Original articles
Dynamical complexity of FitzHugh–Nagumo neuron model driven by Lévy noise and Gaussian white noise

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Abstract

In this paper, on the basis of information theory measures (statistical complexity and normalized Shannon entropy), the dynamical complexity of FitzHugh–Nagumo (FHN) neuron model under the co-excitation of Lévy noise and Gaussian white noise is studied. Because the potential function of the neuron system is asymmetric, we consider not only the total residence time interval of the system, but also the residence time interval of the left and right potential wells respectively. Here, we use Bandt–Pompe algorithm to calculate the three interval sequences, and obtain the statistical complexity and normalized Shannon entropy of the total system as well as the left and right potential wells. Finally, the effects of additive noise intensity, multiplicative noise intensity and system parameter on complexity of system are analyzed. We find that the total dynamical complexity of the system is obviously different from that of a single potential well. In addition, Gaussian white noise and Lévy noise have different effects on the complexity of the system.

Introduction

In nature, system is always disturbed by various external noises inevitably. Therefore, considering the influence of noise on the system has become the main direction of dynamics. Although the noise will affect the normal operation of the system sometimes, it also has its positive side, such as noise induced phase transition, stochastic resonance and noise enhanced stability [1], [5], [11], [13], [29], [34]. So the complexity of nonlinear system under noise for scientific research is important particularly. Generally speaking, the complexity of the system can be measured by the complexity measures. The statistical complexity was first proposed by Lopez-Ruiz [21] in 1995, which is based on the probability distribution function corresponding to the system time series. Subsequently, a series of studies on statistical complexity have been developed and applied rapidly. Anteneodo [3] and others further studied some properties of statistical complexity in detail and applied them. Martin et al. [22] have improved the definition of statistical complexity further, replacing the original Euclidean distance with Wootters distance, and validating the substitution by Logistic mapping. Lamberti et al. [20] applied Jensen–Shannon divergence to statistical complexity firstly, replaced Euclidean distance, and obtained a new definition of statistical complexity. Rosso et al. [26], [27] introduced statistical complexity into dynamics for the first time as an indicator of stochastic resonance. He et al. [14], [15] used statistical complexity to study the stochastic resonance and the dynamic complexity of bistable system excited by color noise. The effects of correlation time of color noise, amplitude and frequency of signal on stochastic resonance and dynamic complexity were discussed respectively. At the same time, in order to verify the effectiveness of this method, the expression of signal-to-noise ratio is also derived.

Neurons are the basic functional units of the biological nervous system. The action potential produced by neurons is the basis of the transmission and expression of nerve information. Nowadays, there are many kinds of mathematical models to describe the neuron system. In the 1950s, Hodgkin and Huxley [16] established a Hodgkin–Huxley (HH) neuron system model based on ion channel theory to study the discharge characteristics and synchronous behavior of neurons. Then FitzHugh and Nagumo [12] proposed a two-dimensional FHN neuron model, which simplified the original model. Alarcon and others [2] further simplified the two-dimensional FHN neuron model and obtained one-dimensional neuron system. Over the last few decades, the FHN neuron model has continuously attracted much attention of many researchers [10], [23], [24], [28], [30]. For example, Ref. [24] studied the effects of colored noise on stochastic resonance in sensory neurons. Ref. [28] studied the determination of firing times for the stochastic FHN neuronal model. Ref. [10] investigated the Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays. Ref. [30] discussed the stochastic resonance of FHN system under Lévy noise.

However, the current research still has some shortcomings. On the one hand, there are still few studies on Lévy noise. As a typical non-Gaussian noise, Lévy noise can reflect some random disturbances that may exist in real life more accurately. In recent years, the special effects of Lévy noise have been realized in physics, biology, economics and other fields, so the research of Lévy noise is very important [6], [7], [8], [9], [17], [18], [19], [30], [31], [32], [33], [35], [36], [37], [38]. On the other hand, there are few literatures about the dynamical complexity based on information theory measures, especially for asymmetric systems.

In this paper, the statistical complexity and normalized Shannon entropy are used to study the dynamical complexity of one-dimensional FHN neuron model under the co-excitation of Lévy noise and Gaussian white noise. Firstly, the basic theory of FHN neuron system and Lévy noise is introduced; then the definition of statistical complexity, normalized Shannon entropy and Bandt–Pompe algorithm used in this paper are explained in detail; next, numerical simulation methods of two noise are introduced and the method of solving the system is given; finally, according to the definition mentioned in this paper, the statistical complexity and Shannon entropy of the system are calculated. The effects of amplitude A, multiplicative noise intensity D, stability index α, skewness parameter β and additive noise intensity Q on the dynamical complexity of the system are analyzed.

Section snippets

FHN neuron model and Lévy noise

Consider a simplified FHN neuron model driven by Lévy noise and Gaussian white noise [10], [23], [24], [28], [30]: dvdt=v(a−v)(v−1)−bγv+Acos(ωt)+vε(t)+η(t)

Here, v represents the membrane voltage variable, 0<a<1 is the threshold, a reflects the degree of systematic fast change, b and γ are the positive constants, Acos(ωt) is the applied current with the amplitude A and the frequency ω. Owing to the simplified FHN neuron model omits the slow variable equation, the noise must be considered in

Normalized Shannon entropy and statistical complexity

In 1948, Shannon has put forward the concept of “information entropy” to solve the problem of quantifying information. Shannon entropy represents the uncertainty of the physical process described by probability distribution P=pi,i=1,…,N. It is one of the classical information theory measures, based on the following definition. S[P]=−∑i=1Npilnpi

If SP=0, the system is in a “perfect order” state, and our knowledge of the process described by the probability distribution P is maximal; on the

Dynamical complexity of FHN neuron model

Since the one-dimensional neuron system may have bistable structure only when the system parameters satisfy the condition of bγ<a−122, So according to references [10], [23], [24], [28], [30], we take a=0.5, b=0.01, γ=1, the following analysis is based on these parameters.

Conclusions

In this paper, the statistical complexity and normalized Shannon entropy are used to study the dynamical complexity of one-dimensional FHN neuron system under the co-excitation of Lévy noise and Gaussian white noise. The basic points include the following three steps. Firstly, according to the numerical simulation methods in this paper, the system’s response is obtained, and the continuous residence time interval of the system and the continuous residence time interval of the left and right

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11672207) and the Tianjin Natural Science Foundation of China (Grant No. 17JCYBJC15700)

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