At each timepoint, the accumulator memory a (black trace) represents an estimate of the ‘Right’ vs ‘Left’ evidence accrued so far. At stimulus end, the model decides ‘Right’ if a > Þ, the decision boundary, and ‘Left’ otherwise, where Þ is a free parameter. Light grey traces indicate alternate runs with different instantiase.
a The decision variable. Right ↑ (left ↓) pulses change the value of a by positive (negative) impulses of magnitude C.
parameterizes noise in the initial value of a.
a diffusion constant, parameterizing noise in a.
parameterizes noise when adding the evidence from a Right or Left pulse: variance is added to the amplitude C of the evidence contributed by each click.
λ parameterizes consistent drift in the memory a. In the ‘leaky’ or forgetful case (λ < 0, illustrated), drift is towards a = 0, and later pulses impact the decision more than earlier pulses. In the ‘unstable’ or impulsive case (λ > 0), drift is away from a = 0, and earlier pulses impact the decision more than later pulses. The memory's time constant τ = 1/λ.
B the height of the ‘sticky’ decision bounds and parameterizes the amount of evidence necessary to commit to a decision.
φ, τϕ parameterize sensory adaptation by defining the dynamics of C. Immediately after a click, the magnitude C is multiplied by φ. C then recovers towards an unadapted value of 1 with time constant τϕ. Facilitation is thus represented by ϕ > 1, while depression is represented by ϕ < 1 (inset).
Þ the decision boundary. These properties are implemented by the following equations: if |a| ≥ B then da/dt = 0; else
δt,tR,L are delta functions at the times of the pulses.
η are i.i.d. gaussian variables drawn from .
dW is a white noise Wiener process.
The initial condition a(t = 0) is drawn from the gaussian .
Adaptation dynamics are given by:
In addition, a lapse rate parameterizes the fraction of trials on which a random response is made.
Ideal performance (a = #right clicks − #left clicks) would be achieved by Þ = 0.
© 2013 AAAS. All Rights Reserved. Figure 2—figure supplement 1 and legend text reproduced from Brunton BW, Botvinick MM, Brody CD. 2013. Rats and humans can optimally accumulate evidence for decision-making. Science 340, 95–98. doi:10.1126/science.1233912. Reprinted with permission.