Stability and Hopf Bifurcation Analysis of a Continuous Neural Network with Mixed Delays

A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.


I. INTRODUCTION
In recent years, neural network models have been widely used in engineering, especially in signal processing, image recognition, pattern recognition, remote sensing technology and so on. As we all know, the mathematical theory of neural network is the premise of its application. Owing to its nonlinear characteristics, neural network models often have rich dynamic properties. Most of the early work focused on the analysis and research of neural network dynamics of non time delay, constant time delay and autonomous neural network models, and obtained many meaningful results, including equilibrium point, periodic solution, bifurcation and chaos [1]- [12].
In the circuit implementation of neural networks and the stickiness triggered between synapses of biological neural networks, there is a common phenomenon that the signal transmission has time delays, which is the primary origin of neural network oscillation and instability. Therefore, as one of the main subjects, the stability analysis of time-delay neural networks has been previously studied [4]- [17]. It was originally represented by the following two-neural networks with two discrete delays proposed by Olien and Bé lair [4] in 1997. x t x t a f x t i Afterward, Wei and Ruan [5] and Huang et al. [9] considered a simple two-neuron network model with two and four time delays, respectively. Song [8] designed a simplified bidirectional associative memory (BAM) with two delays using three neurons and investigated its stability at equilibrium and the properties of bifurcating periodic    

t x t a f F t s x s ds a f x t x t x t a f x t a f F t s x s ds
The stability and bifurcation analysis of system (2) were investigated by regarding the sum of the two discrete delays as a bifurcation parameter. Recently, Wang et al [23] gave the following three-neuron network model with multiple discrete and distributed delays                 as bifurcation parameter to discuss the existence conditions of Hopf bifurcation, the direction of bifurcations and the corresponding stability of the bifurcating periodic solutions.
What the most network systems with mixed delays involving discrete and distributed delays appear to have in common is that the distributed delay of neuron only receives it from itself failing to consider its distributed transmission effect on other neurons, including systems (2) and (3). Inspired by the viewpoint, this work develops a mathematical model of a three-neuron network with mixed delays to describe the response of discrete delays between neurons and the influence of distributed transmission delays of one neuron by itself and to another. Our aim is to give a linear stability analysis for the three-neuron network with discrete time delays and distributed delays. Also, some criteria for analyzing the characteristics of dynamic behavior are performed to decide properties of the bifurcating periodic solutions.
The rest of this article is organized as follows. In Section II, we introduce a three-neuron network with discrete and distributed delays. In Section III, we discuss some conditions to ensure the stability of equilibrium and existence of Hopf bifurcation by considering the characteristic equation of the linearization. In Section IV, the properties of the bifurcating periodic solutions are decided by employing the normal form theory and center manifold theorem. In Section V, we present some examples and numerical simulations to support theoretical analysis. Finally, we come to a conclusion shown in Section VI.

II. MODEL DESCRIPTION AND TRANSFORMATON
A continuous three-neuron network with mixed delays involving discrete and distributed delays is described as the following differential equations        is the mean delay of the kernel and one can see reference [23] for general expressions of kernel functions. It is worth noting that, in previous studies, most of the kernel functions are just expressed in the form of weak.
The simplified network architecture of system (4) is presented in Fig. 1. What is mainly different from system (3) is that the three neurons transmit signals with discrete time delay  , and the neuron 1 x receives a distributed delay input from itself and sends another to 2 x . In addition, in system (4), kernel functions and delays of the distribution terms are different.
For system (4), as a matter of convenience, we introduce Subsequently, we take the transformation system (5) as the model object for analysis. For the sake of argument, we denote that  is the set of all positive integers and paper.

III. STABILITY AND HOPF BIFURCATION ANALYSIS
Our work in this section is to analyze the stability and Hopf bifurcation of system (5). Actually, the equilibrium point , that is, the origin is the equilibrium point of system (5). Accordingly, (0, 0, 0) is the equilibrium point of system (4). At the right end of (5), using Taylor's expansion to expand () f  at * x and take its linear approximation part, we can obtain the following linear approximation system of (5) at the equilibrium point.
x t x t a bx t a bx t x t x t a bx t x t x t a bx t a bx t . Its Jacobian matrix J and corresponding characteristic equation are as follows, respectively.
We denote 55 , Square both sides of (10) and (11) respectively, and then add them together to obtain 12 10 where   2 (14) is equivalent to (15) has at least one positive root. Proof. Denote (12) and (13), 1 arctan  is a solution of equations (12) and (13), and i i  is a pair of pure imaginary roots of (7) with j i   . In other words, (7) has no pure imaginary roots when The following lemma proposed by Ruan and wei [29] is of great importance in determining the distributions of the roots of (7).
Lemma 2 [29] : Consider the exponential polynomial as the root of (8) satisfying We give the assumption below (H2) Thus, the transversality condition is arrived.
These lemmas and transversality condition imply the following main results. (2) If 0   , then the solution * x of system (5) is unstable; (3) The system (5) undergoes a Hopf bifurcation at the equilibrium *

x of system (5) is unstable; (3) The system (5) undergoes a Hopf bifurcation at the equilibrium
Obviously, (26) and (8) are essentially the same characteristic equation, so the discussion will not be repeated.

IV. PROPERITY OF HOPF BIFURCATION
In Section III, the conditions for different cases to guarantee that system (5) undergoes Hopf bifurcation and Hopf bifurcation values are obtained. In this section, we will discuss the direction and the period of bifurcating periodic solutions by employing the normal form theory and center manifold theorem introduced by Hassard et al. [30]. Next, we only take the first case 12 0   in Section III as an example, and the discussion of other cases is similar.
where   In fact, we can take where  is the Dirac delta function. Further, we define the operators Then the equation (27) is equivalent to the following operator differential equation  q q q q q d q q q q q ed D q q q q q q q q q q D q q q q q Be q q q q q q q q q q q q q q q D e b a q a q From (45) Comparing the coefficients in (51) with those in (48) ).
x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t Also, the equilibrium * x of system (66) is the origin (0, 0, 0, 0, 0, 0) . We consider the case for is asymptotically stable, its convergence trend is similar to Fig. 2
Author Name: Preparation of Papers for IEEE Access (February 2017) 12 VOLUME XX, 2017

VI. CONCLUSION
Discrete and distributed delays are usually inevitable in neural networks. Considering the response of discrete delays between neurons and the influence of the distributed transmission delays of neurons themselves and other neurons, this paper constructs a three-neuron network model with mixed delays involving multiple discrete and distributed delays, which is different from the existing models, to describe the transmission state among three neurons. We use two different kernel functions to deform the model equivalently, instead of considering only one kernel function as in a large number of previous literatures. According to different time delays, we discuss the stability of the system and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the roots of the characteristic equations of the linearized system at the equilibrium state of the nonlinear system. So as to the ranges of parameters for the system to remain stable and unstable are determined. Firstly, setting two distributed delays 12   as a bifurcating parameter, we discuss the stability of the system at origin and existence of Hopf bifurcation. It is obtained that the zero solution of the system is asymptotically stable for 1 10 [0, )   and unstable for 1 10   . The system undergoes a Hopf bifurcation at origin at 11 j i   . We find that when 1 0  , the discussion is the same as the first case, so there is no repetition. In addition, when the conditions of Hopf bifurcation theorem are satisfied for 12 0   , we give the calculation formulas to determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions with the help of the central manifold theorem and normal for methods, so as to accurately characterize the existence and stability of periodic solutions. The experimental results show that the oscillation and instability caused by time delays obviously affect the stability of the network.
It can be seen that the advantage of our model is to use continuous distributed delay instead of point delay or discrete delay to more accurately describe the changes of network state and highlight the interaction mechanism between different neurons. Apparently, the treatment of distributed delays will increase the dimension of a network. However, the stability and bifurcation of high-dimensional neural network systems with delays are still limited, because the analysis of the distribution of higher-order exponential polynomial roots with multiple time delays is quite difficult. On the other hand, with the further increase of bifurcating parameters, how it affects the bifurcating solutions, that is, the global bifurcating, need to be further studied. The authors would like to thank the editors and the referees for their helpful suggestions incorporated into this paper.