计算相位响应曲线的方波扰动直接算法

通过相位响应曲线可对具有极限环周期运动的动力系统的性质有更为深入的理解.神经元是一个典型的动力系统,因此相位响应曲线提供了一种研究神经元重复周期放电行为的新思路.本文提出一种求解相位响应曲线的方法,即方波扰动的直接算法,通过Hodgkin-Huxley,Fitz Hugh-Nagumo,Morris-Lecar和Hindmarsh-Rose神经元模型验证该算法可计算周期峰放电、周期簇放电的相位响应曲线.该算法克服了其他算法在运用过程中的局限性.利用该算法计算结果表明:周期峰放电的相位响应曲线类型是由其分岔类型所决定;在Morris-Lecar模型中发现一种开始于Hopf分岔终止于鞍点同宿轨道分岔的阈上周期振荡,其相位响应曲线属于第二类型.通过大量的相位响应曲线的计算发现相位响应的相对大小及正负性仅取决于扰动所施加的时间,而且周期簇放电的相位响应曲线比周期峰放电的相位响应曲线更为复杂.

A direct algorithm with square wave perturbation for calculating phase response curve

Neuron is a typical dynamic system, therefore, it is quite natural to study the firing behaviors of neurons by using the dynamical system theory. Two kinds of firing patterns, i.e., the periodic spiking and the periodic bursting, are the limit cycle oscillators from the point of view of nonlinear dynamics. The simplest way to describe the limit cycle is to use the phase of the oscillator. A complex state space model can be mapped into a one-dimensional phase model by phase transformation, which is helpful for obtaining the analytical solution of the oscillator system. The response characteristics of the oscillator system in the motion state of the limit cycle to the external stimuli can be characterized by the phase response curve. A phase response curve illustrates the transient change in the cycle period of an oscillation induced by a perturbation as a function of the phase at which it is received. Now it is widely believed that the phase response curve provides a new way to study the behavior of the neuron. Existing studies have shown that the phase response curve of the periodic spiking can be divided into two types, which are closely related to the bifurcation mechanism of neurons from rest to repetitive firing. However, there are few studies on the relationship between the phase response curve and the bifurcation type of the periodic bursting. Clearly, the first prerequisite to understand this relationship is to calculate the phase response curve of the periodic bursting. The existing algorithms for computing the phase response curve are often unsuccessful in the periodic bursting. In this paper, we present a method of calculating the phase response curve, namely the direct algorithm with square wave perturbation. The phase response curves of periodic spiking and periodic bursting can be obtained by making use of the direct algorithm, which is verified in the four neuron models of the Hodgkin-Huxley, FitzHugh-Nagumo, Morris-Lecar and Hindmarsh-Rose. This algorithm overcomes the limitations to other algorithms in the application. The calculation results show that the phase response curve of the periodic spiking is determined by the bifurcation type. We find a suprathreshold periodic oscillation starting from a Hopf bifurcation and terminating at a saddle homoclinic orbit bifurcation as a function of the applied current strength in the Morris-Lecar model, and its phase response curve belongs to Type II. A large amount of calculation indicates that the relative size of the phase response and its positive or negative value depend only on the time of imposing perturbation, and the phase response curve of periodic bursting is more complicated than that of periodic spiking.