具有动态阈值的混合神经元模型中的阈下振荡与放电
Subthreshold oscillations and spiking in hybrid neuron models with a dynamic threshold
- Gdańsk University of Technology（格但斯克理工大学）
- University of Castilla-La Mancha（卡斯蒂利亚-拉曼恰大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究分析具有动态阈值和重置机制的Meng-Huguet-Rinzel混合神经元模型，证明非混合系统存在唯一全局吸引周期轨道，混合系统至多有一个无重置周期轨道且可共存放电吸引子，为动态阈值机制提供严格数学框架。
AI中文摘要：
我们研究一类二维混合非自治周期受迫神经元模型，即Meng-Huguet-Rinzel神经元模型，该模型具有动态阈值和重置机制。我们关注连续的阈下动力学与离散的放电诱导重置之间的相互作用。该模型由一个线性电压方程与一个由膜电位（电压）的非线性函数控制的阈值变量耦合而成，并包含周期性的外部强迫，其形式为脉冲或整流正弦电流。我们分析了周期解的存在性和唯一性，并证明了该系统的非混合版本具有一个唯一的全局吸引周期轨道。对于混合系统，我们证明了至多存在一个无重置的周期轨道，而数值模拟表明无重置周期轨道与周期放电吸引子可能共存。我们还证明了混合系统相空间中所包含的一些自然多边形构成该系统的紧致正不变全局吸引集。这些结果为神经元模型中动态阈值机制的分析提供了严格的数学框架，并有助于从理论上理解在时间周期强迫下阈下振荡与放电行为之间的转变。
英文摘要：
We investigate a class of two-dimensional hybrid non-autonomous periodically forced neuron models known as the Meng-Huguet-Rinzel neuron model, with a dynamic threshold and reset mechanism. We focus on the interplay between the continuous subthreshold dynamics and discrete spike-induced resets. The model consists of a linear voltage equation coupled with a threshold variable governed by a nonlinear function of the membrane potential (voltage), and incorporates periodic external forcing in the form of either pulse or rectified sinusoidal currents. We analyze the existence and uniqueness of periodic solutions and prove that the non-hybrid version of the system possesses a unique globally attracting periodic orbit. For the hybrid system, we show that there exists at most one periodic orbit without resets, while numerical simulations indicate the possibility of coexistence of a reset-free periodic orbit and periodic spiking attractors. We also prove that some natural polygons contained in the phase space of the hybrid system form compact positively invariant globally attracting sets of this system. These results provide a rigorous mathematical framework for the analysis of dynamic threshold mechanisms in neuron models and contribute to the theoretical understanding of transitions between subthreshold oscillations and spiking behavior under a time-periodic forcing.
