A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit
A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit
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DOI：
10.1137/22m1482913
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
J. Carrillo;X. Dou;Zhennan Zhou
中科院分区：
文献类型：
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作者：
J. Carrillo;X. Dou;Zhennan Zhou
神经元集体行为的电压-电导动力学模型已经被科学家和数学家研究了二十年，但尽管在各种情况下有大量的数值证据，但对其解结构的严格分析只得到了部分结果。在这项工作中，我们考虑一个简化的电压电导模型，其中电压变量中的速度场是可分离的形式。简化模型的长期行为得到充分研究，导致以下二分法：要么密度函数收敛到全局平衡，要么随着时间趋向无穷大，发射率发散。此外，还对快速电导渐进极限进行了论证和分析，其中极限模型的解要么在有限时间内爆炸，要么全局存在导致时间周期解。这些结果的一个重要含义是，不可分离的速度场，或者物理上的泄漏机制，是基于可用数值证据的原始模型中出现周期解的关键要素。
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.