{"title":"慢-快动力学皮质扩张性去极化模型中的行波","authors":"David Reyner-Parra, C. Bonet, T. Seara, G. Huguet","doi":"10.1063/5.0160509","DOIUrl":null,"url":null,"abstract":"Cortical spreading depression and spreading depolarization (CSD) are waves of neuronal depolarization that spread across the cortex, leading to a temporary saturation of brain activity. They are associated with various brain disorders such as migraine and ischemia. We consider a reduced version of a biophysical model of a neuron–astrocyte network for the initiation and propagation of CSD waves [Huguet et al., Biophys. J. 111(2), 452–462, 2016], consisting of reaction-diffusion equations. The reduced model considers only the dynamics of the neuronal and astrocytic membrane potentials and the extracellular potassium concentration, capturing the instigation process implicated in such waves. We present a computational and mathematical framework based on the parameterization method and singular perturbation theory to provide semi-analytical results on the existence of a wave solution and to compute it jointly with its velocity of propagation. The traveling wave solution can be seen as a heteroclinic connection of an associated system of ordinary differential equations with a slow–fast dynamics. The presence of distinct time scales within the system introduces numerical instabilities, which we successfully address through the identification of significant invariant manifolds and the implementation of the parameterization method. Our results provide a methodology that allows to identify efficiently and accurately the mechanisms responsible for the initiation of these waves and the wave propagation velocity.","PeriodicalId":340975,"journal":{"name":"Chaos: An Interdisciplinary Journal of Nonlinear Science","volume":"20 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Traveling waves in a model for cortical spreading depolarization with slow–fast dynamics\",\"authors\":\"David Reyner-Parra, C. Bonet, T. Seara, G. Huguet\",\"doi\":\"10.1063/5.0160509\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Cortical spreading depression and spreading depolarization (CSD) are waves of neuronal depolarization that spread across the cortex, leading to a temporary saturation of brain activity. They are associated with various brain disorders such as migraine and ischemia. We consider a reduced version of a biophysical model of a neuron–astrocyte network for the initiation and propagation of CSD waves [Huguet et al., Biophys. J. 111(2), 452–462, 2016], consisting of reaction-diffusion equations. The reduced model considers only the dynamics of the neuronal and astrocytic membrane potentials and the extracellular potassium concentration, capturing the instigation process implicated in such waves. We present a computational and mathematical framework based on the parameterization method and singular perturbation theory to provide semi-analytical results on the existence of a wave solution and to compute it jointly with its velocity of propagation. The traveling wave solution can be seen as a heteroclinic connection of an associated system of ordinary differential equations with a slow–fast dynamics. The presence of distinct time scales within the system introduces numerical instabilities, which we successfully address through the identification of significant invariant manifolds and the implementation of the parameterization method. Our results provide a methodology that allows to identify efficiently and accurately the mechanisms responsible for the initiation of these waves and the wave propagation velocity.\",\"PeriodicalId\":340975,\"journal\":{\"name\":\"Chaos: An Interdisciplinary Journal of Nonlinear Science\",\"volume\":\"20 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Chaos: An Interdisciplinary Journal of Nonlinear Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0160509\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos: An Interdisciplinary Journal of Nonlinear Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1063/5.0160509","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
皮层扩张性抑制和扩张性去极化(CSD)是神经元去极化的波,它在皮层上扩散,导致大脑活动暂时饱和。它们与偏头痛和缺血等各种脑部疾病有关。我们考虑了CSD波起始和传播的神经元-星形胶质细胞网络生物物理模型的简化版本[Huguet et al., Biophys]。[j] .化学工程学报,2016,36(2):559 - 562。简化的模型只考虑神经元和星形细胞膜电位和细胞外钾浓度的动力学,捕获了涉及这些波的煽动过程。我们提出了一个基于参数化方法和奇异摄动理论的计算和数学框架,给出了波解存在性的半解析结果,并结合其传播速度对其进行了计算。行波解可以看作是一个具有慢-快动力学的常微分方程相关系统的异斜连接。系统中不同时间尺度的存在引入了数值不稳定性,我们通过识别重要的不变流形和实现参数化方法成功地解决了这一问题。我们的结果提供了一种方法,可以有效和准确地确定这些波的起始机制和波的传播速度。
Traveling waves in a model for cortical spreading depolarization with slow–fast dynamics
Cortical spreading depression and spreading depolarization (CSD) are waves of neuronal depolarization that spread across the cortex, leading to a temporary saturation of brain activity. They are associated with various brain disorders such as migraine and ischemia. We consider a reduced version of a biophysical model of a neuron–astrocyte network for the initiation and propagation of CSD waves [Huguet et al., Biophys. J. 111(2), 452–462, 2016], consisting of reaction-diffusion equations. The reduced model considers only the dynamics of the neuronal and astrocytic membrane potentials and the extracellular potassium concentration, capturing the instigation process implicated in such waves. We present a computational and mathematical framework based on the parameterization method and singular perturbation theory to provide semi-analytical results on the existence of a wave solution and to compute it jointly with its velocity of propagation. The traveling wave solution can be seen as a heteroclinic connection of an associated system of ordinary differential equations with a slow–fast dynamics. The presence of distinct time scales within the system introduces numerical instabilities, which we successfully address through the identification of significant invariant manifolds and the implementation of the parameterization method. Our results provide a methodology that allows to identify efficiently and accurately the mechanisms responsible for the initiation of these waves and the wave propagation velocity.