Contribution
A systematic mathematical comparison of six two-choice decision models, the DDM, O-U, race, mutual inhibition, feedforward inhibition and pooled inhibition, which locates each relative to the SPRT. It then adds a theory of how the threshold should be set under four criteria (Bayes risk, reward rate, and two accuracy-weighted variants of reward rate), derives optimal performance curves as predictions, and extends both analyses to biased priors and biased rewards.
Key insight
The question “which model of choice is right?” and the question “what would an optimal decider do?” have the same answer in a precise regime. Integrating the difference of the evidence for two options is the SPRT. Leaky, competing neural units compute that difference when their leak and their mutual inhibition cancel, and they do so in one dimension when both are strong enough to pin activity to a line. Inhibition is what turns two accumulators into one comparison; leak balanced against it is what makes the comparison lossless. Optimality then moves from the mechanism to its settings: a threshold that trades speed for accuracy, and a starting point that encodes prior odds.
Assumptions
- Two alternatives, evidence drawn i.i.d. within a trial from one of two distributions; drift fixed within a trial.
- Linearised models. All network models are linearised (threshold-linear floors ignored), with activity measured from baseline so that y < 0 need not mean negative firing.
- No cost for sampling in the optimality of the SPRT and Neyman–Pearson test (p. 703); time enters only through the reward criteria.
- Extended equivalences require the total input I₁ + I₂ to be constant across trials and the two starting points to be anticorrelated (y₁(0) = −y₂(0)); the authors note there is no cortical evidence yet for the latter.
- Pooled inhibition reduction requires the inhibitory population’s decay to be fast relative to the excitatory one, an assumption they say needs physiological test (footnote 7).
- Rewards follow each correct response, and delays D (response to next stimulus) and D_p (extra penalty delay after errors) are fixed per block.
Key results
- DDM formulas (Eqs. 5–9). dx = A dt + c dW. Under interrogation at T, ER = Φ(−A√T/c). In free response with thresholds ±z, ER = 1/(1 + e^{2Az/c²}) and DT = (z/A) tanh(Az/c²). ER and DT depend only on ratios of A, z and c².
- Model reductions (Fig. 4; Eqs. 18–38). In rotated coordinates the linearised mutual inhibition model decouples into x₁ (difference), an O-U process with λ = w − k, and x₂ (sum), always a stable O-U process with rate k + w. With k = w the difference is a pure DDM, with drift (I₁ − I₂)/√2 and threshold z = √2 Z − (I₁ + I₂)/(√2(k + w)) (Eq. 26). The feedforward inhibition model equals the DDM exactly when u = 1. The pooled inhibition model equals a mutual inhibition model with decay k + ww′/k_inh − v and inhibition ww′/k_inh. The race model is not reducible.
- Interrogation optimum. Error rate is minimised at λ = 0, whatever the magnitude of k and w, even k = w = 0. As Usher and McClelland observed, error depends only on |λ|. With variable starting points, λ < 0 (leak dominant, recency) can do better, because it discounts the noisy start.
- Free-response optimum (Fig. 10). At fixed ER, DT is shortest when k = w, and it falls toward the DDM’s value as k = w grows. The race model is slowest.
- Unbounded accuracy (Table 1). Under interrogation, ER → 0 as T → ∞ only for λ = 0; otherwise a finite floor depends only on |λ| (Eq. 39). In free response, ER → 0 for λ ≤ 0, but for λ < 0 decision times diverge. With drift variability no model reaches zero error; the floor is the fraction of trials with negative drift.
- Optimal thresholds (Fig. 12). For every criterion the optimal threshold is zero when drift is zero, rises, then falls as drift dominates noise; it rises with noise to a limit and rises with delay. For reward rate the condition is e^{2z̃ã} − 1 = 2ã(D_total − z̃), with z̃ = z/A and ã = (A/c)², and a unique solution. It depends on D and D_p only through their sum (any RR-maximising mechanism shares this; Appendix A).
- Optimal performance curves (Fig. 13). For reward rate, DT/D_total is a function of ER alone: peak DT about 20% of the maximum inter-decision interval, at ER about 18%. For Bayes risk (Edwards 1965), peak DT about 0.136q at ER about 13.5%.
- Threshold learning (Fig. 15). Reward rate falls more steeply below the optimal threshold than above it, so overestimating costs less than underestimating, and gradient learning finds the upward direction more easily. Adaptive learners should therefore set thresholds too high. The authors offer this as an explanation, in reward-rate terms alone, of people’s apparent bias toward accuracy.
- Biased priors (Eqs. 62–63, Fig. 16). The optimal starting point is proportional to the log prior odds (Edwards); Link’s rule, minimising ER at fixed threshold, is half that. Estimates from Laming, Link and Van Zandt et al. lie closer to Edwards’s rule, significantly so for Laming’s data (paired t-test, p = 0.04). With constant signal strength, only the start should move; with mixed difficulty, drift should be biased too. Above a critical bias or below a critical delay, the RR-optimal policy is to skip integration and always choose the likelier option.
- Biased rewards (Fig. 18). The optimal start depends on delay: like probability bias for short D_total, halfway to it for long D_total.
- Neural fit. Platt and Glimcher’s prestimulus LIP activity grew roughly linearly with both prior probability and reward share from 20% to 80%, as the optimal start points do over that range.
- Experiment. 20 participants, dot motion at fixed coherence, four delay conditions. One representative participant’s extended-DDM fit is reported (mA = 1, sA = 0.31, sx = 0.14, c = 0.33; thresholds rising from 0.16 to 0.26 with delay; T₀ = 0.37 s). The full analysis is deferred to a later report.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The pure DDM implements the SPRT (free response) and Neyman–Pearson test (interrogation), so it is the optimal two-choice decider | strong (classical theorems, re-derived) | p. 703–704, Appendix A |
| C2 | The linearised mutual inhibition model reduces to the DDM when leak equals inhibition and both are large | strong for the linear model (derivation) plus simulation | Eqs. 18–26, Figs. 6–8 |
| C3 | Among mutual inhibition models, balanced and strong leak and inhibition maximise performance | strong in the linear model; simulation for the extended model | Figs. 9–10, Table 1 |
| C4 | The reward-rate-optimal threshold is unique and depends on delays only through their sum | strong (derivation) | Eq. 51, Appendix A |
| C5 | The optimal performance curve relating DT/D_total to ER is parameter-free | strong as mathematics; its empirical test is deferred | Eq. 58, Fig. 13 |
| C6 | Threshold-learning asymmetry predicts a bias toward over-cautious thresholds | moderate (analysis), untested here | Fig. 15, Inequality 61 |
| C7 | Optimal starting points under bias follow log prior odds, and data favour this over Link’s rule | moderate: three data sets, one significant | Fig. 16b |
| C8 | Prestimulus LIP activity conforms to RR-optimal starting points | weak to moderate: qualitative, near-linear range, and slope difference not significant | Platt & Glimcher comparison |
Concepts
- interrogation vs free-response paradigm: decision forced at a set time versus made when the participant chooses.
- balanced mutual inhibition model: decay equals inhibition (λ = 0).
- decision line: the attracting line in the (y₁, y₂) plane along which the difference evolves once the sum has settled.
- reward rate (RR): the proportion of correct trials divided by the average time between decisions, which includes RT, the delay D and, after errors, the penalty delay D_p (Gold & Shadlen’s definition).
- optimal performance curve: the relation between ER and normalised DT that must hold if the threshold is optimal for a criterion.
- signal-to-noise ratio ã = (A/c)², and normalised threshold z̃ = z/A.
Connections
- Usher & McClelland 2001 (LIT-565). The source of the mutual inhibition model, and of the observations that λ’s sign does not affect accuracy and that the balanced model is fastest at fixed accuracy (their 1995 technical report). This paper derives them and adds the requirement of large leak and inhibition.
- Ratcliff & McKoon 2008 (LIT-576). The empirical counterpart. This paper uses Ratcliff’s extended DDM, and Ratcliff and Tuerlinckx’s quantile fitting; Ratcliff and McKoon later reject its account of threshold setting as unable to explain verbal calibration.
- Schurger et al. 2012 (LIT-605). Their model is the leak-dominated (λ < 0) single accumulator with a constant input. In this paper’s terms it is a stable O-U process with an interior attractor, whose crossings at a threshold above the attractor are driven by noise. That is the regime this paper shows gives long and variable decision times. The mapping is mine; neither paper draws it.
- Wiecki et al. 2013 (LIT-563). Estimates the extended DDM whose parameters this paper interprets as controlled quantities: drift (attention), starting point (expectancy) and threshold (caution).
Bearing on the record
- It supplies a normative reading of
behavioral-integrationat the level of one decision: accumulate the difference, and set threshold and start by the utility at stake. Its last section recasts cognitive control as that tuning. That THEORY is filed as THEORY-061, and the balance of leak and inhibition as THEORY-068. - No ML instruction. Although the SPRT and reward-rate optimisation are shared with machine learning, the paper is about human and animal decisions and gives no ML practice. Nothing here belongs in the anthology.
Limitations
- Two alternatives, stationary drift. Multi-alternative optimality (MSPRT) and within-trial changes of signal are pointed to, not treated.
- Linearised models. Thresholding nonlinearities are dropped; their effects are discussed elsewhere (Brown et al. 2005).
- Variability is not optimal. The extended DDM fits better than the pure one, but its drift and starting-point variability are by construction suboptimal; the paper explains them by attention and history, as adaptive on a broader scale, without a model.
- Accuracy floors. That people do not reach perfect accuracy at long times is waved to Ratcliff’s (1988) bounded diffusion, confirmed “but we do not treat this issue here”.
- The experiment is reported for one participant; the predictions about delays and optimal performance curves are not tested here.
Open questions
- Do people’s decision times and errors fall on the parameter-free optimal performance curve, and does performance depend on delays only through their sum?
- When starting point varies, can unbalanced (leak-dominant) networks be optimal in practice?
- How do neuromodulatory systems (locus coeruleus is suggested) implement gain changes for interrogation?
Corrections
- none to a seeded skim (there was no seed)