell-level firing-rate function, say

*s*(

*H*), to the smooth and suitably generic population-level sigmoid

*S*(

*H*) that appears in the model dynamics:

*S*() are largely exchangeable with changes in coupling parameters (e.g.,

*γ,*

*α*); what matters is the effects on the gain and/or offset of signals in the fast mutual inhibition feedback loop and/or the slow adaptation loop.

_{ i}-controlled priming

*X*

_{ i}asymmetry will locally “bridge” together a pair of same-colored sectors (red for

*X*

_{1}>

*X*

_{2}) and generally bias the percept choice (see Figure A2). Note that

*X*

_{ i}asymmetry occurs not only when the current stimulus is unbalanced but also when there is “gain” imbalance in the (not bistability-generating) stages preceding the stage we model explicitly. Familiar sources of such gain imbalance are adaptation with unambiguous stimuli or “attention”. This simple mechanism explains how percept-choice repetition can be overridden by a normal aftereffect and how percept repetition is reduced by unambiguous stimuli but enhanced by ambiguous stimuli (Pearson & Clifford, 2005). It also captures how attention can bias the choice process via gain imbalance in the preprocessing and/or the choice-producing stage.

*T*

_{OFF}regime around where these two types of sequence cross over into another, but it falls short of capturing the known occurrence of reasonably reliable percept repetition for much longer

*T*

_{OFF}values. As soon as

*T*

_{OFF}exceeds a few times the timescale of the linear adaptation decay we have assumed so far, exponential decay of both

*A*

_{ i}will reduce the offsets

*βA*

_{ i}/(1 +

*A*

_{ i}) in the

*H*

_{ i}or

*h*

_{ i}fixed points that determine the next percept choice. The choice mechanism (which always predicts repetition in this regime) becomes unreliable when the difference between the two

*βA*

_{ i}/(1 +

*A*

_{ i}) terms decays below the effective noise level of the local field ( Figure A3)—the actual noise level is unknown, but this affects only logarithmically the maximal

*T*

_{OFF}at which reliable percept repetition still occurs, if the

*A*

_{ i}decay exponentially.

^{−2}to 10

^{2}s (Fairhall, Lewen, Bialek, & de Ruyter van Steveninck, 2001). Explicit representation of all the contributing mechanisms, most of which are actually unknown, is not required—the net effect, replacing the usual exponential decay of adaptation by a long-tailed (say, hyperbolic) decay, can be captured most simply by replacing the linear decay term in the adaptation dynamics by a quadratic term, that is, replacing Equation 2 by

*A*

_{i}. Its effect on the percept-choice dynamics is small in the small-

*T*

_{OFF}regime where the system behavior crosses over from repeat to alternation sequences. This is because large (here, ≈1) values of the relevant

*A*

_{i}occur in this regime (see Figure 2), making the effective decay rate (here,

*A*

_{i}) roughly as large as it was in our first linear approximation of the

*A*

_{i}dynamics (Equation 2). The long-tailed adaptation manifests itself at small

*A*

_{i}, thus increasing the longest

*T*

_{OFF}values where percept repetition occurs with large probability.

*X*

_{1}=

*X*

_{2}, the Repeat 1 and Repeat 2 sequence attractors are equally stable; hence, occasional noise-induced choice “errors” lead to stochastic hopping between these two coexistent sequence attractors. Here, we stress that coexistence (and hopping) occurs also between attractors that are not symmetry related, for example, between the repeat and alternate sequences (see Figure A4a), over a substantial range of timing parameters along regime boundaries such as in Figure 3c. In many experiments, these boundaries will therefore appear as fragmented, irreproducible zones (e.g., as in Figure A4b), and their position will depend on the particular sequence of parameters used (hysteresis). A full analysis of this suprasequence temporal structure is beyond the scope of this article, but we stress the richness of choice-sequence behaviors that emerges from our minimal neural model.

*T*

_{OFF},

*T*

_{ON}) plane (up to 44 points) in random sequence, and this sampling was repeated twice per subject. The stimuli (in separate sessions) were (a) a depth/rotation ambiguous sphere, rendered by 40 dots/(deg)

^{2}, randomly replaced at each stimulus onset, or (b) a binocularly rivalrous dichoptic pair of gratings (orientation, ±45°; spatial frequency, 1.75 cycles/deg, with a Gaussian envelope of

*σ*= 0.5°). We asked our four subjects to score percepts during the on-interval, as soon as it could be classified as either of the two competing percepts.

- We avoided sampling the long
*T*_{ON}regime where*spontaneous*percept switching interferes with the onset-locked choice process that we focus on here. The long*T*_{ON}regime yields interesting but more complicated dynamics, an investigation of which is beyond the scope of this study and, hence, necessitates further research. For the timing conditions used in this study, spontaneous percept switches begin to occur (but remain below 10%), only at the largest on-times (2 s) we tested, and even then, this hardly affected the interruption-locked alternation, as predicted by the top parts of our phase diagram ( Figure 3c). - For very small
*T*_{OFF}(here, roughly <1/10 s), temporal smearing by the preprocessing stages prevents*X*_{ i}from decaying sufficiently close to zero to enable our choice process to occur. This causes a crossover to the slow, spontaneous percept-switching characteristic of uninterrupted (*T*_{OFF}= 0) stimuli. Partial loss of interruption locking at the smallest*T*_{OFF}registers in our plots as a formal increase in repetition probabilities. This regime is irrelevant to our model, but it serves to differentiate our*loss*of choice alternation for short*T*_{OFF}from the recently found (Kanai et al., 2005)*increase*in alternation when a short, strong flash is added to a continuous bistable stimulus: Our*T*_{OFF}is then about as long as their flashes; thus, the opposite nature of the two effects excludes lumping them as due to some nonspecific effect of stimulus transients. However, if we assume (in line with the proposal of Kanai et al., 2005) that their strong flashes cause temporary (≈1/4 s) shifts in attention, that is, dips in the neural gain preceding (or within) the stage we model, then this converts the short flashes into effective*X*_{i}interruptions with a*T*_{OFF}for which our model can predict percept alternation. As noted before (Kanai et al., 2005), such assumed gain dips also fit the fact that their flashes can induce temporary disappearance of nonambiguous percepts.

*X*= 1,

*α*= 5,

*γ*= 10/3,

*τ*= 1/50, and sigmoid function

*S*(

*z*> 0) =

*z*

^{2}/(1 +

*z*

^{2});

*S*(

*z*≤ 0) = 0 throughout and chose

*β*= 4/(3

*α*) except in plotting the

*β*dependence of the choice dynamics in Figure 2a. All plots were composed by combining the

*H*

_{ i}null-cline equations with

*H*

_{ i}(

*t*) and

*A*

_{ i}(

*t*) trajectories computed by adaptive numerical integration (Mathematica “Dsolve”). In our classification of percept sequences ( Figure 3c), the numerical integration for each of 128

^{2}points in (

*T*

_{OFF},

*T*

_{ON}) space was carried out over the first seven cycles of the stimulus sequence, using the last two on-intervals at each point to determine the sequence types shown. All sequence types shown are stable, and the “run-in” length used suffices for reducing transient-induced errors on the boundary positions to below the plot resolution.