Eggert J, van Hemmen J L
Physik Department, Technische Universität München, D-85747 Garching bei München, Germany.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Feb;61(2):1855-74. doi: 10.1103/physreve.61.1855.
Starting from single, spiking neurons, we derive a system of coupled differential equations for a description of the dynamics of pools of extensively many equivalent neurons. Contrary to previous work, the derivation is exact and takes into account microscopic properties of single neurons, such as axonal delays and refractory behavior. Simulations show a good quantitative agreement with microscopically modeled pools of spiking neurons. The agreement holds both in the quasistationary and nonstationary dynamical regimes, including fast transients and oscillations. The model is compared with other pool models based on differential equations. It turns out that models of the graded-response category can be understood as a first-order approximation of our pool dynamics. Furthermore, the present formalism gives rise to a system of equations that can be reduced straightforwardly so as to gain a description of the pool dynamics to any desired order of approximation. Finally, we present a stability criterion that is suitable for handling pools of neurons. Due to its exact derivation from single-neuron dynamics, the present model opens simulation possibilities for studies that rely upon biologically realistic large-scale networks composed of assemblies of spiking neurons.
从单个发放脉冲的神经元出发，我们推导了一组耦合微分方程，用于描述大量等效神经元集群的动力学。与之前的工作不同，该推导是精确的，并考虑了单个神经元的微观特性，如轴突延迟和不应期行为。模拟结果表明，与微观建模的发放脉冲神经元集群具有良好的定量一致性。这种一致性在准静态和非静态动力学状态下均成立，包括快速瞬态和振荡。将该模型与基于微分方程的其他集群模型进行了比较。结果表明，分级响应类别的模型可以理解为我们集群动力学的一阶近似。此外，目前的形式体系产生了一个方程组，可以直接简化，从而获得对集群动力学的任意期望近似阶的描述。最后，我们提出了一个适用于处理神经元集群的稳定性判据。由于它是从单个神经元动力学精确推导出来的，本模型为依赖于由发放脉冲神经元集合组成的生物学现实大规模网络的研究开辟了模拟可能性。
