Drift Diffusion Model

What is drift diffusion model

The drift diffusion model (DDM) is an extension of the Diffusion Model, specifically tailored for modeling decision-making processes.
It adds a drift component to the diffusion process, representing a systematic bias or tendency towards one decision or another.

mathematical representation

d x (t) = v d t + \sqrt{D} d W (t)

where:

$x (t)$ is the state variable (e.g., accumulated evidence)
$D$ is the diffusion coefficient
$W (t)$ is a Wiener process (representing random noise)
$v$ is the drift rate (the average rate of evidence accumulation), which determines the direction and speed of the evidence accumulation, influencing how quickly a decision is made

key feature

drift:
- the model includes a drift term that represents the systematic accumulation of evidence in favor of one decision over another
both stochastic and deterministic:
- the model combines both random fluctuations (diffusion) and systematic tendencies (drift)

Application of drift diffusion models

in cognitive science

decision marking
- it can model binary decision-making tasks where an individual has to choose between two alternatives, and the decision is made when it reaches a pre-defined threshold

Understand drift diffusion model with Sequential Probability Ratio Test

The log likelihood ratio at a time step ( $L_{T}$ ) will equal the ratio at the previous time step ( $L_{T - 1}$ ) plus the ratio for the measurement at that time step, given by $Δ_{T}$ :

\begin{array}{r} L_{T} = L_{T - 1} + Δ_{T} \end{array}

Adding $Δ_{t}$ over time gives

L_{T} \sim N (2 \frac{μ^{2}}{σ^{2}} T, 4 \frac{μ^{2}}{σ^{2}} T) = N (b T, c^{2} T)

as claimed.

The log-likelihood ratio $L_{t}$ is a biased random walk --- normally distributed with a time-dependent mean and variance. This is the Drift Diffusion Model.

The $b$ is a drift component, and the $c$ is a diffusion component.