Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany.
Bernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany.
Phys Rev E. 2020 Aug;102(2-1):022407. doi: 10.1103/PhysRevE.102.022407.
The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects and nonstationary responses of the population activity to rapidly changing stimuli. Here we derive low-dimensional firing-rate models for homogeneous populations of neurons modeled as time-dependent renewal processes. The class of renewal neurons includes integrate-and-fire models driven by white noise and has been frequently used to model neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues characterizing the timescales of the firing rate dynamics and the Laplace transform of the interspike interval density, for which explicit expressions are available for many renewal models. Retaining only the first eigenmode already yields a reliable low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness and other renewal models that admit an explicit analytical calculation of the eigenvalues. The eigenmode expansion presented here provides a systematic framework for alternative firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.
使用低维发放率或神经群体模型，可以对大量神经元的宏观动力学进行数学分析。然而，这些模型无法捕捉到尖峰同步效应以及群体活动对快速变化的刺激的非平稳响应。在这里，我们为同质神经元群体推导了低维发放率模型，这些神经元被建模为依赖时间的更新过程。更新神经元的类别包括由白噪声驱动的积分-触发模型，并且经常被用于模拟神经元不应期和尖峰同步动力学。该推导基于相关不应期密度方程的本征模展开，它将先前的福克-普朗克方程的谱方法推广到任意更新模型。我们发现了表征发放率动力学时间尺度的特征值与尖峰间隔密度的拉普拉斯变换之间的简单关系，对于许多更新模型，都有其明确的表达式。仅保留第一个本征模就可以得到发放率动力学的可靠低维近似，该近似可以捕捉到尖峰同步效应以及刺激起始时的快速瞬态动力学。我们明确地证明了我们的模型对于具有绝对不应期的大同质泊松神经元群体以及其他允许明确计算特征值的更新模型的有效性。这里提出的本征模展开为基于具有不应期的尖峰神经元动力学的计算神经科学中的替代发放率模型提供了一个系统框架。
