神经元网络中自发放电的几何结构
The geometry of spontaneous spiking in neuronal networks
摘要：
本文提出了在小噪声作用下可兴奋神经元电耦合网络中斑图形成的数学理论。使用Freidlin-Wentzell随机扰动动力系统的大偏差理论和代数图论的元素，我们确定和分析的主要制度在网络动力学的关键控制参数：兴奋性，耦合强度，和网络拓扑结构。分析揭示了电耦合网络中自发动力学的几何结构。具体来说，我们表明，在单位n-立方体的表面上的某个连续函数的最小值的位置编码的网络产生的最有可能的活动模式。通过研究该函数的极小值在耦合强度变化下的演化，我们描述了网络动力学中的主要变换。最小化问题也用于定量描述的主要动力学制度和它们之间的过渡。特别是，对于弱耦合和强耦合制度，我们提出的网络活动率的函数的耦合强度和网络的程度的渐近公式。变分分析的补充，在强耦合制度的同步状态的稳定性分析。稳定性估计揭示了网络的连通性和与网络图相关的循环子空间的性质对其同步性质的贡献。这项工作的动机是实验和建模研究的整体神经元的蓝斑，在脑干中的核参与调节认知性能和行为。
英文摘要：
The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior.
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