论文标题
磁滞分叉和应用于延迟的Fitzhugh-Nagumo神经系统
Hysteresis bifurcation and application to delayed Fitzhugh-Nagumo neural systems
论文作者
论文摘要
在大量的生物学实验研究中已经描述了滞后动力学。许多这样的研究都是现象学的，数学上的欣赏并没有引起足够的关注。在本文中，我们通过使用分叉和扰动理论从动力学系统的角度探讨了滞后的性质，并从动力学系统的角度进行了研究。我们首先根据不同类型的吸引子之间的系统行为进行滞后分类。然后，我们专注于数学上的可观情况，在数学上，平衡点和极限周期之间的滞后运动是由亚临界型霍夫夫分叉和极限周期的鞍节点分叉启动的。我们通过使用多个尺度的方法来提出一个分析框架，以使正常形式达到第五阶。将理论结果与时域模拟和数值延续进行比较，显示出良好的一致性。尽管我们考虑了论文中时间删除的Fitzhugh-Nagumo神经系统，但对于其他系统或参数，应该清楚地概括。我们在论文中提出的一般框架可以自然地扩展到神经科学中爆发活动的概念，在神经科学中，滞后是产生爆发振荡的主要机制。
Hysteresis dynamics has been described in a vast number of biological experimental studies. Many such studies are phenomenological and a mathematical appreciation has not attracted enough attention. In the paper, we explore the nature of hysteresis and study it from the dynamical system point of view by using the bifurcation and perturbation theories. We firstly make a classification of hysteresis according to the system behaviours transiting between different types of attractors. Then, we focus on a mathematically amenable situation where hysteretic movements between the equilibrium point and the limit cycle are initiated by a subcritical Hopf bifurcation and a saddle-node bifurcation of limit cycles. We present a analytical framework by using the method of multiple scales to obtain the normal form up to the fifth order. Theoretical results are compared with time domain simulations and numerical continuation, showing good agreement. Although we consider the time-delayed FitzHugh-Nagumo neural system in the paper, the generalization should be clear to other systems or parameters. The general framework we present in the paper can be naturally extended to the notion of bursting activity in neuroscience where hysteresis is a dominant mechanism to generate bursting oscillations.
