Tóth T I, Crunelli V
Physiology Unit, Cardiff School of Biosciences, University of Wales Cardiff, PO Box 911, Cardiff, UK.
J Neurosci Methods. 1999 Mar 1;87(2):119-36. doi: 10.1016/s0165-0270(98)00153-8.
The propagation of excitation along the dendrites and the axon of a neurone is described by a partial differential equation which is nonlinear when voltage-gated conductances are present. In this case, numerical methods are employed to obtain a solution: the evolution of the membrane potential in space and time. Even when the membrane is passive (linear), numerical methods might still be preferred to analytical ones that are often too cumbersome to obtain. In this paper, we present the Chebyshev pseudospectral or collocation method as an alternative to the hitherto commonly used finite difference schemes (compartmental models) that are based on sufficiently fine equidistant subdivisions of the spatial structure (dendrites or axon). In the Chebyshev method, solutions are approximated by finite Chebyshev series. The solutions have uniform, usually high, numerical accuracy at any spatial point, not only at the original collocation points. Often, truncation errors become negligible, hence, the total error is essentially the rounding error of the computations. Furthermore, quantities involving spatial derivatives, and in particular the axial current, can be computed exactly from the solution, i.e. the membrane potential. Space-dependent parameter distributions (channel densities, non-uniform dendritic geometries), as well as mixed linear boundary conditions can easily be implemented, and can be chosen from the large class of piecewise smooth functions.
神经元树突和轴突上兴奋的传播由一个偏微分方程描述，当存在电压门控电导时，该方程是非线性的。在这种情况下，采用数值方法来获得解：即膜电位在空间和时间上的演化。即使膜是无源的（线性的），数值方法可能仍然比解析方法更受青睐，因为解析方法通常过于繁琐而难以获得。在本文中，我们提出了切比雪夫伪谱或配置法，作为迄今为止常用的有限差分格式（房室模型）的替代方法，这些格式基于空间结构（树突或轴突）足够精细的等距细分。在切比雪夫方法中，解由有限切比雪夫级数近似。这些解在任何空间点都具有均匀的、通常很高的数值精度，而不仅仅是在原始配置点。通常，截断误差可以忽略不计，因此，总误差基本上是计算的舍入误差。此外，涉及空间导数的量，特别是轴向电流，可以从解（即膜电位）精确计算出来。与空间相关的参数分布（通道密度、非均匀树突几何形状）以及混合线性边界条件可以很容易地实现，并且可以从一大类分段光滑函数中选择。
