Formal model

Example of ten evidence accumulation sequences for the DDM, where the true result is assigned to the upper threshold. Due to the addition of noise, two sequences have produced an inaccurate decision.

The accumulation of evidence in the DDM is governed according to the following formula:

${\displaystyle dx=Adt+cdW\ ,\ x(0)=0}$^[8]

At time zero, the evidence accumulated, x, is set equal to zero. At each time step, some evidence, A, is accumulated for one of the two possibilities in the 2AFC. A is positive if the correct response is represented by the upper threshold, and negative if the lower. In addition, a noise term, cdW, is added to represent noise in input. On average, the noise will integrate to zero.^[8] The extended DDM^[14] allows for selection of ${\displaystyle A}$ and the starting value of ${\displaystyle x(0)}$ from separate distributions – this provides a better fit to experimental data for both accuracy and reaction times.^[22][23]

Ornstein–Uhlenbeck model

The Ornstein–Uhlenbeck model^[15] extends the DDM by adding another term, ${\displaystyle \lambda }$, to the accumulation that is dependent on the current accumulation of evidence – this has the net effect of increasing the rate of accumulation towards the initially preferred option.

${\displaystyle dx\ =\ (\lambda x+A)dt\ +\ cdW}$^[8]

Race model

In the race model,^[12][13][24] evidence for each alternative is accumulated separately, and a decision made either when one of the accumulators reaches a predetermined threshold, or when a decision is forced and then the decision associated with the accumulator with the highest evidence is chosen. This can be represented formally by:

${\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ I_{\text{1}}dt\ +\ cdW_{\text{1}}\\dy_{\text{2}}\ =\ I_{\text{2}}dt\ +\ cdW_{\text{2}}\end{aligned}},\quad y_{\text{1}}(0)\ =\ y_{\text{2}}(0)=0}$^[8]

The race model is not mathematically reducible to the DDM,^[8] and hence cannot be used to implement an optimal decision procedure.

Mutual inhibition model

The mutual inhibition model^[17] also uses two accumulators to model the accumulation of evidence, as with the race model. In this model the two accumulators have an inhibitory effect on each other, so as evidence is accumulated in one, it dampens the accumulation of evidence in the other. In addition, leaky accumulators are used, so that over time evidence accumulated decays – this helps to prevent runaway accumulation towards one alternative based on a short run of evidence in one direction. Formally, this can be shown as:

${\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ (-ky_{\text{1}}-wy_{\text{2}}+I_{\text{1}})dt\ +\ cdW_{\text{1}}\\dy_{\text{2}}\ =\ (-ky_{\text{2}}-wy_{\text{1}}+I_{\text{2}})dt\ +\ cdW_{\text{2}}\end{aligned}},\quad y_{\text{1}}(0)\ =\ y_{\text{2}}(0)=0}$^[8]

Where ${\displaystyle k}$ is a shared decay rate of the accumulators, and ${\displaystyle w}$ is the rate of mutual inhibition.

Feedforward inhibition model

The feedforward inhibition model^[25] is similar to the mutual inhibition model, but instead of being inhibited by the current value of the other accumulator, each accumulator is inhibited by a fraction of the input to the other. It can be formally stated thus:

${\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ I_{\text{1}}dt\ +\ cdW_{\text{1}}\ -\ u(I_{\text{2}}dt\ +\ cdW_{\text{2}})\\dy_{\text{2}}\ =\ I_{\text{2}}dt\ +\ cdW_{\text{2}}\ -\ u(I_{\text{1}}dt\ +\ cdW_{\text{1}})\end{aligned}},\quad y_{\text{1}}(0)\ =\ y_{\text{2}}(0)=0}$^[8]

Where ${\displaystyle u}$ is the fraction of accumulator input that inhibits the alternate accumulator.

Pooled inhibition model

Wang^[26] suggested the pooled inhibition model, where a third, decaying accumulator is driven by accumulation in both of the accumulators used for decision making, and in addition to the decay used in the mutual inhibition model, each of the decision driving accumulators self-reinforce based on their current value. It can be formally stated thus:

${\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ (-ky_{\text{1}}-wy_{\text{3}}+vy_{\text{1}}+I_{\text{1}})dt\ +\ cdW_{\text{1}}\\dy_{\text{2}}\ =\ (-ky_{\text{2}}-wy_{\text{3}}+vy_{\text{2}}+I_{\text{2}})dt\ +\ cdW_{\text{2}}\\dy_{\text{3}}\ =\ (-k_{\text{inh}}y_{\text{3}}+w'(y_{\text{1}}+y_{\text{2}}))dt\end{aligned}}}$^[8]

The third accumulator has an independent decay coefficient, ${\displaystyle k_{\text{inh}}}$, and increases based on the current values of the other two accumulators, at a rate modulated by ${\displaystyle w'}$.