Multistability and bifurcations for polyrhythmic Central Pattern Generators

该项目将发展多节奏活动爆发模式的多稳定性动力学原理及其对多功能中心模式发生器的控制。多稳定性增强了神经系统的灵活性，在运动控制、动态记忆、信息处理和决策方面具有深远的意义。研究者和他的学生将在单个和网络中间神经元的现实模型中识别和研究爆发节律的一般非局部分支，并为cpg的爆发发生创建一个动态系统分类。研究小组将基于动力系统和全局分岔理论创建一套新的方法和计算工具，以检查高阶霍奇金-赫胥黎型模型和网络中爆发模式的复杂转换。研究者和他的学生将通过创建透明的计算工具来检测和预测多时间尺度神经元模型中复杂振荡解的转换，从而增强现有的数学技术。这包括将神经元动力学简化为一个完整的、无方程的膜电位on poincar<s:1>映射家族的新方法，以及破裂CPG电路的相位差映射。这种简化将使我们对高阶、多时间尺度神经元模型的动态有一个清晰的理解，并通过揭示控制多功能CPG网络全局动态的隐藏中心，提供对多稳定性的控制。对网络破坏中间神经元的动力学特性有了广泛的了解，将使团队能够推导出精确的相位模型来复制他们的高维模型的动力学。这些简化的模型将用于检查特定兴奋-抑制性CPG电路的更大和更复杂的现实模型。不同解剖回路的能力，如中枢模式发生器，产生多种神经活动模式来控制几种运动类型，如心脏跳动，清醒，游泳等，在脊椎动物和无脊椎动物中广泛存在。理解神经元连通性进化的一般机制以及不同神经活动模式之间的转换，并对这些过程进行建模，是应用数学和计算神经科学面临的基本挑战。这个项目是一个真正的跨学科研究，将最先进的数学，更具体地说是应用动力系统和非局部分岔的理论，与生命科学联系起来。它将扩展和概括我们对神经系统动力学原理的理解；特别是调节多功能中枢模式发生器多节律的机制。多稳定性增强了神经系统的灵活性，在人类和动物的运动控制、动态记忆、信息处理和决策方面具有深远的意义。研究者和他的学生将在单个和网络中间神经元的现实模型中识别和研究爆发节律的一般分支，并为多功能神经回路中的爆发发生创建一个动态系统分类。

The project will develop the dynamical principles of multistability of bursting patterns of polyrhythmic activity and its control for multifunctional Central Pattern Generators. Multistability enhances the flexibility of nervous systems and has far reaching implications for motor control, dynamic memory, information processing, and decision making. The Investigator and his students will identify and study generic nonlocal bifurcations of bursting rhythms in realistic models of single and networked interneurons, as well as create a dynamical systems classification for the bursting genesis in CPGs. The research team will create a suite of new methods and computational tools based on the theory of dynamical systems and global bifurcations to examine complex transformations of bursting patterns in high-order Hodgkin-Huxley type models and networks. The Investigator and his students will enhance the existing mathematical technique by creating transparent computational tools for the detection and prediction of transformations of complex oscillatory solutions in neuronal models with multiple time scales. This includes the novel approaches of reducing neuronal dynamics to a complete, equation-free family of onto Poincaré mappings for membrane potentials, and the phase-difference mappings for bursting CPG circuits. The reduction will yield a clear understanding of the dynamics of a high-order, multiple-time scale neuron model, as well as provide with a control of the multistability by revealing the hidden centers that govern globally the dynamics of a mutlifunctional CPG network. Having the extensive knowledge of dynamical properties of networked busting interneurons will allow the team to derive precise phase models to replicate the dynamics of their high-dimensional models. These reduced models will be used to examine larger and more complex realistic models of the specific excitatory-inhibitory CPG circuits. The ability of distinct anatomical circuits, like Central Pattern Generators, to generate multiple patterns of neural activity to control several locomotion types, like cardiac beating, waking, swimming etc, is widespread among vertebrate and invertebrate species. Understanding generic mechanisms of the evolution of neuronal connectivity and transitions between different patterns of neural activity and modeling these processes are the fundamental challenges for applied mathematics and computational neuroscience. This project is a genuinely cross-disciplinary research, bridging state-of the art mathematics, more specifically the theory of applied dynamical systems and nonlocal bifurcations, with life sciences. It shall extend and generalize our understanding of dynamical principles of neural systems; specifically mechanisms regulating polyrhythms of multifunctional Central Pattern Generators. Multistability enhances the flexibility of nervous systems and has far reaching implications for motor control, dynamic memory, information processing, and decision making of humans and animals. The Investigator and his students will identify and study generic bifurcations of bursting rhythms in realistic models of single and networked interneurons, as well as create a dynamical systems classification for the bursting genesis in multifunctional neural circuits.