Marpeau Fabien, Barua Aditya, Josić Kresimir
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA.
J Comput Neurosci. 2009 Jun;26(3):445-57. doi: 10.1007/s10827-008-0121-7. Epub 2008 Dec 9.
The stochastic integrate and fire neuron is one of the most commonly used stochastic models in neuroscience. Although some cases are analytically tractable, a full analysis typically calls for numerical simulations. We present a fast and accurate finite volume method to approximate the solution of the associated Fokker-Planck equation. The discretization of the boundary conditions offers a particular challenge, as standard operator splitting approaches cannot be applied without modification. We demonstrate the method using stationary and time dependent inputs, and compare them with Monte Carlo simulations. Such simulations are relatively easy to implement, but can suffer from convergence difficulties and long run times. In comparison, our method offers improved accuracy, and decreases computation times by several orders of magnitude. The method can easily be extended to two and three dimensional Fokker-Planck equations.
随机积分发放神经元是神经科学中最常用的随机模型之一。虽然有些情况可以进行解析处理，但全面分析通常需要进行数值模拟。我们提出了一种快速且准确的有限体积法来近似相关福克 - 普朗克方程的解。边界条件的离散化带来了特殊挑战，因为标准的算子分裂方法未经修改无法应用。我们使用平稳和时变输入来演示该方法，并将其与蒙特卡罗模拟进行比较。这种模拟相对容易实现，但可能存在收敛困难和运行时间长的问题。相比之下，我们的方法提高了精度，并将计算时间减少了几个数量级。该方法可以轻松扩展到二维和三维福克 - 普朗克方程。
