The stochastic FitzHugh-Nagumo (FHN) model considered here is a two-dimensional nonlinear stochastic differential equation with additive degenerate noise, whose first component, the only one observed, describes the membrane voltage evolution of a single neuron. Due to its low dimensionality, its analytical and numerical tractability, and its neuronal interpretation, it has been used as a case study to test the performance of different statistical methods in estimating the underlying model parameters. Existing methods, however, often require complete observations, non-degeneracy of the noise or a complex architecture (e.g., to estimate the transition density of the process, "recovering" the unobserved second component), and they may not (satisfactorily) estimate all model parameters simultaneously. Moreover, these studies lack real data applications for the stochastic FHN model. Here, we tackle all challenges (non-globally Lipschitz drift, non-explicit solution, lack of available transition density, degeneracy of the noise, and partial observations) via an intuitive and easy-to-implement sequential Monte Carlo approximate Bayesian computation algorithm. The proposed method relies on a recent computationally efficient and structure-preserving numerical splitting scheme for synthetic data generation, and on summary statistics exploiting the structural properties of the process. We succeed in estimating all model parameters from simulated data and, more remarkably, real action potential data of rats. The presented novel real-data fit may broaden the scope and credibility of this classic and widely used neuronal model.

翻译：本文所考虑的随机FitzHugh-Nagumo（FHN）模型是一个具有加性退化噪声的二维非线性随机微分方程，其第一个分量（也是唯一被观测的分量）描述了单个神经元的膜电压演化。得益于其低维性、解析与数值易处理性以及神经科学解释，该模型常被用作案例研究，以检验不同统计方法在估计底层模型参数方面的性能。然而，现有方法通常需要完整观测数据、噪声的非退化性，或复杂的架构（例如，为估计过程的转移密度而“重构”未观测的第二个分量），且可能无法（令人满意地）同时估计所有模型参数。此外，现有研究缺乏针对随机FHN模型的真实数据应用。在此，我们通过一种直观且易于实现的序贯蒙特卡洛近似贝叶斯计算算法，应对所有挑战（非全局Lipschitz漂移项、无显式解、转移密度不可得、噪声退化性以及部分观测）。所提方法依赖于一种近期提出的、计算高效且保持结构特性的数值分裂方案来生成合成数据，并利用过程的结构特性构建摘要统计量。我们成功从模拟数据中估计出所有模型参数，更显著的是，亦从大鼠的真实动作电位数据中实现了参数估计。这一新颖的真实数据拟合结果有望拓展这一经典且广泛使用的神经元模型的应用范围与可信度。