Department of Mathematics and Statistics, Center for BioDynamics, Boston University, Boston, MA, 02215, USA.
J Math Neurosci. 2012 Feb 21;2(1):3. doi: 10.1186/2190-8567-2-3.
Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which we show arises in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.
快速动作电位产生 - 尖峰 - 以及尖峰和静止交替的爆发 - 是神经元活动中常见的两种动态模式。在神经元系统的计算模型中，从尖峰到爆发的转变通常表现出复杂的分岔结构。一种类型的转变涉及环突转折，我们表明它出现在广泛的具有三种不同类别的爆发动力学的众所周知的计算神经元模型中：亚霍普夫/折叠周期爆发、圆/折叠周期爆发和折叠/折叠周期爆发。这些模型共享的基本特征是多个时间尺度，自然导致分解为慢系统和快系统、快系统中周期轨道的鞍节点以及全系统中的环面分岔。我们表明，每个模型系统中从尖峰到爆发的转变由环突转折的爆炸给出。基于这些例子以及新兴理论，我们提出环突转折是分离尖峰和爆发活动状态的常见动态现象。
