Rafal Bogacz, Eric Brown, Jeff Moehlis, Philip Holmes and Jonathan D. Cohen (2006), Psychological Review 113(4):700–765 — DOI-10.1037/0033-295X.113.4.700
Key takeaways
- The sequential probability ratio test (SPRT) of Barnard and Wald is the fastest test for a given error rate, and its continuum limit is the pure drift diffusion model (DDM). So the DDM is the optimal two-choice decision process, both in free response (fastest for fixed accuracy) and under interrogation (most accurate at a fixed time).
- Biologically motivated models are optimal exactly when they reduce to the DDM. In the Usher–McClelland mutual inhibition model the difference between the two units is an Ornstein–Uhlenbeck process with λ = w − k. With leak equal to inhibition it is a diffusion process, and with both large, activity collapses fast onto a “decision line”, so the two-dimensional model behaves like the one-dimensional DDM. The feedforward inhibition model is exactly the DDM when inhibition equals excitation. Wang’s pooled inhibition model reduces to mutual inhibition when inhibitory neurons are fast. The race model never reduces.
- How cautious a decider should be is itself optimisable. For reward rate there is a unique optimal threshold, which depends on the delays only through their sum, and the resulting relation between error rate and normalised decision time is parameter-free: decision time peaks at about 20% of the inter-decision interval, at an error rate of about 18%. With unequal priors the optimal starting point is proportional to the log prior odds, which matches the prestimulus LIP activity reported by Platt and Glimcher.
Standing in the record
Filed on 2026-10-03 at the owner’s request, as the paper in the batch that relates the models to each other and to the SPRT. It extends Usher and McClelland (LIT-565): it takes their leaky, competing accumulator, linearised as the “mutual inhibition model”, and proves analytically what they showed by simulation, namely that the difference of the two accumulators is an Ornstein–Uhlenbeck process that becomes the diffusion model when leak equals inhibition. It adds what they lacked: the balanced model is optimal only if leak and inhibition are also large, and then it is optimal by every reward criterion it considers. It also answers one of their arguments. Usher and McClelland cited people’s imperfect accuracy at long times as a challenge to the DDM; Bogacz et al. reply (footnote 10) that Ratcliff’s 1988 bounded diffusion explains it. They state but do not develop that reply, so it is not declared as a correction.
Ratcliff and McKoon (LIT-576), writing later, object that this paper’s account of threshold setting cannot explain humans who calibrate from verbal instructions or without feedback.
For the record, this is the formal sense in which “integration” is a word
about optimality and not only about mechanism. A system that compares
evidence for competing options by accumulating the difference is doing
statistically the best that can be done, and the paper’s last section
recasts cognitive control as the tuning of drift, starting point and threshold
to maximise a utility function. That gives behavioral-integration a
normative standard. It does not give a theory of how an organism’s many
systems are integrated: the analysis is of one decision between two options.
Its multi-alternative extension (the MSPRT, which Bogacz and Gurney map onto
cortex–basal ganglia circuits) is pointed to, not developed.
It is not a model of emotion or motive, and it does not bear on the record’s theory of competing motive states (THEORY-056) beyond the formal analogy noted for LIT-565. No instruction for machine-learning practice; nothing here belongs in the anthology.