Contribution
A review that states the diffusion model completely, explains how each parameter shapes accuracy and the correct and error RT distributions, and illustrates the three canonical manipulations on new human data from the motion-discrimination task used in monkey neurophysiology. It then surveys the model as a measurement tool: individual differences, aging, aphasia, coupling to encoding models, and the mapping to neural firing rates.
Key insight
Two-choice behaviour carries more information than mean RT and accuracy: the full shape of the correct and error RT distributions. A noisy accumulator between two boundaries predicts that shape from geometry alone. Its parameters therefore act as a decomposition of performance into evidence quality, caution, bias and non-decision time. It can be falsified, because each manipulation has to land on one parameter, and the shape of the distributions must stay the same across conditions while their location and spread change.
Assumptions
- Scope (§2): fast two-choice decisions (mean RT under about 1,000–1,500 ms) made in a single stage, not reasoning tasks.
- Process (§2): evidence starts at z and accumulates with drift v and within-trial SD s (a scaling parameter) until it reaches 0 or a. Drift is constant within a trial.
- Across-trial variability (§2): drift ~ Normal(v, η); starting point uniform with range s_z; non-decision time uniform with mean Ter and range s_t. Boundary variability is assumed to be absorbed by starting-point variability.
- Contaminants (§2.6): a proportion p_o of uniform contaminant responses is modelled, after cutting short and long outliers (usually 2–3%).
- Fitting (§2.6): χ² over the proportions between the .1, .3, .5, .7 and .9 quantiles of correct and error RTs, minimised by SIMPLEX. In this paper fits are to data averaged over participants.
Key results
- Distribution geometry (§2.1). Lowering drift slows the .9 quantile about four times as much as the .1. Narrowing boundaries changes both, in roughly a 2:1 ratio, and trades speed for accuracy.
- Bias, two ways (§2.2). Moving the starting point shifts both the leading edge and the tail. Moving the drift criterion (the zero point of drift) changes mainly the tail, which is the analogue of moving the criterion in signal detection theory.
- Correct versus error RTs (§2.3). Across-trial drift variability makes errors slower than correct responses; starting-point variability makes them faster; together they cover the observed crossovers (Ratcliff & Rouder 1998; Ratcliff et al. 1999).
- Experiment 1 (15 participants, six coherences from 5% to 50%). Accuracy ran from .58 to .94 and mean RT from about 660 to 550 ms. Only drift varied across conditions (0.042 to 0.369; Table 2). The tails spread by up to 300 ms and the leading edge by under 40 ms. Quantile–quantile plots were nearly linear: the shape of the distribution was invariant.
- Experiment 2 (14 participants, speed versus accuracy instructions). Instructions changed accuracy by 0–6% but median RT by 120–200 ms, the .1 quantile by 40–100 ms and the .9 by 250–550 ms. All of it was fitted by boundary separation alone (a = 0.109 vs 0.152). Separate Ter values for the two instructions differed by 6 ms and were dropped.
- Experiment 3 (17 participants, 75:25 left–right proportions). The starting point moved to about a third of the boundary distance toward the likelier response and accounted for most of the effect. Letting the drift criterion vary improved χ² by only 1%. There were systematic misses on the .9 quantiles for errors.
- Human versus monkey distributions (§4). Human RT distributions in this task were right-skewed with roughly exponential tails, unlike the nearly symmetric distributions of Roitman and Shadlen’s monkeys that motivated Ditterich’s model with time-varying drift.
- Drift and coherence (§4.4.1). Drift is nearly linear in coherence, with a slight bend toward 50%, which supports Palmer et al.’s linear assumption without requiring it a priori.
- Response signal and go/no-go (§5). These are accounted for with implicit boundaries. The boundary-free version fails on response-signal data fitted jointly with standard RT data.
- Individual differences (§8). Across 18 data sets of 30–40 participants each, accuracy correlated with drift and mean RT with boundary separation, and drift and boundary separation were uncorrelated. Across four tasks, individuals’ boundaries (r = .32), Ter (r = .47) and drift (r = .37) correlated. Older adults’ slowing is mostly conservative boundaries; drift is mostly not worse. Aphasic patients set wider boundaries and have longer Ter, with small drift deficits.
- Neural correlates (§10). In superior colliculus buildup cells (Ratcliff, Cherian et al. 2003) the average simulated path matched the average firing rate, including the time shifts between fast, middle and slow thirds of responses, which the noise predicts. A dual diffusion model, with leaky accumulators per alternative as in Usher and McClelland, fits the two cell types separately. Wong and Wang’s spiking model reduces to two units with self-excitation and mutual inhibition.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Difficulty maps to drift, speed–accuracy instructions to boundary separation, and stimulus proportion mainly to starting point | moderate to strong for these experiments; fits are to group-averaged data | Exps. 1–3, Tables 1–2 |
| C2 | RT distribution shape is invariant across conditions, and the model predicts it | moderate | Q–Q plots, Fig. 8; earlier unpublished analyses cited |
| C3 | Across-trial drift and starting-point variability jointly explain slow and fast errors | strong (cited) | §2.3, Ratcliff & Rouder 1998, Ratcliff et al. 1999 |
| C4 | Older adults’ slowing is mostly caution, not poorer evidence | moderate, from the authors’ own earlier studies | §8.2.1 |
| C5 | Averaged diffusion paths track superior colliculus firing rates | moderate (one study, aggregated cells) | §10 |
| C6 | Likelihood-ratio decision models are implausible for humans who calibrate from verbal instruction in one trial | argument, not test | §7.1 |
| C7 | Diffusion models are “as near to” a solution to simple decision making as behavioural science allows | assertion | §11 |
Concepts
- drift rate (v): the quality of evidence from the stimulus, positive toward one boundary.
- drift criterion: the zero point dividing positive from negative drift, analogous to the criterion of signal detection theory.
- boundary separation (a): the evidence required for a response; caution.
- Ter: mean duration of all non-decision components (encoding u plus response output w).
- quantile probability plot: RT quantiles of correct and error responses plotted against response proportion, for every condition at once.
Connections
- Usher & McClelland 2001 (LIT-565). Placed in the accumulator subclass. It and Bogacz et al. are cited as “more recent accumulator models” that “may be as successful as” the single-process model, but have been tested on fewer paradigms (§6). §7.1 adds that for leaky models such as the LCA it is unclear whether likelihood-based and distance-from-criterion accounts coincide.
- Bogacz et al. 2006 (LIT-569). Cited for criterion-setting proposals that, the authors say, cannot explain calibration from verbal instruction or without feedback (§7.1). Bogacz et al. also fit the extended model to RT quantiles and error rates, but by weighted least squares rather than χ².
- Wiecki, Sofer & Frank 2013 (LIT-563). Implements this model, with the same across-trial variability parameters, in hierarchical Bayesian form. It replaces the group-averaged χ² fits used here with partial pooling.
- Schurger et al. 2012 (LIT-605). Uses the linear mean–SD relation of first-passage times, a property of drift-diffusion, as a check on its own leaky accumulator.
Bearing on the record
- The model says what an evidence-accumulation decision is, with parameters that dissociate. That makes it the measurement backbone for any THEORY the record might hold about deliberation, caution or bias; THEORY-061 is the first to rest on it.
- No ML instruction. Nothing here belongs in the anthology.
Limitations
- Group-averaged fits. The authors report that individual and averaged parameter values were within 2 SEs “with only one or two exceptions”, and that χ² values here are not interpretable in the standard way.
- Scope. The model applies only to fast single-stage two-choice decisions. It is silent on how criteria are set (§7.1), which the authors attribute to decision history.
- Variability parameters are poorly estimated (§4.4); η and s_z depend on the relatively few error RTs.
- Neural evidence relies on aggregated firing rates and on difference between cell types, not single cells, for the single-process model.
Open questions
- How do people set and calibrate boundaries from instructions alone?
- How does encoding produce drift, for coherence and other stimuli? The model is offered as the meeting point where such encoding models can be tested.
- Can likelihood-based models be distinguished from distance-from-criterion models for leaky or dual-diffusion processes (§7.1)?
Corrections
- none to a seeded skim (there was no seed)