Mathematics for Real-World Systems Centre for Doctoral Training, University of Warwick, Coventry, United Kingdom.
Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, Berlin, Germany.
PLoS Comput Biol. 2020 Apr 20;16(4):e1007175. doi: 10.1371/journal.pcbi.1007175. eCollection 2020 Apr.
Analytical forms for neuronal firing rates are important theoretical tools for the analysis of network states. Since the 1960s, the majority of approaches have treated neurons as being electrically compact and therefore isopotential. These approaches have yielded considerable insight into how single-cell properties affect network activity; however, many neuronal classes, such as cortical pyramidal cells, are electrically extended objects. Calculation of the complex flow of electrical activity driven by stochastic spatio-temporal synaptic input streams in these structures has presented a significant analytical challenge. Here we demonstrate that an extension of the level-crossing method of Rice, previously used for compact cells, provides a general framework for approximating the firing rate of neurons with spatial structure. Even for simple models, the analytical approximations derived demonstrate a surprising richness including: independence of the firing rate to the electrotonic length for certain models, but with a form distinct to the point-like leaky integrate-and-fire model; a non-monotonic dependence of the firing rate on the number of dendrites receiving synaptic drive; a significant effect of the axonal and somatic load on the firing rate; and the role that the trigger position on the axon for spike initiation has on firing properties. The approach necessitates only calculating the mean and variances of the non-thresholded voltage and its rate of change in neuronal structures subject to spatio-temporal synaptic fluctuations. The combination of simplicity and generality promises a framework that can be built upon to incorporate increasing levels of biophysical detail and extend beyond the low-rate firing limit treated in this paper.
神经元发放率的解析形式是分析网络状态的重要理论工具。自 20 世纪 60 年代以来，大多数方法都将神经元视为电紧凑的，因此是等电位的。这些方法为研究单细胞特性如何影响网络活动提供了重要的见解；然而，许多神经元类型，如皮质锥体细胞，是电延伸的物体。在这些结构中，计算由随机时空突触输入流驱动的复杂电活动流动，提出了一个重大的分析挑战。在这里，我们证明了 Rice 的越界方法的扩展，以前用于紧凑的细胞，为具有空间结构的神经元的发放率提供了一个通用的框架。即使对于简单的模型，所得到的解析近似也显示出惊人的丰富性，包括：对于某些模型，发放率与电紧张长度无关，但形式与点状漏电积分和放电模型不同；发放率与接收突触驱动的树突数量呈非单调关系；轴突和体细胞负荷对发放率有显著影响；以及触发位置对轴突起始放电的影响。该方法只需要计算受时空突触波动影响的神经元结构中非阈值电压及其变化率的均值和方差。该方法的简单性和通用性相结合，有望成为一个可以进一步构建的框架，以纳入越来越多的生物物理细节，并扩展到本文处理的低发放率限制之外。
