Nonsmooth dynamics in spiking neuron models
Nonsmooth dynamics in spiking neuron models
复制标题
DOI：
10.1016/j.physd.2011.05.012
复制
发表时间：
2012-11-15
影响因子：
4
通讯作者：
Wedgwood, K. C. A.
中科院分区：
文献类型：
--
作者：
Coombes, S.;Thul, R.;Wedgwood, K. C. A.
对尖峰神经网络的大规模研究是理解生物神经组织动力学的现代方法的关键部分。计算神经科学中的一种方法是考虑神经元的详细电生理特性，并建立大量的计算区段模型。另一种选择是开发最小的尖峰神经元模型，同时降低参数和变量空间的维度，以促进更有效的模拟研究。在后一种情况下，选择的单神经元模型通常是由非光滑动力系统描述的经典积分-点火模型的变体。在本文中，我们回顾了这一类中一些比较流行的尖峰模式，并描述了它们可以产生的尖峰模式的类型(从强直到爆发式激发)。我们表明，最初为研究碰撞振子而开发的一些技术与它们的分析直接相关，特别是那些用于处理掠过分叉的技术。重要的是，我们强调了一个特定的单一神经元模型，能够产生真实的尖峰序列，既计算便宜，又易于分析。这是一个平面非线性积分-点火模型，具有分段线性矢量场和尖峰时的状态相关重置。我们称之为PWL-IF模型，并从单个神经元和网络两个层面对其进行分析。非光滑动力系统的技术和术语被用来充实单神经元模型的分叉结构，以及发展Lyapunov指数的概念。我们还展示了如何构造该系统的相响应曲线，强调数学神经科学中的技术也可以转换回非光滑动力系统领域。使用尖峰时间的线性稳定性分析来评估周期尖峰轨道的稳定性。在网络层面，我们考虑了电压变量之间的线性耦合，就像在具有缝隙连接耦合的神经生物网络中所发生的那样，并展示了如何分析异步和同步状态的性质(存在性和稳定性)。在前一种情况下，我们使用了一种适用于任何大的全局耦合极限环振子系统的相密度技术，而在后一种情况下，我们发展了一种新的技术，可以处理尖峰时模型的非光滑重置。最后，我们讨论了神经科学建模的其他方面，这些方面可能受益于从不断增长的非光滑动力学知识体系中进一步翻译思想。(C)2011爱思唯尔B.V.保留所有权利。
Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a nonsmooth dynamical system. In this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate (ranging from tonic to burst firing). We show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis, particularly those for treating grazing bifurcations. Importantly we highlight one particular single neuron model, capable of generating realistic spike trains, that is both computationally cheap and analytically tractable. This is a planar nonlinear integrate-and-fire model with a piecewise linear vector field and a state dependent reset upon spiking. We call this the PWL-IF model and analyse it at both the single neuron and network level. The techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model, as well as to develop the notion of Lyapunov exponents. We also show how to construct the phase response curve for this system, emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems. The stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times. At the network level we consider linear coupling between voltage variables, as would occur in neurobiological networks with gap-junction coupling, and show how to analyse the properties (existence and stability) of both the asynchronous and synchronous states. In the former case we use a phase-density technique that is valid for any large system of globally coupled limit cycle oscillators, whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking. Finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics. (C) 2011 Elsevier B.V. All rights reserved.