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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 689319, 17 pages
Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling
1Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, China
2Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China
Received 27 February 2012; Accepted 26 May 2012
Academic Editor: Yuriy Rogovchenko
Copyright © 2012 Changjin Xu and Peiluan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results.
In recent years, a number of different classes of neural networks with or without delays, including Hopfield networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associate memory neural networks have been active research topic as , and substantial efforts have been made in neural network models, for example, Huang et al.  studied the global exponential stability and the existence of periodic solution of a class of cellular neural networks with delays, Guo and Huang  investigated the Hopf bifurcation natures of a ring of neurons with delays, Yan  analyzed the stability and bifurcation of a delayed tri-neuron network model, Hajihosseini et al.  made a discussion on the Hopf bifurcation of a delayed recurrent neural network in the frequency domain, and Liao et al.  did a theoretical and empirical investigation of a two-neuron system with distributed delays in the frequency domain. Agranovich et al.  considered the impulsive control of a hysteresis cellular neural network model. For more information, one can see [8–24]. In 1986 and 1987, Babcock and Westervelt [25, 26] had investigated the stability and dynamics of the following simple neural network model of two neurons with inertial coupling: where is the input voltage of the th neuron, denotes the output of the th neuron, is the damping factor, and is the overall gain of the neuron which determines the strength of the nonlinearity. For a more detailed interpretation of the parameters, one can see [25, 26]. In 1997, Lin and Li  made a detail discussion on the bifurcation direction of periodic solution for system (1.1).
From applications point of view, considering that there is a time delay (we assume that it is ) in the response of the output voltages to changes in the input, that is, there exists a feedback delay of the input voltage of the th neuron to the growth of the output of the th neuron, then we modify system (1.1) as follows: It is well known that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation is very critical. When delays are incorporated into the network models, stability and Hopf bifurcation analysis become more difficult. To obtain a deep and clear understanding of dynamics of neural network model of two neurons with inertial coupling, we will make a discussion on system (1.2), that is, we study the stability, the local Hopf bifurcation for system (1.2).
The remainder of this paper is organized as follows. In Section 2, we investigate the stability of the equilibrium and the occurrence of local Hopf bifurcations. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results.
2. Stability of the Equilibrium and Local Hopf Bifurcations
In this section, we shall study the stability of the equilibrium and the existence of local Hopf bifurcations. For simplification, we only consider the zero equilibrium. One can check that if the following condition: holds, then (1.2) has a unique equilibrium . The linearization of (1.2) at is given by whose characteristic equation takes the form of that is, Multiplying on both sides of (2.4), it is easy to obtain In order to investigate the distribution of roots of the transcendental equation (2.5), the following lemma is helpful.
Lemma 2.1 (see ). For the following transcendental equation: as vary, the sum of orders of the zeros of in the open right half-plane can change, and only a zero appears on or crosses the imaginary axis.
For is a root of (2.5) if and only if Separating the real and imaginary parts gives Then, we obtain In view of , then, we have which is equivalent to where , , , .
Denote Since and , then we can conclude that (2.14) has at least one positive root. Without loss of generality, we assume that (2.14) has sixteen positive roots, denoted by . Then, by (2.12), we have where , then are a pair of purely imaginary roots of (2.4) with . Define The above analysis leads to the following result.
Let be a root of (2.5) near , , and . Due to functional differential equation theory, for every , there exists such that is continuously differentiable in for . Substituting into the left hand side of (2.5) and taking derivative with respect to , we have Noting that where , . ,
Theorem 2.3. If and hold, then the equilibrium of system (1.2) is asymptotically stable for . Under the conditions and , if the condition holds, then system (1.2) undergoes a Hopf bifurcation at the equilibrium when , ; .
3. Direction and Stability of the Hopf Bifurcation
In the previous section, we obtained conditions for Hopf bifurcation to occur when . In this section, we shall obtain the explicit formulae for determining the direction, stability, and periods of these periodic solutions bifurcating from the equilibrium at these critical value of , by using techniques from normal form and center manifold theory . Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcation at the equilibrium for , and then are corresponding purely imaginary roots of the characteristic equation at the equilibrium .
For convenience, let and , where is defined by (2.16) and , drop the bar for the simplification of notations, then system (2.2) can be written as an FDE in as where and , and are given by respectively, where .
From the discussion in Section 2, we know that if , then system (3.1) undergoes a Hopf bifurcation at the equilibrium (0, 0, 0, 0) and the associated characteristic equation of system (3.1) has a pair of simple imaginary roots .
By the representation theorem, there is a matrix function with bounded variation components such that In fact, we can choose where is the Dirac delta function.
For , define Then, (3.1) is equivalent to the following abstract differential equation: where . For , define For and , define the following bilinear form: where , and the and are adjoint operators. By the discussions in Section 2, we know that are eigenvalues of , and they are also eigenvalues of corresponding to and , respectively. By direct computation, we can obtain where Furthermore, and .
Define on the center manifold , and we have where and and are local coordinates for center manifold in the direction of and . Noting that is also real if is real, we consider only real solutions. For solutions of (3.1), we have That is, where Hence, we have and we obtain Thus, we derive the following values: which determine the quantities of bifurcation periodic solutions of (3.1) on the center manifold at the critical value . Summarizing the results obtained above leads to the following theorem.
Theorem 3.1. The periodic solution is forward (backward) if . The bifurcating periodic solutions on the center manifold are orbitally asymptotically stable with asymptotical phase (unstable) if . The periods of the bifurcating periodic solutions increase (decrease) if.
4. Numerical Examples
In this section, we present some numerical results of system (1.2) to verify the analytical predictions obtained in the previous section. Let us consider the following system: which has an equilibrium and satisfies the conditions indicated in Theorem 2.3. The equilibrium is asymptotically stable for . Using the software MATLAB (here take for example), we derive . Thus by algorithm (3.20) derived in Section 3, we have , , , . Furthermore, it follows that and . Thus, the equilibrium is stable when . Figures 1(a)–1(j) show that the equilibrium is asymptotically stable when . It is observed from Figures 1(a)–1(j) that the input voltage of the th neuron and the output of the th neuron converge to their steady states in finite time. If we gradually increase the value of and keep other parameters fixed, when passes through the critical value , the equilibrium loses its stability and a Hopf bifurcation occurs, that is, the input voltage of the th neuron the output of the th neuron will keep an oscillary mode near the equilibrium . Due to and , the direction of the Hopf bifurcation is , and these bifurcating periodic solutions from at are stable. Figures 1(j)–2(d) suggest that Hopf bifurcation occurs from the equilibrium when .
In this paper, we have studied the bifurcation natures of a delayed neural network model of two neurons with inertial coupling. Regarding delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the equilibrium , we proposed the conditions to define the parameters for the occurrence of Hopf bifurcation and the oscillatory solutions of the models equations. It is shown that if conditions and hold, the equilibrium of system (1.2) is asymptotically stable for all . Under conditions and , if condition is satisfied, as the delay increases and crosses a threshold value , the equilibrium loses its stability and the delayed network model of two neurons with inertial coupling enters into a Hopf bifurcation. In addition, using the normal form method and center manifold theorem, explicit formulaes for determining the properties of periodic solutions are worked out. Simulations are included to verify the theoretical findings. The obtained findings are useful in applications of network control.
This work is supported by National Natural Science Foundation of China (no. 60902044), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J20122100) and Doctoral Foundation of Guizhou University of Finance and Economics (2010), Governor Foundation of Guizhou Province (2012), and the Science and Technology Program of Hunan Province (no. 2010FJ6021).
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