Analytically-Tractable Visual Rivalry Model

• The paper introduces a quantitative model of visual rivalry that reproduces dominance durations through state memory during stimulus-off intervals.
• It employs dynamical systems with event-driven resets to link neural adaptation and inhibitory competition, yielding measurable hysteresis effects.
• The approach provides closed-form predictions matching gamma/lognormal statistics, offering actionable insights into neural population dynamics.

Analytically-tractable models of visual rivalry provide a quantitative framework for understanding the mechanisms underlying perceptual alternations between competing stimuli. Such models seek not only to reproduce observed behavioral distributions—such as the probability distribution of dominance durations (PDDD)—but also to yield closed-form or algorithmically explicit predictions linking neural population dynamics, adaptation, and experimental variables. Recent research has revealed constraints on acceptable model architectures through novel experimental paradigms and advances in dynamical systems analysis, putting emphasis on the need for memory-retaining, resettable perceptual states, and quantitatively accounting for hysteresis and adaptation-induced timescale separations.

1. Experimental Foundations: Recurrence and Resetting in Dominance Duration Distributions

A pivotal empirical observation derives from direct measurement of dominance durations under both continuous and periodic (on–off) stimulation regimes in binocular rivalry tasks (Manousakis, 2010). When a bistable stimulus is presented continuously, dominance durations are well-characterized by a directly measured PDDD, denoted as $C(t)$. When the same stimulus is periodically interrupted (on intervals $T_\mathrm{on}$, off intervals $T_\mathrm{off}$), the resulting PDDD, $D(t)$, is remarkably well-approximated by temporally "slicing" $C(t)$ and inserting blank gaps: $$D(t) = \begin{cases} C(t - nT_\mathrm{off}) & nT < t < nT + T_\mathrm{on} \ 0 & \text{otherwise} \end{cases}$$ where $n = \lfloor t/T \rfloor$ and $T = T_\mathrm{on} + T_\mathrm{off}$.

This finding strongly implicates that the perceptual state remains essentially unchanged during off intervals—a phenomenon sometimes described as "perceptual time stands still." Supporting evidence comes from adjustments for reaction delay ($\delta t \approx 0.3$ sec, Lorentzian width $\epsilon \approx 0.03$ sec): with these corrections, the piecewise reconstruction yields a nearly perfect match to measured intermittent PDDD. Thus, analytically tractable models must explicitly preserve internal state in the absence of external input and support instantaneous resets upon re-presentation.

2. Dynamical Systems Approaches: Adaptation, Competition, and Timescale Separation

Mathematically explicit rivalry models typically describe two mutually inhibitory neural populations, each driven by its respective monocular input and subject to slow adaptation. Canonical forms include:

• Wilson's model (with separate neural populations for perception and inhibition):

$$\begin{aligned} \tau\, dE_1/dt &= -E_1 + \left[100\cdot(V_1 - gI_2)_+^2\right]/\left\{(10 + H_1)^2 + (V_1 - gI_2)_+^2\right\} \ \tau_H\, dH_1/dt &= -H_1 + hE_1 \ \tau_I\, dI_1/dt &= -I_1 + E_1 \end{aligned}$$

with analogous equations for the other eye (Murthy, 2018).

• Minimal quantitative hysteresis model for tCFS (Whyte et al., 20 Oct 2025):

$$\begin{aligned} \tau_E\, \dot{E}_m &= -E_m + f(M + \epsilon E_m - a E_s - g_m H_m) \ \tau_H\, \dot{H}_m &= -H_m + E_m \ \tau_E\, \dot{E}_s &= -E_s + f(S + \epsilon E_s - a E_m - g_s H_s) \ \tau_H\, \dot{H}_s &= -H_s + E_s \end{aligned}$$

Here, $f(x) = \max(x, 0)$, $\tau_H \gg \tau_E$ ensures a separation of timescales between neural activity and adaptation.

For adaptation-driven switching, spectral decomposition or phase plane reduction techniques may be used to produce closed-form solutions for dominance and suppression durations. In the case of tracking continuous flash suppression (tCFS), threshold crossings for awareness and suppression translate to transcendental equations in time, yielding dominance/suppression durations and hysteresis depth in terms of the Lambert $W_0$ function: $$T_{\mathrm{dominated}},\, T_{\mathrm{suppressed}} = \text{explicit expressions in } W_0(\ldots)$$

3. Memory and Reset Mechanisms: Implications for Analytical Tractability

Experimental findings necessitate models that maintain robust long-term memory for perceptual state during off intervals. Adaptation and inhibition-driven models not explicitly designed to “freeze” or conserve slow variables during stimulus absence fail to replicate the slice-and-gap property of the PDDD observed with periodic interruption (Manousakis, 2010). Analytically, this is realized either by introducing a time-varying term or an explicit switch in the differential equations that suppresses adaptation/excitation dynamics during blanks:

• $dA/dt = 0,\, dH/dt = 0$ for $t \in$ [stimulus off].

Consequently, models that naturally embody such state conservation—through piecewise-constant or event-driven updates—are analytically tractable, permitting explicit mapping from continuous to intermittent dominance distributions and providing mechanistic clarity for the observed “halt” in perceptual time.

4. Hysteresis and Parameter Dependence: Quantitative Law-Like Predictions

Recent advances demonstrate that alternation dynamics in rivalry are often hysteretic, especially under dynamically varying contrast paradigms such as tCFS (Whyte et al., 20 Oct 2025). Key outcomes include:

• The threshold for breakthrough (emergence into awareness) exceeds that for suppression, establishing a measurable hysteresis width.
• Explicit parameter dependences in the analytic model allow predictions such as:
  • Increased interocular inhibition parameter $a$ yields deeper hysteresis (wider separation between suppression and breakthrough thresholds).
  • Equilibrated adaptation across populations (e.g., in paradigms without mask/target asymmetry) reduces hysteresis depth.
• The equilibrium distribution of dominance and suppression durations is predicted—and validated experimentally—to be approximately equal; fits to gamma/lognormal distributions reveal indistinguishable statistics for both phases.

These outcomes are formalized in closed-form solutions, mapping model parameters to observable psychophysical metrics, and grounded in empirically testable behavioral predictions.

5. Theoretical and Modeling Constraints: Challenging Adaptation-Inhibition-Only Models

The reproducibility of intermittent PDDD by slicing the continuous distribution (with time-independent, state-preserving intervals) challenges broad classes of adaptation-inhibition rivalry models—e.g., those by Wilson or Noest et al.—that predict ongoing evolution during stimulus removal. An analytically valid framework must reconcile the capacity for rapid perceptual resets with robust memory over multi-second intervals. Specifically, admissible models must:

Requirement	Source	Implication for Model Design
State memory during blanks	(Manousakis, 2010)	Adaptation must be “frozen” during off-time
Hysteresis under tCFS/ramp task	(Whyte et al., 20 Oct 2025)	Need for timescale separation, explicit ramp coupling
Law-like alternation durations	(Cohen et al., 2018, Whyte et al., 20 Oct 2025)	Closed-form/statistical matches to measured PDDD

Successful analytically tractable models thus typically employ event-driven, hybrid approaches or adjustment of ODEs by a stimulus-present/vacant gating variable.

6. Empirical and Computational Implications

Analytically-tractable rivalry models serve as benchmark systems for testing neural hypotheses and evaluating model adequacy against large-scale, high-fidelity behavioral datasets. Their value lies in providing:

• Direct, compact mappings between model parameters (adaptation, inhibition, noise strength) and measurable psychophysical quantities (mean/sd of dominance durations, hysteresis width, etc.).
• Quantitative, falsifiable behavioral predictions (e.g., equality of duration distributions, scaling of hysteresis with pharmacological or stimulus manipulations).
• Modeling blueprints for more advanced neural algorithms and for constraining model class in systems neuroscience.

These models also offer critical mechanistic insight into why classical adaptation/inhibition models may fall short under interrupted or dynamic viewing protocols and pin down the requisite design features for future, more comprehensive frameworks.

---

In sum, an analytically-tractable model of visual rivalry must incorporate robust mnemonic retention of perceptual state during stimulus-off periods, support explicit event or gating-driven resets of the alternation dynamics, and yield closed-form predictions for dominance/suppression statistics and parametric hysteresis. Such models both accommodate and quantitatively constrain the mechanisms underlying stochastic perceptual switching as revealed by behavioral and psychophysical experiments (Manousakis, 2010, Whyte et al., 20 Oct 2025).