Tracking Continuous Flash Suppression (tCFS)

• Tracking continuous flash suppression (tCFS) is a psychophysical paradigm that uses dynamically ramped target contrast against a constant mask to study visual rivalry.
• It employs coupled nonlinear differential equations and adaptation mechanisms to model perceptual hysteresis and predict breakthrough and suppression thresholds.
• The model provides closed-form predictions validated by empirical studies, highlighting the impact of ramp rate and inhibition on perceptual state durations.

Tracking continuous flash suppression (tCFS) refers to a psychophysical paradigm that investigates the transitions between perceptual awareness and suppression when competing images are presented to the two eyes, leading to visual rivalry. In tCFS, the contrast of a target stimulus presented to one eye is dynamically ramped, while a flashing mask presented to the other eye maintains constant contrast. This framework allows for continuous measurement of the thresholds at which a suppressed stimulus becomes perceptually dominant ("breakthrough") and, conversely, at which a dominant stimulus is suppressed from awareness. The key empirical finding in tCFS is the presence of perceptual hysteresis: the contrast threshold required to break through suppression is substantially higher than the threshold needed to lose perceptual dominance, revealing asymmetry in perceptual transition points that can be quantitatively modeled using systems of coupled nonlinear differential equations and neural adaptation mechanisms.

1. Mathematical Model of tCFS Dynamics

The tCFS paradigm is formalized by modeling the visual system as two reciprocally inhibiting monocular populations, with neural activity variables $E_S$ and $E_M$ corresponding to the stimulus and mask channels. These populations interact via four coupled nonlinear ordinary differential equations:

$$\begin{align*} \tau_E\,\dot{E}_M &= -E_M + f(M + \epsilon E_M - a E_S - g_M H_M) \ \tau_H\,\dot{H}_M &= -H_M + E_M \ \tau_E\,\dot{E}_S &= -E_S + f(S + \epsilon E_S - a E_M - g_S H_S) \ \tau_H\,\dot{H}_S &= -H_S + E_S \ \end{align*}$$

where $f(x) = \max(x,0)$, $M$ is the mask contrast (held constant), $S$ is the target contrast (dynamically ramped), $\epsilon$ is self-excitation strength, $a$ is competitive inhibition strength, $g$ represents adaptation current strengths, and $\tau_E$, $\tau_H$ are the time constants for neural activity and adaptation, respectively.

Critically, the contrast $S$ of the target stimulus evolves according to a piecewise ODE dependent on perceptual state:

$$\dot{S} = \begin{cases} -\gamma, & E_S > E_M \ (\text{stimulus dominant}) \ +\gamma, & E_S < E_M \ (\text{stimulus suppressed}) \ \end{cases}$$

with ramp rate parameter $\gamma$ governing the speed of stimulus modulation.

2. Hysteresis Mechanism in Perceptual Transitions

Hysteresis in tCFS reflects that breakthrough and suppression transitions occur at systematically different contrast thresholds. This asymmetry is attributed to the interplay of competitive inhibition and adaptation. Perceptual switches are governed by the aggregate synaptic drives for each population, e.g.,

$D_S = S + \epsilon E_S - a E_M - g_S H_S$

Transitions occur when $D_S$ (or $D_M$) crosses zero. Two principal mechanisms trigger state switches:

• Adaptation-driven switches: Slow integration of inhibitory current ($H_S$, $H_M$) diminishes the dominant channel's efficacy until a switch is inevitable.
• Stimulus-driven switches: Explicit changes in $S$, as dictated by the ramp, can abruptly render the previously suppressed population dominant.

The separation of timescales ($\tau_H \gg \tau_E$) enables an analytic decoupling of fast neural responses from slow adaptation, facilitating derivation of precise switching conditions.

3. Closed-Form Quantitative Predictions

Closed-form solutions for dominance (breakthrough) and suppression durations rely on asymptotic analysis involving the Lambert W function, which solves $y = xe^x$ for transcendental equations arising from the adaptation terms. Let $S_B$ and $S_S$ denote the breakthrough and suppression thresholds, respectively. Then duration expressions are:

$$T_{\text{Suppressed}} = \frac{1}{\gamma} \left(\frac{a M}{1+g_M-\epsilon} - S_S\right) + \tau_H W_0\left(\frac{g_S}{\gamma\tau_H}\frac{S_S}{1+g_S-\epsilon} e^{ -\frac{\frac{a M}{1+g_M-\epsilon} - S_S}{\gamma\tau_H} }\right)$$

$$T_{\text{Dominant}} = \frac{1}{\gamma} \left(S_B - \frac{(1+g_S-\epsilon)M}{a}\right) + \tau_H W_0\left(\frac{g_M M (1+g_S-\epsilon)}{a\gamma\tau_H(1+g_M-\epsilon)} e^{ -\frac{S_B - \frac{(1+g_S-\epsilon)M}{a}}{\gamma\tau_H} }\right)$$

The principal branch $W_0$ captures the nonlinearity introduced by adaptation. The depth of hysteresis is defined as:

$$\mathcal{H}_{S \to B} = \gamma \cdot T_{\text{Suppressed}}, \quad \mathcal{H}_{B \to S} = \gamma \cdot T_{\text{Dominant}}$$

In steady-state (after initial transients), these hysteresis depths become quantitatively identical. The ramp rate $\gamma$ critically modulates hysteresis: higher $\gamma$ leads to increased hysteresis because adaptation has less time to counteract the changing stimulus.

4. Behavioral Predictions and Experimental Validation

The analytical model provides the following key predictions:

a. Equivalence of Durational Distributions:

The distributions of time intervals spent in dominant versus suppressed perceptual states are predicted to converge and overlap once equilibrium is reached. Empirical data from psychophysical studies (Alais et al., 2024) corroborate this: two-sample Kolmogorov–Smirnov tests did not distinguish the distributions, and maximum likelihood fits for lognormal and gamma distributions yielded nearly identical parameters for both states.

b. Dependence on Ramp Rate:

Hysteresis depth $\mathcal{H}$ increases with ramp rate $\gamma$. Rapid ramps enhance the suppression-to-breakthrough difference, due to insufficient adaptation recovery time within a cycle.

c. Effects of Inhibition and Adaptation:

Increasing competitive inhibition parameter $a$, for instance via GABA agonists, deepens hysteresis as stationary inhibition becomes more pronounced. When adaptation is equalized across both eyes—such as in a tracking binocular rivalry (tBR) variant (with both eyes receiving dynamic inputs)—the model predicts hysteresis is reduced by roughly a factor of two, reflecting increased adaptation in both channels.

5. Comparison with Related Paradigms

tCFS is situated within the broader context of visual rivalry research, notably differing from traditional binocular rivalry and flash suppression in its use of dynamically modulated target contrast. The inclusion of a ramp rate introduces nontrivial dynamical features, such as precisely quantifiable hysteresis and equivalence of state durations, that are absent in paradigms with static contrasts or noise-driven alternations.

The model extends classical rivalry frameworks by combining dynamically tracked stimuli, adaptation, and inhibition in a mathematically tractable form. The use of the Lambert W function for closed solutions distinguishes the approach, allowing explicit and predictive mapping between model parameters and experimental outcomes.

6. Implications and Future Directions

The quantitative framework developed for tCFS not only explains empirical observations but also offers a set of clear predictions for further validation. Manipulations targeting inhibition (pharmacological or otherwise) and adaptation (via stimulus design) are expected to effect predictable changes in hysteresis. The identification of hysteresis as a signature of the balance between adaptation and inhibition may generalize to other multistable perception phenomena.

A plausible implication is that similar analytic strategies could elucidate the dynamics of awareness transitions in other sensory modalities or in disorders of perceptual stability. The model's exactness and tractability present an opportunity for integrating rigorous psychophysical design with neural-level modeling, advancing mechanistic understanding of conscious perception.