In this study, we extend the Morris-Lecar (M-L) neuron model to design a neuronal network model with field effects. Using Lyapunov theory and the master stability function to evaluate the stability of synchronous manifolds. The Hamilton energy function of a single neuron is derived using Helmholtz's theorem to equivalently describe its internal field energy, and bioelectric activities are analyzed in conjunction with synchronization error functions. A comprehensive analysis was conducted using various numerical methods, including time series diagrams, bifurcation diagrams, two-parameter plane synchronization error diagrams, and similarity function diagrams. The observations indicate that variations in conductivity and equilibrium potential directly impact the firing patterns and synchronization states of neurons. In various coupling methods, an increase in coupling strength induces different degrees of oscillation within the coupled system. The effects of three coupling methods on the synchronization of the neuronal network are examined using spatiotemporal evolution diagrams and energy evolution diagrams. Global synchronization error and synchronization factors are introduced to quantify the level of synchrony. Numerical results indicate that an appropriate coupling strength and ionic conductance enhance network synchrony. Numerical results indicate that appropriate coupling strength promotes network synchrony. The insights gained from this study contribute to providing a more coherent framework for constructing neuronal network models under field-coupled conditions, thereby enhancing the understanding of fundamental principles in neuroscience and offering new perspectives for the treatment and rehabilitation of neurological disorders.

中文翻译：

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电磁激励下 Morris-Lecar 模型与场耦合的动力学和同步


在这项研究中，我们扩展了 Morris-Lecar （M-L） 神经元模型，以设计具有场效应的神经元网络模型。使用 Lyapunov 理论和主稳定性函数来评估同步流形的稳定性。使用亥姆霍兹定理推导出单个神经元的哈密尔顿能量函数，以等效地描述其内部场能，并结合同步误差函数分析生物电活动。使用各种数值方法进行了综合分析，包括时间序列图、分叉图、双参数平面同步误差图和相似函数图。观察结果表明，电导率和平衡电位的变化直接影响神经元的放电模式和同步状态。在各种耦合方法中，耦合强度的增加会在耦合系统内引起不同程度的振荡。使用时空进化图和能量进化图研究了三种耦合方法对神经元网络同步的影响。引入了全局同步误差和同步因子来量化同步水平。数值结果表明，适当的耦合强度和离子电导增强了网络同步性。数值结果表明，适当的耦合强度促进了网络同步。从这项研究中获得的见解有助于为在场耦合条件下构建神经元网络模型提供更连贯的框架，从而增强对神经科学基本原理的理解，并为神经系统疾病的治疗和康复提供新的视角。