NOTE-449: The Time Course of Perceptual Choice: The Leaky, Competing Accumulator Model

Contribution

The paper adds two principles to stochastic accumulation models of choice: leakage of accumulated information, and competition between alternatives by lateral inhibition. Recurrent self-excitation and a threshold-linear nonlinearity are included, but mostly to balance leak and to stop negative activations from propagating. It shows that the classical diffusion model is a limiting case (balanced leak and inhibition), and it tests the new principles on time–accuracy curves, latency–discriminability functions, RT distributions, a new sequence-integration experiment, Hick’s law and an interaction in word identification.

Key insight

A choice between alternatives can be both a race and a comparison at once. If every alternative has its own accumulator and each accumulator suppresses the others, the winner’s activation comes to reflect its evidence relative to the rest, while the decision can still be triggered by a fixed absolute level. The dynamics of the difference between two such units depend on one number: leak minus inhibition. When the two balance, the system integrates perfectly, as the diffusion model does. When leak wins, old evidence fades and late evidence dominates. When inhibition wins, an early lead is amplified and early evidence dominates. In both unbalanced cases, accuracy given unlimited time is capped.

Assumptions

• Units as populations (pp. 554–555): each accumulator is a neural population. Its activation is the population input current and its output the mean firing rate, related by a threshold-linear function f(x) = x for x ≥ 0 and 0 otherwise.
• Noise: zero-mean Gaussian noise added to each unit’s input, justified by the central limit theorem over spiking populations (p. 555).
• Dynamics (Eq. 4, p. 559): dx_i = [ρ_i − k x_i − β Σ_{j≠i} x_j] dt/τ + ξ_i √(dt/τ), then x_i → max(x_i, 0). Here k = λ − α is net leak (passive decay minus self-excitation), β is lateral inhibition, and ρ_i is non-negative feed-forward input. Appendix A shows by simulation that truncating at 0 approximates the version with separate fast decay below 0 and slow net decay above it.
• Inputs sum to one in two-choice continuum tasks, ρ₁ + ρ₂ = 1, so v = ρ₁ − ρ₂ (p. 559).
• Fixed delays for encoding and response output, lumped into T₀; no trial-to-trial variability in T₀ or in the input in the main fits.
• Response rules: in time-controlled tasks the most active unit at the signal is chosen; in RT tasks the first unit to reach criterion θ.
• Hick’s law simulations: the criterion is adjusted per set size to hold accuracy fixed (95% or 99%), and input to each accumulator does not depend on the number of alternatives.

Key results

• Reduction to an OU process (Eqs. 5–8, pp. 560–561). For two units in the linear regime, x = x₁ − x₂ obeys dx = (v − Kx) dt/τ + ε√(dt/τ), with K = k − β. Starting from 0, x(t) is Gaussian with μ(t) = (v/K)[1 − e^{−Kt}] and SD(t) = (σ/√K)√(1 − e^{−2Kt}). K = 0 is the classical diffusion process, with μ = vt and SD = σ√t.
• Time–accuracy function (Eqs. 9–10). d′(t) = d_asy (1 − e^{−Kt}) / √(1 − e^{−2Kt}), and d_asy = (2v/σ)√(1/K). Accuracy is bounded for K > 0. Simulations of the full nonlinear model show the curves depend on |K| only: β = 0 and β = 0.4 with k = 0.2 give “virtually identical” curves (Fig. 4).
• Trajectories differ where accuracy does not (Fig. 5). Pure leak keeps the distribution of x₁ − x₂ bounded. Self-enhancement makes it diverge. Leak plus inhibition makes it bimodal, because the losing unit is suppressed to zero and the winner then feels only leak. The authors read this as the system producing “essentially a binary decision”.
• Experiment 1 (time–accuracy, response signal; 3 participants, lags 0 to 2,000 ms). Fitting the OU and drift-variance (DDV) curves with three parameters each per curve, 7 of 9 likelihood ratios favour OU, and their product is 62.02 (Table 1). With one rate and one offset per participant, OU is favoured for all three, with a combined ratio of 43.14 (Table 2). Using lag alone as integration time, OU is favoured 197 to 1 and 106 to 1. Fitted OU time constants lie between about 100 and 400 ms. The full nonlinear model, simulated with parameters derived from the OU fit, matches Participant 2’s curves (Fig. 9).
• Latency–discriminability (Vickers 1970, 5 participants). Fitted with six parameters and no drift or starting-point variability, the model captures accuracy and RT mean and SD for 3 participants. It reproduces errors slower than correct responses and rising skew and kurtosis, which it was not fitted to. It misses the dips for Participants 1 and 5, which the authors call anomalous for every model they know. Lateral inhibition is what makes the latency–probability functions U-shaped (Fig. 15).
• Experiment 2 (RT distributions; 2 participants, 960 trials per level). Fitted to accuracy, RT mean, SD and skew, the model reproduces the shapes of the correct RT distributions. Skew rises with discriminability (1.3, 1.8, 2.2 for Participant 1).
• Experiment 3 (information arrival time; 6 participants, 16-item S/H sequences at 60 per second, 256 ms). On balanced sequences ending in a 4-item cluster, 2 participants favoured the end cluster (recency), 2 opposed it (primacy) and 2 were neutral (Table 4). Fitted leak and inhibition account for each pattern, and the same parameters predict accuracy on the 40-item background trials. The neutral, balanced participants were the most accurate, as the model predicts.
• Hick’s law (Fig. 23). With the criterion adjusted to keep accuracy fixed, RT is linear in log N for the classical accumulator, the LCA and a max-minus-next accumulator. With a fixed criterion none of them gives a logarithmic law. The authors say this does not support the LCA over the others.
• Word identification (McClelland & O’Regan 1981). Lateral inhibition implements an “intersection principle”. One weak source (context or visual preview) activates m candidates whose mutual inhibition cancels its benefit. Two independent weak sources leave the target with double input, and for 0.5 < β < 1 its competitors are suppressed to zero. Simulated: 42.9, 42.8 and 34.7 mean steps (Table 5).
• Confidence. Because suppression is partial, the activation difference at response time can serve as confidence. It falls with longer RT in free response and rises with time in time-controlled tasks, as Vickers’s confidence data require.

Claims

id	claim	strength	support
C1	In the linear two-choice case the accumulator difference is an OU process with net leak K = k − β, and K = 0 is the classical diffusion model	strong (derivation)	Eqs. 5–8
C2	Asymptotic accuracy is bounded whenever K ≠ 0, and the time–accuracy curve depends only on |K|	strong for the linear case; checked by simulation for the nonlinear one	Eqs. 9–10, Fig. 4
C3	Leakage fits empirical time–accuracy curves better than drift variance	weak to moderate: 3 participants, likelihood ratios mostly small, and the DDV compared lacks decision boundaries, which the authors concede could close the gap	Tables 1–2, p. 568
C4	The balance of leak and inhibition sets primacy versus recency, and imbalance costs accuracy	moderate: confirmed on 6 participants, but alternative attentional accounts not excluded	Exp. 3, Fig. 21
C5	Lateral inhibition reproduces asymmetric, U-shaped latency–probability functions without drift variance	moderate: 3 of 5 Vickers participants fitted; fits not optimal by the authors’ account	Figs. 11–15
C6	Hick’s law follows from holding accuracy fixed across set sizes, not from a capacity limit	moderate in simulation; not discriminating between models	Fig. 23
C7	The LCA reconciles relative-evidence effects with confidence data that favour absolute criteria	moderate: simulation reported in prose only	pp. 586

Concepts

• leaky, competing accumulator: an accumulator per alternative, subject to leakage and to lateral inhibition from the others.
• net leakage k: passive decay minus recurrent self-excitation, λ − α.
• differential leakage K: k − β, the net leak of the difference between two units.
• diffusion with drift variance (DDV): Ratcliff’s 1978 diffusion model with trial-to-trial variability in drift.
• time-controlled vs information-controlled tasks: response-signal tasks, where time is fixed by the experimenter, versus standard RT tasks, where the participant’s criterion fixes it.
• intersection principle: support from two independent weak sources singles out the one candidate both favour.

Connections

• Bogacz et al. 2006 (LIT-569). Linearises this model as the “mutual inhibition model”, proves the reduction to the OU and diffusion models analytically, and adds what this paper lacks: that the balanced version is optimal, and that optimality also needs leak and inhibition to be large.
• Ratcliff & McKoon 2008 (LIT-576). The rival single-process view. It credits accumulator models of this kind with possibly comparable fits, but disputes how they could relate to likelihood-based decisions (its §7.1).
• Schurger, Sitt & Dehaene 2012 (LIT-605). Uses a single leaky stochastic accumulator from this paper, with a constant input as urgency, to model when spontaneous movements are initiated.
• Frijda et al. (LIT-540) and THEORY-056. These argue that emotion regulation is one action readiness checking another, not a separate regulator. The LCA gives a formal model of mutually suppressing response tendencies with no supervisory unit. That is my analogy, not the paper’s claim; it is about perceptual choice only.

Bearing on the record

• It anchors behavioral-integration at the level of a single decision: many candidate responses resolved into one by competition plus an absolute criterion. The THEORY on balanced integration is filed as THEORY-068.
• No ML instruction. Nothing here belongs in the anthology.

Limitations

• Small samples. The new experiments have 3, 2 and 6 participants.
• The leak-versus-drift-variance contest is unresolved by its own account. The authors concede that DDV with starting-point variance and decision boundaries “tend[s] to bring” its curves close to the shifted exponential, and that “both leakage and drift variance are factors” (pp. 568, 585).
• Fits are not optimal. The latency–discriminability fits ran only 8,000 Metropolis swaps, and noise in the fitting is acknowledged.
• Neural motivation is analogy. Chelazzi et al.’s IT recordings show competition consistent with lateral inhibition, but the model is a single layer with fixed delays, and its parameters are not measured in tissue.
• Hick’s law holds only for criterion values below the expected asymptote of the correct unit; with unbalanced K and high thresholds the linear relation breaks down (p. 582).

Open questions

• Which dominates in a given task, leakage or drift variance? The authors expect the answer to be task-dependent (memory tasks favouring drift variance).
• Do ambiguous primes give a reduced but non-zero benefit, as partial lateral inhibition predicts, or none, as a best-minus-next criterion predicts?
• How do leak and inhibition play out in multilayer architectures? This is left for later work.

Corrections

• none to a seeded skim (there was no seed)
• Citation. The DOI as printed on the article is 10.1037//0033-295X.108.3.550 (double slash), which is APA’s older form. Crossref resolves 10.1037/0033-295X.108.3.550.