First passage time models describe the time it takes for a random process to exit a region of interest and are widely used across various scientific fields. Fast and accurate numerical methods for computing the likelihood function in these models are essential for efficient statistical inference of model parameters. Specifically, in computational cognitive neuroscience, generalized drift diffusion models (GDDMs) are an important class of first passage time models that describe the latent psychological processes underlying simple decision-making scenarios. GDDMs model the joint distribution over choices and response times as the first hitting time of a one-dimensional stochastic differential equation (SDE) to possibly time-varying upper and lower boundaries. They are widely applied to extract parameters associated with distinct cognitive and neural mechanisms. However, current likelihood computation methods struggle in common application scenarios in which drift rates dynamically vary within trials as a function of exogenous covariates (e.g., brain activity in specific regions or visual fixations). In this work, we propose a fast and flexible algorithm for computing the likelihood function of GDDMs based on a large class of SDEs satisfying the Cherkasov condition. Our method divides each trial into discrete stages, employs fast analytical results to compute stage-wise densities, and integrates these to compute the overall trial-wise likelihood. Numerical examples demonstrate that our method not only yields accurate likelihood evaluations for efficient statistical inference, but also considerably outperforms existing approaches in terms of speed.

翻译：首达时模型描述了随机过程离开感兴趣区域所需的时间，广泛应用于多个科学领域。在这些模型中，快速且精确地计算似然函数的数值方法对于模型参数的高效统计推断至关重要。具体而言，在计算认知神经科学中，广义漂移扩散模型（GDDMs）是一类重要的首达时模型，用于描述简单决策场景背后的潜在心理过程。GDDMs将选择与反应时间的联合分布建模为一维随机微分方程（SDE）首次到达可能随时间变化的上、下边界的时间。这些模型被广泛应用于提取与不同认知及神经机制相关的参数。然而，当前的似然计算方法在常见应用场景中面临困难，这些场景中漂移率在试验内随外生协变量（如特定脑区活动或视觉注视）动态变化。本研究提出了一种基于满足Cherkasov条件的大类SDE的快速灵活算法，用于计算GDDMs的似然函数。我们的方法将每个试验划分为离散阶段，采用快速解析结果计算阶段内密度，并通过积分这些密度得到整个试验的似然。数值示例表明，该方法不仅能为高效统计推断提供精确的似然评估，而且在速度上显著优于现有方法。