Abstract
Diffusion models with constant boundaries and constant drift function have been successfully applied to model phenomena in a wide range of areas in psychology. In recent years, more complex models with time-dependent boundaries and space-time-dependent drift functions have gained popularity. One obstacle to the empirical and theoretical evaluation of these models is the lack of simple and efficient numerical algorithms for computing their first-passage time distributions. In the present work we use a known series expansion for the first-passage time distribution for models with constant drift function and constant boundaries to simplify the Kolmogorov backward equation for models with time-dependent boundaries and space-time-dependent drift functions. We show how a simple Crank–Nicolson scheme can be used to efficiently solve the simplified equation.
Highlights
Diffusion models have been introduced in psychology to account for behavioural data from two-alternative forced choice tasks
Similar to the Ornstein–Uhlenbeck model, the approximation EN is visually indistinguishable from the high-precision approximation for N > 2
Similar to the previous two example models, the approximation EN is visually indistinguishable from the high-precision approximation for all
Summary
Diffusion models have been introduced in psychology to account for behavioural data from two-alternative forced choice tasks. The matrix method is equivalent to a Euler-forward/finite difference approximation of the solution of a Fokker–Planck equation with homogeneous boundary conditions and a Dirac delta function as initial data This reveals the problems involved in this approach: firstly, the wild behaviour of the initial data leads to large errors in the calculated transition probabilities. By solving multiple Fokker–Planck PDEs we obtain the distribution function of the first passage time for a range of time points Both Fokker–Planck equations have properties that inhibit efficient numerical approximation: in the matrix method, the initial data is ill-behaved, in the method of Voss and Voss (2008), one has a discontinuity in the initialand boundary conditions. For the sake of clarity of the exposition, and as the numerical experiments suggest that both methods behave we restrict ourselves to the Crank–Nicolson method here and refrain from discussing the results in Boehm et al (2021) in further detail
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