Ando Hiroyasu, Suetani Hiromichi, Kurths Jürgen, Aihara Kazuyuki
RIKEN Brain Science Institute, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jul;86(1 Pt 2):016205. doi: 10.1103/PhysRevE.86.016205. Epub 2012 Jul 9.
We investigate the entrainment of a neuron model exhibiting a chaotic spiking-bursting behavior in response to a weak periodic force. This model exhibits two types of oscillations with different characteristic time scales, namely, long and short time scales. Several types of phase synchronization are observed, such as 1:1 phase locking between a single spike and one period of the force and 1:l phase locking between the period of slow oscillation underlying bursts and l periods of the force. Moreover, spiking-bursting oscillations with chaotic firing patterns can be synchronized with the periodic force. Such a type of phase synchronization is detected from the position of a set of points on a unit circle, which is determined by the phase of the periodic force at each spiking time. We show that this detection method is effective for a system with multiple time scales. Owing to the existence of both the short and the long time scales, two characteristic phenomena are found around the transition point to chaotic phase synchronization. One phenomenon shows that the average time interval between successive phase slips exhibits a power-law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength. The other phenomenon shows that Kuramoto's order parameter before the transition exhibits stepwise behavior as a function of the driving force strength, contrary to the smooth transition in a model with a single time scale.
我们研究了一个表现出混沌尖峰-爆发行为的神经元模型在弱周期力作用下的同步情况。该模型呈现出两种具有不同特征时间尺度的振荡，即长时间尺度和短时间尺度。观察到了几种类型的相位同步，例如单个尖峰与力的一个周期之间的1:1锁相，以及爆发背后的慢振荡周期与力的l个周期之间的1:1锁相。此外，具有混沌放电模式的尖峰-爆发振荡可以与周期力同步。这种相位同步类型是从单位圆上一组点的位置检测到的，该位置由每个尖峰时刻的周期力相位确定。我们表明，这种检测方法对具有多个时间尺度的系统是有效的。由于短时间尺度和长时间尺度的同时存在，在向混沌相位同步的转变点附近发现了两种特征现象。一种现象表明，连续相位滑移之间的平均时间间隔对驱动力强度呈现幂律缩放，并且缩放指数对驱动力强度的变化具有不光滑的依赖性。另一种现象表明，转变前的Kuramoto序参量作为驱动力强度的函数呈现阶梯状行为，这与具有单个时间尺度的模型中的平滑转变相反。
