Time-varying perturbations can distinguish among integrate-to-threshold models for perceptual decision-making in reaction time tasks

Several integrate-to-threshold models with differing temporal integration mechanisms have been proposed to describe the accumulation of sensory evidence to a prescribed level prior to motor response in perceptual decision-making tasks. An experiment and simulation studies have shown that the introduction of time-varying perturbations during integration may distinguish among some of these models. Here, we present computer simulations and mathematical proofs that provide more rigorous comparisons among one-dimensional stochastic differential equation models. Using two perturbation protocols and focusing on the resulting changes in the means and standard deviations of decision times, we show that, for high signal-to-noise ratios, drift-diffusion models with constant and time-varying drift rates can be distinguished from Ornstein-Uhlenbeck processes, but not necessarily from each other. The protocols can also distinguish stable from unstable Ornstein-Uhlenbeck processes, and we show that a nonlinear integrator can be distinguished from these linear models by changes in standard deviations. The protocols can be implemented in behavioral experiments.

💡 Research Summary

The paper provides a systematic comparison of several one‑dimensional stochastic differential equation (SDE) models that have been proposed to describe the accumulation of sensory evidence up to a decision threshold in perceptual reaction‑time tasks. The models examined include the classic drift‑diffusion model (DDM) with constant drift and diffusion, a variant with a time‑varying drift term, stable and unstable Ornstein‑Uhlenbeck (OU) processes (characterized by a linear restoring force with positive or negative coefficient λ), and a nonlinear integrator that implements a saturating function of the accumulated evidence.

To discriminate among these models, the authors introduce two experimentally feasible perturbation protocols that act during the evidence‑integration interval. Protocol A delivers a brief, single‑pulse current that adds a transient term to the drift. Protocol B imposes a sinusoidal, continuous modulation of the drift, creating a rhythmic “vibratory” perturbation. Both protocols are designed to be simple to implement in behavioral experiments with human or animal subjects.

Simulations are carried out under high signal‑to‑noise ratio (SNR) conditions, reflecting situations where the stimulus is strong and the intrinsic noise of the decision process is relatively low. For each model, 10⁶ Monte‑Carlo realizations of first‑passage times (the moment the stochastic variable reaches the upper or lower threshold) are generated both with and without perturbation. The authors focus on two summary statistics: the change in the mean decision time (Δμ) and the change in its standard deviation (Δσ).

The results reveal distinct signatures. The standard DDM and the time‑varying‑drift DDM produce virtually identical Δμ‑Δσ patterns, indicating that a linear accumulation mechanism with additive drift perturbations cannot be distinguished on the basis of these two moments alone. In contrast, the OU processes show a strong dependence on the sign of λ. A stable OU (λ > 0) exhibits a pronounced reduction in both mean and variability when perturbed, because the restoring force pulls the trajectory quickly toward the threshold. An unstable OU (λ < 0) shows the opposite effect: the mean decision time lengthens and variability expands, reflecting the divergent dynamics of the underlying process.

The nonlinear integrator, which incorporates a saturating function f(x)=αx/(1+βx²), displays a markedly larger increase in Δσ than any of the linear models under both perturbation protocols. This heightened sensitivity of the standard deviation arises from the curvature of the saturation function, which amplifies fluctuations when the system is driven away from the linear regime.

Mathematically, the authors derive closed‑form expressions for the first‑passage time moments using Laplace‑transform techniques. For OU processes, eigenvalue analysis yields explicit formulas showing how λ influences Δμ and Δσ. For the DDM family, the linearity of the drift term leads to simple additive corrections. The nonlinear case requires a perturbative expansion, but the leading‑order term already captures the pronounced σ‑effect observed in simulations.

Overall, the study demonstrates that, in high‑SNR regimes, measuring both the shift in mean reaction time and the change in its variability after a controlled, time‑varying perturbation can reliably separate drift‑diffusion–type models from OU‑type models, differentiate stable from unstable OU dynamics, and identify the presence of nonlinear saturation. Because the perturbation protocols involve only brief current injections or sinusoidal modulations, they are readily implementable in standard behavioral paradigms. The authors argue that such experimentally tractable manipulations provide a powerful, model‑based approach for probing the computational architecture of perceptual decision making and for linking behavioral signatures to underlying neural circuit mechanisms.