Multiple Time Scales in Neuronal Models

研究者研究具有多个时间尺度的动力系统作为神经系统的模型。来自小型无脊椎动物神经网络（龙虾的口胃神经节）的数据和模型指导了这项工作并提供了案例研究。在口胃神经节中观察到的现象，如爆发振荡和尖峰频率适应，可以最好地用多时间尺度的动力系统来建模。以往对多时间尺度动力系统定性性质的数学分析主要处理发生在低维的局部现象。神经系统的动力学提出了现有的多时间尺度系统理论没有解决的问题。本文的目的是通过对具有两个时间尺度的系统的全局动力学和分岔的定性特征进行分类来扩展这一理论。这是一项雄心勃勃的努力，旨在扩展过去三十年来发展起来的非线性动力系统和常微分方程奇摄动系统的理论。以数值研究为指导，描述了在这种情况下出现的模式字典，并对其分析性质进行了表征。这项工作在很大程度上借鉴了动力系统的分岔理论和混合动力系统的模型，结合了连续和离散组件。根据需要，数值算法的发展，方便模拟和分析多时间尺度系统。数学工作的最初重点是有两个慢变量和两个快变量的系统。对基于传导的口胃神经节模型也进行了数值研究。这些研究结果与数据进行了比较，并用于指导模型的改进。动物的神经系统调节和控制肌肉活动，如运动。在这些过程中，发展良好的理论可以建立神经膜电特性的模型，但对生物体使用的动力学原理知之甚少。该项目研究了一个由14个神经元组成的小神经系统的动力学模型，该系统控制着龙虾前肠的节律运动。使用这个系统是因为它足够小，系统内每个神经元的独特属性已经被确定，但足够大，神经元之间相互作用的网络架构也很重要。该系统显示了丰富的节奏感行为。这个项目的重点是涉及不同时间尺度的行为特征。数学理论已经成功地描述了在一系列令人震惊的物理和自然系统中观察到的动力学的普遍特性，但这些理论需要扩展到具有多个时间尺度的系统。目标是构建动态模式的分类，这些模式是神经系统复杂行为形成的组成部分。这些系统的模型也很复杂。需要计算研究来预测它们的动力学行为。该项目旨在实现算法，提高我们从模型中提取有用信息的能力，并指导改进它们的保真度。所开发的理论分析和数值方法涵盖了具有多时间尺度的动力系统的所有模型，并且可以远远超出神经系统的范围，这是本研究的重点。

Guckenheimer 9705780 The investigator studies dynamical systems with multiple time scales as models of neural systems. Data and models from a small invertebrate neural network, the stomatogastric ganglion of lobsters, guide the work and provide case studies. Phenomena observed in the stomatogastric ganglion, such as bursting oscillations and spike frequency adaptation, can best be modeled as dynamical systems with multiple time scales. Previous mathematical analyses of qualitative properties of multiple time scale dynamical systems have dealt mainly with local phenomena that occur in low dimensions. The dynamics of neural systems raise questions that are not addressed by existing theories of multiple time scale systems. The aim here is to extend the theory by classifying qualitative features of the global dynamics and bifurcations for systems with two time scales. This is an ambitious endeavor to extend theories of nonlinear dynamical systems and singularly perturbed systems of ordinary differential equations that have been developed over the past thirty years. Using numerical investigations as a guide, the dictionary of patterns that occur in this setting is described and their analytical properties are characterized. This work draws heavily upon the theories of bifurcations of dynamical systems and models of hybrid dynamical systems that combine continuous and discrete components. As needed, numerical algorithms are developed that facilitate the simulation and analysis of multiple time scale systems. The initial emphasis of the mathematical work is upon systems that have two slow variables and two fast variables. Numerical investigations of conductance-based models for the stomatogastric ganglion also are performed. The results of these studies are compared with data and used to guide the refinement of the models. Nervous systems of animals regulate and control muscular activi ty such as locomotion. Well developed theories enable the construction of models for the electrical properties of nerve membranes in these processes, but there is little understanding of the dynamical principles used by organisms. This project investigates dynamical models of a small neural system consisting of fourteen neurons that control rhythmic motions of the foregut of lobsters. This system is used because it is small enough that unique properties of each neuron within the system have been identified, but large enough that the network architecture of the interactions among neurons is also important. The system displays a rich repertoire of rhythmic behavior. The focus of this project is on features of the behavior that involve different time scales. Mathematical theories have been successful at describing universal properties of the dynamics observed in an astounding array of physical and natural systems, but these theories need to be extended to systems with more than one time scale. The goal is to construct classifications of dynamical patterns that are the components from which the complex behaviors of neural systems are formed. Models of these systems are also complex. Computational investigations are required to predict their dynamical behavior. This project seeks to implement algorithms that improve our ability to extract useful information from the models and guide the improvement of their fidelity. Both the theoretical analysis and the numerical methods that are developed encompass all models of dynamical systems with multiple time scales and can be used far beyond the context of the neural system that is the focus of this study.