Sun Yi, Zhou Douglas, Rangan Aaditya V, Cai David
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
J Comput Neurosci. 2009 Dec;27(3):369-90. doi: 10.1007/s10827-009-0151-9. Epub 2009 Apr 29.
We present an efficient library-based numerical method for simulating the Hodgkin-Huxley (HH) neuronal networks. The key components in our numerical method involve (i) a pre-computed high resolution data library which contains typical neuronal trajectories (i.e., the time-courses of membrane potential and gating variables) during the interval of an action potential (spike), thus allowing us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations; (ii) an algorithm of spike-spike corrections within the groups of strongly coupled neurons to account for spike-spike interactions in a single large time step. By using the library method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity, such as average firing rate, interspike interval distribution, power spectra of voltage traces. Moreover, our large time steps using the library method can break the stability requirement of standard methods (such as Runge-Kutta (RK) methods) for the original dynamics. We compare our library-based method with RK methods, and find that our method can capture very well phase-locked, synchronous, and chaotic dynamics of HH neuronal networks. It is important to point out that, in essence, our library-based HH neuron solver can be viewed as a numerical reduction of the HH neuron to an integrate-and-fire (I&F) neuronal representation that does not sacrifice the gating dynamics (as normally done in the analytical reduction to an I&F neuron).
我们提出了一种基于库的高效数值方法来模拟霍奇金-赫胥黎（HH）神经元网络。我们数值方法的关键组成部分包括：（i）一个预先计算的高分辨率数据库，其中包含动作电位（尖峰）间隔期间的典型神经元轨迹（即膜电位和门控变量的时间进程），从而使我们能够避免详细解析尖峰，并使用较大的数值时间步长来演化HH神经元方程；（ii）一种在强耦合神经元组内进行尖峰-尖峰校正的算法，以在单个大时间步长中考虑尖峰-尖峰相互作用。通过使用库方法，我们可以使用比不使用该库解析轨迹时使用的典型时间步长大一个数量级的时间步长来演化HH网络，同时在网络活动的统计量化方面实现可比的分辨率，例如平均放电率、峰峰间隔分布、电压轨迹的功率谱。此外，我们使用库方法的大时间步长可以打破标准方法（如龙格-库塔（RK）方法）对原始动力学的稳定性要求。我们将基于库的方法与RK方法进行比较，发现我们的方法能够很好地捕捉HH神经元网络的锁相、同步和混沌动力学。需要指出的是，从本质上讲，我们基于库的HH神经元求解器可以被视为将HH神经元数值简化为积分发放（I&F）神经元表示，而不会牺牲门控动力学（这与通常在解析简化为I&F神经元时所做的不同）。
