In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.
翻译:在神经科学中,决定时间的分配模式是单维的Fokker-Planck方程式,具有取决于时间的边界和空间-时间的漂移。需要对这一方程式的解决办法进行有效近似,例如模型评估和参数的安装。然而,规定的边界条件导致强烈的单一性,从而减缓数字近似值的趋同。在本篇文章中,我们证明,解决办法可能与通过转换和减减减已知功能,在具有单一初始和边界条件的矩形空间时域上使用抛射式PDE的解决办法有关。我们核实,新PDE的解决方案确实比原PDE的解决方案更经常,并着手使用空间-时间最低残留法将新的PDE分离。我们还表明,该解决办法取决于确定边界和漂移的参数,因此有理由使用稀有的沙子产品内插法来估计各种参数范围的PDE解决办法。最低残余方法和内插法方法的预测趋同率得到数字模拟的支持。