Binocular rivalry is what you see when each eye is shown a different image: perception does not blend them — it alternates between one and the other every few seconds, entirely on its own. This is the same reciprocal-inhibition rate model (Wilson 2003; Laing & Chow 2002) as the 3D version, but rendered natively in 2D as a phase-plane portrait instead of a flattened 3D scene: the state (E₁,E₂) is a single point moving through the unit square, steered by an instantaneous vector field that itself tilts over time as adaptation builds.
τ dE_i/dt = -E_i + f(I - β·E_j - g·A_i + ξ)
τ_a dA_i/dt = -A_i + E_i
f(x) = 1 / (1 + e^(-4x)) (sigmoid firing-rate response)
The left panel plots E₁ on the horizontal axis and E₂ on the vertical axis. The pale arrows are the instantaneous flow field ∂(E₁,E₂)/∂t (noise off, A held at its current value) — literally the direction the state would move right now. The two dashed curves are the E₁- and E₂-nullclines (where dE₁/dt=0 or dE₂/dt=0 respectively); their crossings are the fixed points the dynamics orbit. Drag to pan the plane, scroll to zoom. The bright dot is the live (E₁,E₂) trajectory with a fading trail. The strip chart on the right scrolls E₁, E₂ (solid) and their adaptation variables A₁, A₂ (faint) through time, and the seesaw bar beneath it shows which percept currently dominates.
- Input drive I — raises both populations' base excitability (like stimulus contrast). Its effect on switching speed is not simply monotonic: over the low-to-mid part of the slider, more drive means faster escapes from adaptation and shorter dominance periods, but push I high enough and both populations saturate near their firing ceiling, adaptation has less relative pull, and dominance periods lengthen again — try sweeping the slider end to end while watching the strip chart.
- Mutual inhibition β — how hard the winning percept suppresses the other. Below a critical β, the field has one stable mixed fixed point (no switching); above it, the mixed point becomes unstable and two dominant fixed points appear, so the trajectory alternates between them.
- Adaptation g — how strongly fatigue undermines the dominant population; watch the flow field tilt away from the current fixed point as its A_i climbs, until the state is pushed across the boundary into the rival's basin.
- Noise σ — random fluctuations, visible as jitter on the trajectory, that can trigger an early escape even before adaptation forces one.
Real-world relevance: this exact class of model is used in visual neuroscience to explain not only binocular rivalry but also ambiguous-figure reversals (Necker cube, Rubin's vase) and motion-direction rivalry — cases where a stable external stimulus nonetheless produces a switching subjective percept, one of the clearest empirically tractable windows into the neural correlates of consciousness.