Source
- Bogacz, Brown, Moehlis, Holmes & Cohen (2006), LIT-569, read in NOTE-461: the SPRT and the DDM (pp. 703–704), the reductions of the mutual inhibition model (Eqs. 18–26, Figs. 4–10) and Table 1. Appendix A, where the proofs are written out, was read only by its section titles.
- Usher & McClelland (2001), LIT-565, read in NOTE-449: the reduction to an Ornstein–Uhlenbeck process (Eqs. 5–10), Figure 4, and Experiment 3.
What was actually shown
The standard of optimality. Bogacz et al. (LIT-569) restate the classical theorems. The sequential probability ratio test (Wald) is the fastest test for a given error rate, and the Neyman–Pearson test is the most accurate for a given time. Both accumulate the log likelihood ratio, and in the continuum limit that is a drift diffusion process. So a two-choice decider that integrates the difference of the evidence for the two options, perfectly and from a fixed start, is doing the best that can be done.
The network and its difference process. Usher and McClelland (LIT-565) give each alternative a leaky accumulator and let each inhibit the other. Linearised, the difference x₁ − x₂ obeys dx = (v − Kx) dt/τ + noise, with K = k − β, leak minus inhibition (their Eqs. 5–8). K = 0 is the classical diffusion model. For K ≠ 0 the discriminability under unlimited time is finite, d_asy = (2v/σ)√(1/K) (Eq. 10). Bogacz et al. derive the same decomposition in rotated coordinates, writing λ = w − k (inhibition minus leak, the opposite sign). The difference is an O-U process with rate λ, and the sum is a stable O-U process with rate k + w. With k = w the difference is a pure diffusion. With k and w both large, the sum settles fast onto a “decision line” and the two-dimensional network behaves as the one-dimensional DDM.
Which balance is optimal, and where. Bogacz et al. find, for the linear model, that:
- under interrogation (a decision forced at time T), error is minimised at λ = 0, whatever the magnitude of leak and inhibition;
- in free response, at fixed error, decision time is shortest at k = w and falls toward the DDM’s as k = w grows;
- as T → ∞, error goes to zero only for λ = 0. For λ ≠ 0 it has a floor that depends on |λ| alone (Table 1).
Which evidence is weighted. In the linear O-U process an input arriving at time s is carried to the decision at t with weight e^{−K(t−s)}. That is the standard solution of the equation, written here by the record; the sources state its consequence. When leak dominates, old evidence decays and late evidence counts most (recency). When inhibition dominates, an early lead is amplified and early evidence counts most (primacy). Usher and McClelland show by simulation of the full nonlinear model that the time–accuracy curve depends only on |K|, while the trajectories differ (Fig. 4–5). So two networks with equal accuracy can weight the stream oppositely.
The first three findings and the weighting are mathematics about a stated model, which is why this account is Active. What could have come out otherwise is the reduction: the race model never reduces to the DDM (LIT-569), and a network can be built that way.
Empirical contact, not part of the claim. Usher and McClelland’s Experiment 3 (6 participants) found 2 who favoured a late cluster of evidence, 2 who favoured an early one and 2 who were neutral. Fitted leak and inhibition accounted for each pattern, and the neutral participants were the most accurate, as the account predicts. That is consistent with it, on six people, and attentional explanations were not excluded (NOTE-449, C4).
What this does not say
- It does not say brains are balanced. It says what balance buys. Whether a given circuit is balanced is an empirical question this account cannot answer, and Usher and McClelland’s own evidence that leak matters (a likelihood ratio of about 62 over three participants) is modest and contested by drift-variance accounts (LIT-576).
- Two alternatives, stationary evidence, linearised units. The multi-alternative case (the MSPRT) is pointed to and not treated. The threshold-linear floor is dropped. The extended equivalences need the total input to be constant across trials, and Bogacz et al. note there is no cortical evidence for the anticorrelated starting points they also need.
- Imbalance is not always worse in practice. Bogacz et al. report by simulation that with variable starting points a leak-dominant network can do better, because it discounts a noisy start. And in free response, error can still be driven to zero for λ < 0, but only with decision times that diverge. “Bounds accuracy” in the title is about a fixed time.
- Optimality here is statistical, not about what is at stake. Setting the threshold for a utility (reward rate, Bayes risk) is a further step, which Bogacz et al. also take. The companion account THEORY-061 uses it.
What would show it wrong
As mathematics, an error in the derivations. Appendix A of LIT-569 has not been read line by line here, so a first-hand check of the proofs of the O-U error rates and of the free-response optimum would close that gap. The claim would be shown not to apply, rather than wrong, by evidence that the choice circuits it is used to describe do not compare alternatives by mutual inhibition at all.
Connections
- The unity it is about. This is
behavioral-integrationat the scale of one decision: how two candidate responses come down to one. It is not a claim about phenomenal unity or about any self (ADR-024). - THEORY-056 (regulation as one motive state checking another, with no regulator above them). Mutual inhibition between accumulators is a formal case of tendencies that suppress one another with no supervisory unit, and this account says when such suppression is lossless. Neither source is about emotion or motive, so no relation is declared.
- THEORY-059 (the readiness potential) uses the leak-dominant, single-accumulator regime of the same process.
Note of 2026-10-25: what Wald’s paper itself says
Wald’s 1945 paper (LIT-1523, read in NOTE-1165 from a copy the owner supplied) qualifies two things this account takes from LIT-569.
- Optimality. “Fastest for a given error rate” is, in Wald 1945, a theorem only up to the overshoot: no sequential test with the same error probabilities needs fewer observations on average than his lower bounds, which the SPRT attains when the walk stops on its boundaries (§4.7). Exact optimality is his footnote 12’s conjecture, proved later by Wald and Wolfowitz (1948, not held). The simulated ASN lay 0.6–1.5% above the bound at the test’s own realized errors.
- The continuum limit. Wald never says diffusion, Brownian motion or limit. He says the boundaries would be exact “if fractional observations were possible”, and his overshoot-free OC, ASN and distribution of the sample number are exactly those of Brownian motion with drift (THEORY-951, THEORY-952). So the drift-diffusion model is the case in which Wald’s approximations are exact, which is why the limit LIT-569 proves holds. The limit stays sourced through LIT-569.
The account is unchanged.