Neural mass models provide a useful framework for modelling mesoscopic neural dynamics and in this article we consider the Jansen and Rit neural mass model (JR-NMM). We formulate a stochastic version of it which arises by incorporating random input and has the structure of a damped stochastic Hamiltonian system with nonlinear displacement. We then investigate path properties and moment bounds of the model. Moreover, we study the asymptotic behaviour of the model and provide long-time stability results by establishing the geometric ergodicity of the system, which means that the system—independently of the initial values—always converges to an invariant measure. In the last part, we simulate the stochastic JR-NMM by an efficient numerical scheme based on a splitting approach which preserves the qualitative behaviour of the solution.

中文翻译：

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Jansen和Rit神经质量模型的随机版本：分析和数值。

神经质量模型为建模介观神经动力学提供了有用的框架，在本文中，我们考虑了Jansen和Rit神经质量模型（JR-NMM）。我们制定了它的随机形式，它是通过合并随机输入而产生的，它具有带非线性位移的阻尼随机哈密顿系统的结构。然后，我们研究路径属性和模型的矩边界。此外，我们通过建立系统的几何遍历性来研究模型的渐近行为并提供长期稳定性结果，这意味着系统（独立于初始值）始终收敛于不变测度。在最后一部分中，我们通过基于拆分方法的有效数值方案来模拟随机JR-NMM，该方法保留了解的定性行为。