Tian Zhong-Qi Kyle, Zhou Douglas
School of Mathematical Sciences, MOE-LSC, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China.
Front Comput Neurosci. 2020 May 8;14:40. doi: 10.3389/fncom.2020.00040. eCollection 2020.
The exponential time differencing (ETD) method allows using a large time step to efficiently evolve stiff systems such as Hodgkin-Huxley (HH) neural networks. For pulse-coupled HH networks, the synaptic spike times cannot be predetermined and are convoluted with neuron's trajectory itself. This presents a challenging issue for the design of an efficient numerical simulation algorithm. The stiffness in the HH equations are quite different, for example, between the spike and non-spike regions. Here, we design a second-order adaptive exponential time differencing algorithm (AETD2) for the numerical evolution of HH neural networks. Compared with the regular second-order Runge-Kutta method (RK2), our AETD2 method can use time steps one order of magnitude larger and improve computational efficiency more than ten times while excellently capturing accurate traces of membrane potentials of HH neurons. This high accuracy and efficiency can be robustly obtained and do not depend on the dynamical regimes, connectivity structure or the network size.
指数时间差分（ETD）方法允许使用大时间步长来有效地演化诸如霍奇金-赫胥黎（HH）神经网络之类的刚性系统。对于脉冲耦合的HH网络，突触尖峰时间无法预先确定，并且与神经元自身的轨迹相互卷积。这给高效数值模拟算法的设计带来了一个具有挑战性的问题。例如，在尖峰区域和非尖峰区域之间，HH方程中的刚性差异很大。在此，我们设计了一种用于HH神经网络数值演化的二阶自适应指数时间差分算法（AETD2）。与常规的二阶龙格-库塔方法（RK2）相比，我们的AETD2方法可以使用大一个数量级的时间步长，并将计算效率提高十多倍，同时能出色地捕捉HH神经元膜电位的精确轨迹。这种高精度和高效率能够稳健地实现，且不依赖于动力学状态、连接结构或网络规模。
