标题： 具有同步记忆依赖性细胞内振荡的细胞-块ODE-PDE模型在$\mathbb{R}^2$中 显示英文标题

标题： Synchronized Memory-Dependent Intracellular Oscillations for a Cell-Bulk ODE-PDE Model in $\mathbb{R}^2$

摘要： 对于$\mathbb{R}^2$中的细胞-体ODE-PDE模型，开发了一种混合渐近-数值理论，以提供一种新的理论和计算高效的途径，用于研究与空间隔离的动态活跃“单元”或“细胞”相关的振荡动力学如何被由整个细胞种群产生和吸收的PDE体扩散场所调节。 对PDE扩散场中振荡器同步的研究是Yoshiki Kuramoto基础工作的初始目标之一。 对于这个细胞-体模型，强局部扰动理论，如扩展到时间依赖设置，用于推导一个新的积分微分ODE系统，该系统表征了在记忆依赖的体扩散场中的细胞内动力学。 对于这个非局部约化系统，开发了一种新颖的快速时间推进方案，部分依赖于\emph{指数和方法}来数值处理卷积积分，以快速且准确地计算长时间间隔内的积分微分系统的数值解。 对于Sel'kov反应动力学的特殊情况，展示了各种范围的流入和流出渗透率参数、体降解率和体扩散率以及细胞的特定空间配置下，大规模振荡动力学行为，包括相位同步、混合模式振荡和群体感应。 我们快速算法的结果，在笔记本电脑上不到一分钟的CPU时间内获得，与使用商业PDE求解器进行的细胞-体模型PDE模拟进行了基准测试，这些模拟的运行时间要大几个数量级。 显示英文摘要

摘要： For a cell-bulk ODE-PDE model in $\mathbb{R}^2$, a hybrid asymptotic-numerical theory is developed to provide a new theoretical and computationally efficient approach for studying how oscillatory dynamics associated with spatially segregated dynamically active ``units" or ``cells" are regulated by a PDE bulk diffusion field that is both produced and absorbed by the entire cell population. The study of oscillator synchronization in a PDE diffusion field was one of the initial aims of Yoshiki Kuramoto's foundational work. For this cell-bulk model, strong localized perturbation theory, as extended to a time-dependent setting, is used to derive a new integro-differential ODE system that characterizes intracellular dynamics in a memory-dependent bulk-diffusion field. For this nonlocal reduced system, a novel fast time-marching scheme, relying in part on the \emph{sum-of-exponentials method} to numerically treat convolution integrals, is developed to rapidly and accurately compute numerical solutions to the integro-differential system over long time intervals. For the special case of Sel'kov reaction kinetics, a wide variety of large-scale oscillatory dynamical behavior including phase synchronization, mixed-mode oscillations, and quorum-sensing are illustrated for various ranges of the influx and efflux permeability parameters, the bulk degradation rate and bulk diffusivity, and the specific spatial configuration of cells. Results from our fast algorithm, obtained in under one minute of CPU time on a laptop, are benchmarked against PDE simulations of the cell-bulk model, which are performed with a commercial PDE solver, that have run-times that are orders of magnitude larger.