分段线性神经质量模型网络
Networks of piecewise linear neural mass models
摘要：
神经质量模型在大规模脑建模中无处不在。在节点级别，它们用一组具有非线性（通常为S形）的常微分方程来表示。使用来自大脑图谱的结构数据，它们可以连接成一个网络，以研究功能动态状态的出现，例如同步。简单限制的经典sigmoidal非线性分段线性漫画，我们表明，著名的威尔逊-考恩神经质量模型可以分析在节点和网络的水平。通过将矩阵指数解拼接在一起来构造节点级的周期轨道，并使用Floquet理论来确定稳定性。对于循环矩阵描述的相互作用的网络，我们表明，同步状态的稳定性可以确定在一个低维Floquet问题参数化的相互作用矩阵的特征值。这个网络Floquet问题很容易解决，使用线性代数，预测的发病时空网络模式所产生的同步不稳定性。我们考虑的情况下，一个不连续的选择节点的非线性，即更换的S形的Heaviside非线性。这就产生了一个连续时间的开关网络。在节点级，这允许存在不稳定的滑动周期轨道，我们构建。周期轨道的稳定性现在处理与修改Floquet理论来处理通过切换流形通过跃变矩阵的小扰动的演变。在网络级，同步状态的稳定性分析更具挑战性。在这里，我们报告使用的想法最初开发的研究玻璃网络治疗的稳定性周期性网络状态的神经质量模型与不连续的相互作用。
英文摘要：
Neural mass models are ubiquitous in large scale brain modelling. At the node level they are written in terms of a set of ODEs with a nonlinearity that is typically a sigmoidal shape. Using structural data from brain atlases they may be connected into a network to investigate the emergence of functional dynamic states, such as synchrony. With the simple restriction of the classic sigmoidal nonlinearity to a piecewise linear caricature we show that the famous Wilson-Cowan neural mass model can be analysed at both the node and network level. The construction of periodic orbits at the node level is achieved by patching together matrix exponential solutions, and stability is determined using Floquet theory. For networks with interactions described by circulant matrices, we show that the stability of the synchronous state can be determined in terms of a low-dimensional Floquet problem parameterised by the eigenvalues of the interaction matrix. This network Floquet problem is readily solved using linear algebra, to predict the onset of spatio-temporal network patterns arising from a synchronous instability. We consider the case of a discontinuous choice for the node nonlinearity, namely the replacement of the sigmoid by a Heaviside nonlinearity. This gives rise to a continuous-time switching network. At the node level this allows for the existence of unstable sliding periodic orbits, which we construct. The stability of a periodic orbit is now treated with a modification of Floquet theory to treat the evolution of small perturbations through switching manifolds via saltation matrices. At the network level the stability analysis of the synchronous state is considerably more challenging. Here we report on the use of ideas originally developed for the study of Glass networks to treat the stability of periodic network states in neural mass models with discontinuous interactions.
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