Two leaky, mutually-inhibiting accumulators race toward a threshold — the Leaky Competing Accumulator (LCA) model of choice (Usher & McClelland, 2001), applied here to the Stroop task (Stroop, 1935; Cohen, Dunbar & McClelland's 1990 connectionist account):
dx_C = (I_C − k·x_C − β·x_W)·dt + σ·√dt·ξ
dx_W = (I_W − k·x_W − β·x_C)·dt + σ·√dt·ξ
first accumulator to reach threshold θ wins → that trial's response
This 2D version plots the exact same stochastic race as a phase-plane trajectory: the horizontal axis is x_C (color-naming activation), the vertical axis is x_W (word-reading activation). Each trial traces a path from the origin; the trial ends the instant the path crosses into the L-shaped region beyond the threshold lines x_C=θ or x_W=θ. This is mathematically identical to the race — just viewed as a state-space diagram instead of two racing bars.
- Congruent — the printed word names the same color as the ink, so the word pathway reinforces I_C: paths bend sharply toward the x_C=θ boundary — faster, near-perfect responses.
- Neutral — a non-color string (e.g. "XXXX"): the word pathway is inert, giving a baseline path with no interference.
- Incongruent — the word names a different color than the ink, so paths drift toward the diagonal and sometimes cross into the x_W=θ boundary first: slower RTs and more errors — the classic Stroop interference effect.
- Automaticity raises the baseline drive of the word channel (skilled readers can't switch reading off); threshold moves the L-boundary outward, trading speed for accuracy; noise controls how jagged each path is, which is why RTs form a distribution rather than one fixed number.
Real-world relevance: this race-to-threshold dynamic is the standard account behind the Stroop task, flanker task and Simon effect, and underlies clinical measures of attentional control and inhibitory function.