Rangan Aaditya V, Cai David
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
J Comput Neurosci. 2007 Feb;22(1):81-100. doi: 10.1007/s10827-006-8526-7. Epub 2006 Jul 28.
We discuss numerical methods for simulating large-scale, integrate-and-fire (I&F) neuronal networks. Important elements in our numerical methods are (i) a neurophysiologically inspired integrating factor which casts the solution as a numerically tractable integral equation, and allows us to obtain stable and accurate individual neuronal trajectories (i.e., voltage and conductance time-courses) even when the I&F neuronal equations are stiff, such as in strongly fluctuating, high-conductance states; (ii) an iterated process of spike-spike corrections within groups of strongly coupled neurons to account for spike-spike interactions within a single large numerical time-step; and (iii) a clustering procedure of firing events in the network to take advantage of localized architectures, such as spatial scales of strong local interactions, which are often present in large-scale computational models-for example, those of the primary visual cortex. (We note that the spike-spike corrections in our methods are more involved than the correction of single neuron spike-time via a polynomial interpolation as in the modified Runge-Kutta methods commonly used in simulations of I&F neuronal networks.) Our methods can evolve networks with relatively strong local interactions in an asymptotically optimal way such that each neuron fires approximately once in [Formula: see text] operations, where N is the number of neurons in the system. We note that quantifications used in computational modeling are often statistical, since measurements in a real experiment to characterize physiological systems are typically statistical, such as firing rate, interspike interval distributions, and spike-triggered voltage distributions. We emphasize that it takes much less computational effort to resolve statistical properties of certain I&F neuronal networks than to fully resolve trajectories of each and every neuron within the system. For networks operating in realistic dynamical regimes, such as strongly fluctuating, high-conductance states, our methods are designed to achieve statistical accuracy when very large time-steps are used. Moreover, our methods can also achieve trajectory-wise accuracy when small time-steps are used.
我们讨论用于模拟大规模积分发放（I&F）神经网络的数值方法。我们数值方法中的重要元素包括：（i）一种受神经生理学启发的积分因子，它将解转化为一个数值上易于处理的积分方程，即使在I&F神经元方程刚性很强的情况下，比如在强烈波动的高电导状态下，也能让我们获得稳定且准确的单个神经元轨迹（即电压和电导随时间的变化过程）；（ii）在强耦合神经元组内进行的尖峰 - 尖峰校正的迭代过程，以考虑在单个大数值时间步长内的尖峰 - 尖峰相互作用；（iii）网络中放电事件的聚类过程，以利用局部化架构，例如强局部相互作用的空间尺度，这在大规模计算模型中经常出现——例如初级视觉皮层的模型。（我们注意到，我们方法中的尖峰 - 尖峰校正比在I&F神经网络模拟中常用的改进龙格 - 库塔方法中通过多项式插值对单个神经元尖峰时间的校正更为复杂。）我们的方法能够以渐近最优的方式演化具有相对强局部相互作用的网络，使得每个神经元在[公式：见原文]次操作中大约放电一次，其中N是系统中的神经元数量。我们注意到，计算建模中使用的量化通常是统计性的，因为在表征生理系统的实际实验中的测量通常是统计性的，比如放电率、峰峰间隔分布和峰触发电压分布。我们强调，解析某些I&F神经网络的统计特性所花费的计算量比完全解析系统内每个神经元的轨迹要少得多。对于处于现实动态状态的网络，比如强烈波动的高电导状态，我们的方法旨在在使用非常大的时间步长时实现统计精度。此外，当使用小时间步长时，我们的方法也能实现逐轨迹精度。
