Oscillatory Behavior on a Three-Node Neural Network Model with Discrete and Distributed Delays

Neural networks are complex and large-scale dynamical systems. Time delay is inevitably encountered in implementation of dynamical neural networks and is frequently a source of oscillation and instability. On the one hand, information transmission from one neuron to another neuron may make the response of networks with discrete delays. On the other hand, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, the distributed delays in neural system need to be considered. In the past two decades, many researches have studied various neural networkswith discrete and distributed delays [1–4]. In [1], Gopalsamy and Leung studied the following neural network model:


Introduction
Neural networks are complex and large-scale dynamical systems.Time delay is inevitably encountered in implementation of dynamical neural networks and is frequently a source of oscillation and instability.On the one hand, information transmission from one neuron to another neuron may make the response of networks with discrete delays.On the other hand, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.Thus, the distributed delays in neural system need to be considered.In the past two decades, many researches have studied various neural networks with discrete and distributed delays [1][2][3][4].In [1], Gopalsamy and Leung studied the following neural network model: ( By means of the Lyapunov functional, the authors obtained some necessary and sufficient conditions for the existence of a globally asymptotically stable equilibrium point of system (1).Ruan and Filfil [2] where the delayed feedback kernel satisfies ∫ ∞ 0   () = const.and ∫ ∞ 0   () < ∞, () = (1/(1 +  − )) ( = 1, 2).Local stability analyses are carried out for model (2).Computer simulations are performed to illustrate the obtained 2 Advances in Artificial Neural Systems results.Liao et al. [3] discussed a two-neuron system with distributed delays in the frequency domain as follows: 2 ()  = −  2 () where () is a strong kernel.For simplicity, the authors set Then system (3) is equivalent to the following model: By applying the frequency domain approach and analyzing the characteristic equation, the Hopf bifurcation occurs when the mean delay  exceeds a critical value.Thus, there is a family of periodic solutions bifurcates from the equilibrium point.Recently, Hajihosseini et al. [4] have investigated the following three-node network model: where  1 and  2 are parameters.The authors also make the change of variables   () = ∫ ∞ 0 ()  ( − ) ( = 1, 2, 3) such that system (6) changes to the following: Under the restrictive condition | 1 +  2 | < 1, the Hopf bifurcation and stability of the bifurcating periodic solutions have been discussed.There are many authors who have studied the bifurcating periodic solutions for various models [5][6][7][8][9][10][11][12][13][14].It is well known that the basic method in studying the bifurcating periodic solution for a time delay system is to discuss an algebraic equation in which the bifurcating value will be determined.Motivated by the above models we shall concern the existence of oscillating solutions for the following three-node network system with discrete and distributed delays: where () =  2  − ( > 0) is a strong kernel function, the passive decay rates  1 ,  2 , and  3 are positive constants,   ,   , and   ( = 1, 2, 3) are constants, and delays  1 ,  2 , and  3 are nonnegative constants.It was emphasized that bifurcating approach is hard to deal with model (8), because it is very difficult to find the bifurcating parameter when  1 ,  2 , and  3 are different nonnegative constants.In order to discuss the existence of oscillating solutions for system (8) we adopt Chafee's criterion [15].A time delay system will generate a limit cycle if the system has a unique unstable equilibrium point and bounded solutions.In other words, there exists an oscillatory solution of the model.System (8) can accord with the demands of Chafee's criterion; we refer the reader to [16, appendix].

Preliminaries
First we assume that the activation functions  and  both are monotone continuous bounded functions.,  ∈  4 (), (0) = (0) = 0, () > 0, and () > 0 for  ̸ = 0.For example, () = tanh(), and () = arctan () satisfy those conditions.For the kernel function () =  2  − ( > 0) we have From ( 9) we get Taking the derivative on both sides of system (8) we obtain Set so ( 11) can be rewritten as follows: Taking the derivative again on both sides of (13) and using (10) give Now by setting 2 , and  () =  2  3 ()/ 2 , we get the following time delay equivalent system of (8): The linearization of system (15) around the zero point is the following: Advances in Artificial Neural Systems 5 System ( 16) can be written as a matrix form: where where is a nonsingular matrix.Then system (8) has a unique equilibrium point.
Proof.Noting that  and  both are monotone continuous bounded functions satisfying (0) = (0) = 0. Then () =   (0) + , () =   (0) + , where  and  both are higher order infinitesimals when  → 0. From ( 8), an equilibrium point is a solution of the following algebraic equation: Advances in Artificial Neural Systems is another equilibrium point of system (8), neglecting of the higher order infinitesimal, we obtain The matrix form of ( 23) is follows: Since  is a nonsingular matrix based on the algebraic knowledge one can have  * −  * = 0 or  * =  * .This means that system (8) has a unique equilibrium point.Obviously, the unique equilibrium point exactly is zero point since (0) = (0) = 0.

Oscillating Solutions Analysis
We adopt the following norms of vectors and matrices: The measure () of the matrix  is defined by () = lim  → 0 + ((‖ + ‖ − 1)/), which for the chosen norms reduces to In order to discuss the instability of equilibrium point for system (8), we consider the equivalent system ( 15) of ( 8).Note that the linearized system of ( 15) is (16).Obviously, if the trivial solution of ( 16) is unstable, it implies that the trivial solution of system ( 15) is unstable and thus the instability of the trivial solution of system (8).Therefore, we first have Theorem 3. Assume that system (8) has a unique equilibrium point and the determinant of matrix  is not equal to zero.Let  1 ,  2 , . . .,  9 ,  1 ,  2 , . . .,  9 be the eigenvalues of the matrixes  and , respectively.If there is at least one Re  > 0 (Im  may be equal to zero) or there exists one positive real eigenvalue   > |  |, for some  ∈ {1, 2, . . ., 9}, then the unique equilibrium point, namely, the trivial solution of system (16) is unstable, implying that the equivalent system (15) or (8) generates oscillating solutions.
Proof.Since  is a singular matrix, set  1 =  2 = ⋅ ⋅ ⋅ =  6 = 0.For given value of , let  1 ,  2 , . . .,  9 be the eigenvalues of the matrix .Thus, the characteristic equation of system ( 16) is as follows: where Since there is at least one Re  > 0, without loss of generality, set Re 1 > 0. From  1 = 0 we have the equation  −  1 = 0.This means that  =  1 , and there is a Re  > 0. So, the unique equilibrium point of system ( 16) is unstable based on the theory of differential equation.If by corresponding some   we have   > |  |  ∈ {1, 2, . . ., 9}, we pointed out that there exists positive real root for transcendental equation  −   −    −  = 0. Let () =  −   −    −  .Then () is a continuous function of .Since (0) = −  −   , from   > |  |, we know that (0) < 0. Obviously, there is a suitable large  * > 0 such that ( * ) =  * −   −    − *   > 0 since  − *   can be suitably small.Thus, by the continuity of (), there is a point, say  ∈ (0, * ) such that () = 0.In other words, there is a positive real eigenvalue.In this case the unique equilibrium point of system ( 16) is also unstable.Since all solutions of the system are bounded, on the basis of Chafee's criterion [15], system (15) generates a limit cycle, implying that there exists an oscillating solution of system (15) and therefore system (8).
Proof.First we consider system (16) We claim that the trivial solution of (31) is unstable.Suppose that this is not true; then the characteristic equation associated with Inequality (36) contradicts the first inequality of (28).Therefore, our claim regarding the instability of the trivial solution is valid.Similarly, consider system (16) in the case  1 =  2 =  3 =  * ; we know that the trivial solution is unstable if the second inequality of (28) holds.Note that  ≤   ≤  * ( = 1, 2, 3).So, the trivial solution of ( 16) is unstable when condition (28) holds, implying that the trivial solution of ( 15) is unstable.According to Chafee's criterion, system (15) generates a limit cycle, suggesting that there is an oscillating solution of system (15) and therefore system (8).and  2 = 0.8,  3 = 0.5, time delays are (0.5, 0.6, and 0.7).The oscillatory solutions also appeared.This means that the restrictive condition (0) = (0) = 0, () > 0, and () > 0 for  ̸ = 0 is only for convenience of the proof.

Conclusion
This paper discusses the oscillatory behavior of the solutions for a three-node network model with discrete and distributed delays.Two theorems are provided to ensure the existence of oscillating solutions for the model.Computer simulations suggested that our theorems are only sufficient conditions.How to find a necessary condition is still an open problem.
in the case  1 =  2 =  3 =  and we easily get: