Department of Physiology and Pharmacology, State University of New York, Downstate Medical Center, Brooklyn, New York, USA.
Neural Comput. 2010 Aug;22(8):2113-36. doi: 10.1162/neco.2010.09-09-1097.
We use high-order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions, where available, and grid convergence. The space derivatives are numerically approximated by means of differentiation matrices. Nonlinearity in the equation arises from the Hodgkin-Huxley dynamics of the gating variables for ion channels. We have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing. A flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak, which furthermore leads, for the passive cable, to a simple, exact solution. We have used three distinct approaches to assess the numerical solutions: comparison with exact solutions in an unbranched passive cable, the convergence of solutions with progressive refinement of the grid in an active cable, and the simulation of spike initiation in a biophysically realistic single-neuron model. The spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second-order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space. It provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes.
我们在非线性电缆方程中使用高阶逼近方案来处理空间导数，并通过使用精确解（如果有的话）和网格收敛来研究数值解误差的行为。空间导数通过差分矩阵进行数值逼近。方程中的非线性来自离子通道门控变量的 Hodgkin-Huxley 动力学。我们特别研究了突触电流分布的影响，并比较了谱解与有限差分的准确性。使用了一种灵活的注入电流形式，可以从非常宽的峰值平滑调整到非常窄的峰值，这对于无源电缆来说导致了一个简单的精确解。我们使用了三种不同的方法来评估数值解：与无分支无源电缆中的精确解进行比较，在有源电缆中随着网格的逐步细化来评估解的收敛性，以及在生物物理上逼真的单个神经元模型中模拟尖峰的起始。谱方法为无源电缆提供了良好的数值解，其准确性可与二阶有限差分法相媲美，并且在由空间上平滑分布的输入驱动的模拟系统中具有更高的准确性。由于更好地逼近传播尖峰，它在有源电缆和真实神经元模型中提供了更快的收敛速度。
