论文标题
用于相互作用神经元及其渐近极限的简化电压传统动力学模型
A simplified voltage-conductance kinetic model for interacting neurons and its asymptotic limit
论文作者
论文摘要
二十年来，科学家和数学家已经研究了神经元集体行为的电压传统动力学模型，但是在各种情况下，尽管有大量的数值证据，但对其溶液结构的严格分析仅是部分获得的。在这项工作中，我们考虑了一个简化的电压传导模型，其中电压变量中的速度场的形式为可分离。简化模型的长时间行为经过全面研究，导致以下二分法：密度函数会收敛到整体平衡，或者随着时间的流逝，点火速率差异。此外，快速电导渐近极限是合理的和分析的，其中对极限模型的解决方案要么在有限的时间内炸毁，要么在全球上存在导致时间周期性解决方案。这些结果的一个重要含义是，基于可用的数值证据，非分离速度场或物理上的泄漏机制是原始模型中周期性解决方案出现的关键要素。
The voltage-conductance kinetic model for the collective behavior of neurons has been studied by scientists and mathematicians for two decades, but the rigorous analysis of its solution structure has been only partially obtained in spite of plenty of numerical evidence in various scenarios. In this work, we consider a simplified voltage-conductance model in which the velocity field in the voltage variable is in a separable form. The long time behavior of the simplified model is fully investigated leading to the following dichotomy: either the density function converges to the global equilibrium, or the firing rate diverges as time goes to infinity. Besides, the fast conductance asymptotic limit is justified and analyzed, where the solution to the limit model either blows up in finite time, or globally exists leading to time periodic solutions. An important implication of these results is that the non-separable velocity field, or physically the leaky mechanism, is a key element for the emergence of periodic solutions in the original model based on the available numerical evidence.
