The diffusion model, most fully developed by Roger Ratcliff (1978), is the dominant quantitative account of speeded two-choice decision making. The model assumes that decisions arise from the continuous accumulation of noisy evidence from a starting point toward one of two absorbing boundaries. The rate of evidence accumulation (drift rate v) reflects the quality of information extracted from the stimulus, the distance between boundaries (boundary separation a) reflects the decision maker's speed-accuracy tradeoff setting, and the nondecision time (Tₑᵣ) captures the duration of encoding and response execution processes that are not part of the decision itself.
The Wiener Diffusion Process
v = drift rate (stimulus quality/difficulty)
a = boundary separation (speed-accuracy tradeoff)
z = starting point (response bias)
Tₑᵣ = nondecision time (encoding + motor)
s = within-trial noise (scaling parameter, typically s = 0.1)
Mean RT ≈ (a/2v) · tanh(va/2s²) + Tₑᵣ
Accuracy = 1 / (1 + e^(−2va/s²))
The diffusion process is a continuous-time random walk: at each infinitesimal moment, evidence moves toward the correct boundary with an average step of v·dt (signal) and a random perturbation of s·dW (noise). The process terminates when it first reaches either the upper boundary (response A) or the lower boundary (response B). The first-passage time distribution gives the predicted RT distribution for each response, and the relative probability of reaching each boundary gives the predicted accuracy. The model thus jointly predicts the full shape of RT distributions and choice probabilities from a single set of parameters.
Full Diffusion Model and Applications
The "full" diffusion model (Ratcliff & McKoon, 2008) adds across-trial variability in drift rate (normally distributed, η), starting point (uniformly distributed, sᵤ), and nondecision time (uniformly distributed, sₜ). These variability parameters are essential for explaining the relative speed of correct vs. error responses: without across-trial variability in drift rate, the model predicts that errors are faster than correct responses, whereas with variability, the model can predict either faster or slower errors depending on the parameter regime.
Each parameter of the diffusion model maps onto a distinct cognitive process, making the model a powerful tool for cognitive decomposition. Drift rate captures the efficiency of information processing and declines with stimulus difficulty, aging, and cognitive impairment. Boundary separation captures response caution and increases under accuracy-emphasis instructions. Nondecision time captures peripheral processing speed and increases with motor dysfunction. This parameter-to-process mapping has been used to study aging, ADHD, anxiety, clinical populations, and individual differences in cognitive ability.
The diffusion model has been applied to lexical decision, recognition memory, perceptual discrimination, numerosity judgment, and many other two-choice tasks. It provides a principled decomposition of observed performance into latent cognitive processes and has become a standard tool in mathematical psychology and cognitive neuroscience. Its success has inspired a family of related models, including the linear ballistic accumulator, the leaky competing accumulator, and neural implementations of drift-diffusion in spiking neural circuits.