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A nonsmooth neuron model with periodic excitation which can reproduce spiking and bursting behavior of cortical neurons is investigated in this paper. Based on nonsmooth bifurcation analysis, the mechanism of the bursting behavior induced by slow-changing periodical stimulation as well as the associated evolution with the variation of the stimulation is explored. The modulating character of the external excitation and the effect of the bifurcation occurring at the switching boundary of the vector field are presented.

Recently, understanding of the brain and its behavior has been a research focus for its importance and various applications. As primary building blocks of brain, innumerable neurons compose the very complex information network. It can encode, transfer, and integrate information by firing activities [

The neuron is one of the most sophisticated nonlinear dynamical systems. A series of dynamical models which can describe the corresponding action potential shapes for different neuronal cell types has been developed [

To explore the influences of periodical stimulation on the dynamical behavior of the neuron, a two-dimensional PWL neuron model is discussed in this paper. We tackle the case with slowly varying periodic excitation and investigate the evolution of the bursting dynamics based on bifurcation analysis.

This model is developed from a PWL modification of the Izhikevich model [^{+} ionic currents and inactivation of Na^{+} ionic currents and it provides negative feedback to

Here, we focus on the case with a periodic stimulation; that is, the stimulation in model (

Periodic bursting and time series of system (

The transient states in these two cases are both eliminated.

Demonstrated as the red curve in Figure

System (

Thus, system (

Since the value of

For this PWL autonomous system, one switching boundary exists, denoted by

Further analysis indicates that the number of equilibrium points is not constant under the above parameters due to piecewise linearity of

Distribution of the equilibria of system (

The stabilities of these equilibria of the associated autonomous systems are determined by the eigenvalues of the related linearization matrix. Here, we take the case of

Obviously, the equilibrium point

Now we focus on the full system (

Due to the linear structure of the system, though nonsmooth, the generalized solution for system (

The details can be described as follows:

(1) For

(2) For

The “solution” expressed by (

The “generalized solution” for system (

The set of the “generalized equilibrium points” is not straight lines but piecewise curves (seen in the magnification of the curves).

It should be pointed out that the classical continuous Jacobian matrix cannot be obtained due to the nonsmoothness of the vector filed of this system. According to differential inclusions, we can use the generalized differential of Clarke to set up a “generalized Jacobian matrix” to explore the bifurcation of the equilibrium at the switching boundary and the “generalized Jacobian matrix” of system is expressed by

Figure

Set of eigenvalues of the generalized Jacobian of the system.

Now we explain the mechanism of the periodic bursting based on nonsmooth bifurcation theory. For this purpose, the phase portrait shown in Figure

Overlap of the “solution” and the phase portraits of system (

In Figure

The “generalized equilibrium points” are stable ones in region (I), and they lose their stability at the switching boundary

Point

The orbit may move down exactly along the stable “solution” with the variation of the stimulation, that is, the angular frequency

We now account for the evolution of the phase portrait with the increases of the stimulation amplitude.

For the purpose of comparison, Figure

Partial enlarged detail of phase portraits of system (

This paper focuses on a nonautonomous piecewise linear neuron model. Bursting behavior in this system as well as the associated evolution induced by the slow-varying periodic stimulation is discussed by means of the dynamical bifurcation analysis. Due to the nonsmoothness of the vector field, discontinuous Hopf bifurcation occurring at the switching boundary is pointed out to induce a much higher frequency relative to the stimulation frequency and to connect the fast spiking state and the slow quiescent state. Periodic stimulation is suggested to be an important element related to bursting phenomenon closely. The method applied in this paper may be considered as a possible way to analyze the effects of periodic input on the information processing pattern of neuron.

The authors declare that they have no competing interests.

This work is supported by the National Natural Science Foundation of China (Grant nos. 11302086, 11374130, and 11474134), Natural Science Foundation of Jiangsu province (Grant no. BK20141296), and Foundation of Six Kinds of High Talents of Jiangsu Province (Grant no. 2015-DZXX-023).