Global Stability, Bifurcation, and Chaos Control in a Delayed Neural Network Model.
In recent years, neural networks (especially Hopfield type, cellular, bidirectional, and recurrent neural networks) have been applied successfully in many areas such as signal processing, pattern recognition, and associative memories. In most of the research works, stability of the designed neural network is an important step of analyzing the dynamics. Chaos plays an important role in human brain cognitive functions related to memory process. For example, chaotic behavior has been observed in nerve membranes by electrophysiological experiments on squid giant axons [1-3] and in measurements of brain electroencephalograms (EEG) [4, 5]. At first, Aihara et al.  introduced chaotic neural network models in order to simulate the chaotic behavior of biological neurons. Both the network and its component neuron are responsible for chaotic dynamics if suitable parameter values are chosen [6,7]. The investigation of chaotic neural networks is of practical importance and many interesting results have been obtained so far (see [8-11] and the references therein). The control of chaotic behavior in chaotic neural networks is an important problem to apply them in information processing . The first chaos control was proposed by Ott et al. (the OGY method) . Since this pioneer work of OGY, various methods such as the occasional proportional feedback (OPF) method , continuous feedback control , and pinning method  have been proposed for chaos control.
Many research works have been done to know the effect of time delays in neural system [17-19]. The malfunctioning of the neural system is often related to changes in the delay parameter causing unmanageable shifts in the phases of the neural signals. In the olfactory system, the phase transition has the appearance of a change in the EEG from a chaotic, aperiodic fluctuation to a more regular nearly periodic oscillation. In fact, neural network with delay can actually synchronize more easily controlling chaotic behavior of the system [10, 20]. Besides, change of neuronal gain parameter causes a change of all connectivity weights and therefore affects the dynamical behavior of the network .
For this reason, we are motivated to study effectiveness of time delay as well as neuronal gain parameters in changing the dynamics of an artificial neural network model. Using Lyapunov functional method, we studied the global stability analysis of the system and obtained sufficient criteria involving synaptic weights. We investigated the system numerically without time delay and with time delay. Chaotic behavior of the system without delay is controlled as the interconnection transmission delay is introduced in the model. Hopf-bifurcation analysis of the system with respect to time delay as parameter is discussed. Analytical results are verified using numerical simulations to show the reliability and effectiveness of the model. Numerical simulations of the model (using time series, phase portraits, and bifurcation diagrams) are presented showing changes of dynamics of the system. Chaotic behavior is observed when a gradual increase of slope of activation functions is made. Finally, some concluding remarks have been drawn on the implication of our results in the context of related work mentioned above.
2. Mathematical Model with Time Delay and Global Stability
We consider an artificial n-neural network model (n [greater than or equal to] 3) of time delayed connections between the neurons using system of delay differential equations:
[mathematical expression not reproducible] (1)
[mathematical expression not reproducible], (2)
where [f.sub.k](Y) is the response functions and [[theta].sub.k] and [[beta].sub.k] are the threshold and the slope (neuronal gain parameter) of the response function of the neuron k, respectively. The value of this sigmoidal type of response function is always nonnegative, bounded by 0 and 1. [W.sub.j1] denotes the weight of the synaptic connection from the first neuron to the jth neuron, [[alpha].sub.k] is the decay or degradation rate parameter, and [[tau].sub.k] is the signal transmission delay and nonnegative constant.
In the following, we assume that each of the relations between the output of the cell f and the state of the cell possesses the following properties.
(H1) f is bounded function in R;
(H2) f is continuous and differentiable nonlinear function.
To clarify our main results, we present the following lemma and proof under more general conditions.
Lemma 1. For the delayed nonlinear system (1), suppose that the output of the cell f satisfies the hypotheses (H1) and (H2) above. Then all solutions of (1) remain bounded for [0, +[infinity]).
Proof. It is easy to verify that all solutions of (1) satisfy differential inequalities of the form
-[[alpha].sub.k][[gamma].sub.k] (t) -[[gamma].sub.k] [less than or equal to] [[??].sub.k] (t) [less than or equal to] - [[alpha].sub.k][y.sub.k](t) + [[gamma].sub.k] (3)
where [[gamma].sub.k] = sup [absolute value of ([f.sub.k] [[[summation].sup.n.sub.i=1] [w.sub.ik], [y.sub.k] (t - [[tau].sub.i])])], k = 1, 2, ..., n; and
[mathematical expression not reproducible]. (4)
Using (3), one can easily prove that solutions of (1) remain bounded on [0, +[infinity]). This completes the proof. ?
By coordinate translation [v.sub.k](t) = [y.sub.k](t) - [y.sup.*.sub.k], k = l, 2, ..., n, and assuming [[tau].sub.1] = [[tau].sub.2] = ... = [[tau].sub.n] = [tau], (1) can be written as
[mathematical expression not reproducible] (5)
[mathematical expression not reproducible]. (6)
Clearly, [(0, 0, ..., 0).sup.T] is an equilibrium of (5). To prove the global asymptotic stability of [y.sup.*] of (1), it is sufficient to prove global asymptotic stability of the trivial solution of (5).
This section derives the criteria for the global asymptotic stability of the trivial equilibrium of the network by constructing a suitable Lyapunov functional.
Theorem 2. If (i) [square root of 2][[alpha].sub.2] > [square root of ([[summation].sup.n.sub.k=1] [[alpha].sub.k])] and (ii) 2 [[summation].sup.n.sub.j=2] [[alpha].sup.2.sub.j] > [[[summation].sup.n.sub.j=2] [[alpha].sub.j] + [[alpha].sub.1] [[summation].sup.n.sub.j=2] [w.sup.2.sub.j1]], then the trivial solution of (5) is globally asymptotically stable for any delay.
Proof. We consider Lyapunov functional defined by
[mathematical expression not reproducible] (7)
As [g.sub.1] and [g.sub.j] (j = 2, 3, ..., n) satisfy the propositions on [f.sub.1] and [f.sub.j], we have the following expressions:
[mathematical expression not reproducible], (8)
where [mathematical expression not reproducible], (9)
where [q.sub.j1] (t) = [[integral].sup.1.sub.0] [g'.sub.j]([rv.sub.1](t))dr, and there exist [p.sup.*], [q.sup.*] [member of] (0,1] such that [p.sub.j1](t) [less than or equal to] [p.sup.*] [less than or equal to] 1, [q.sub.j1](t) [less than or equal to] [q.sup.*] [less than or equal to] 1, j = 2, 3, ..., n.
[mathematical expression not reproducible]. (10)
Similarly, [[summation].sup.n.sub.j=2] [g.sup.2.sub.j][[v.sub.1](t)] [less than or equal to] [v.sup.2.sub.i](t).
[mathematical expression not reproducible]. (11)
So if [mathematical expression not reproducible], then dV/dt < 0 when v [not equal to] 0; hence the trivial solution of (5) is globally asymptotically stable.
3. Bifurcation Analysis
As it is very difficult to study the dynamics of the proposed model analytically, we will study bifurcation analysis for n = 3 only with three distinct delays. Hence our mathematical model will take the form as
[mathematical expression not reproducible]. (12)
For simplicity, the following set of new state variables are introduced:
[x.sub.1](t) = [y.sub.1] (t - ([[tau].sub.2] + [[tau].sub.3])), [x.sub.2](t) = [y.sub.2] (t - [[tau].sub.2]), [x.sub.3](t) = [y.sub.3](t - [[tau].sub.3]). (13)
Also, let [sigma] = [[tau].sub.1] + [[tau].sub.2] + [[tau].sub.3].
Now, system (12) takes the following equivalent form with single delay ([sigma]) as
[mathematical expression not reproducible]. (14)
Linearizing the system about the steady state [E.sup.*] = ([x.sup.*] [x.sup.*.sub.2], [x.sup.*.sub.3]), we get
[mathematical expression not reproducible], (15)
where [u.sub.1] = [x.sub.1] - [x.sup.*.sub.1], [u.sub.2] = [x.sub.2] - [x.sub.2], and [u.sub.3] = [x.sub.3] - [x.sup.*.sub.3].
The characteristic equation associated with the linearized system (14) is
D ([lambda], [tau]) = [[lambda].sup.3] + [A.sub.1][[lambda].sup.2] + [A.sub.2][lambda] + [A.sub.3] - ([A.sub.4][lambda] + [A.sub.5]) [e.sup.-[lambda][sigma]] = 0, (16)
[mathematical expression not reproducible], (17)
and [[beta].sub.1] > 0 so M > 0, N > 0, and P > 0.
Let i[omega] ([omega] > 0) be a root of (16); then
-i[[omega].sup.3] - [A.sub.1][[omega].sup.2] + i[A.sub.2][omega] + [A.sub.3] - (i[A.sub.4][omega] + [A.sub.5]) (cos [omega][sigma] - i sin [omega][sigma]) = 0. (18)
Eliminating the harmonic terms in (18), we have
D ([omega]) = [[omega].sup.6] + ([A.sup.2.sub.1] - 2[A.sub.2])[[omega].sup.4] + ([A.sup.2.sub.2] - 2[A.sub.1] [A.sub.3] - [A.sup.2.sub.4]) [[omega].sup.2] + [A.sup.2.sub.3] - [A.sup.2.sub.5] = 0. (19)
The existence of unique ac is given by
[mathematical expression not reproducible]. (20)
So for different values of [eta], we will get different values of [[sigma].sub.c]. For simplicity, we will denote the minimum value of [[sigma].sub.c] by [[sigma].sub.0] and let [[omega].sub.0] be the positive and simple root of (18). By differentiating [lambda] with respect to [sigma] at [sigma] = [[sigma].sub.c] in (16), we will get
[lambda]' ([[sigma].sub.c]) = [lambda]([A.sub.4] [lambda] + [A.sub.5])/(3[[lambda].sup.2] + 2[A.sub.1][lambda] + [A.sub.2]) [e.sup.[lambda][tau]] - [A.sub.4] + [[sigma].sub.c] ([A.sub.4][lambda] + [A.sub.5]). (21)
Next, based on the above results, a theorem is established.
Theorem 3. [E.sup.*] is asymptotically stable for [sigma] = 0, and it is impossible that it remains stable for all [sigma] > 0. Hence, there exists a [[sigma].sub.c] > 0, such that, for [sigma] < [[sigma].sub.c], [E.sup.*] is asymptotically stable and, for [sigma] > [[sigma].sub.c], [E.sup.*] is unstable and as [sigma] increases through [[sigma].sub.c], [E.sup.*] bifurcates into small amplitude periodic solutions of Hopf type .
4. Numerical Simulation
In this section, the network consisting of five neurons with connections as depicted in Figure 1 will be considered. For bifurcation results, we assume n = 3. Similar results can be obtained also for n = 6. We used Matlab 22.214.171.1249 for simulation of numerical examples.
4.1. Chaos Control. We consider the system n = 5 without any time delay (see Example 1). This system (22) shows chaotic nature (see Figure 2(a)) which is controlled as transmission delay is introduced (see Example 2). Here, we have considered time delay [tau] as system parameter keeping other system parameter values as in system (22). Figure 2(b) shows periodic behavior in presence of delay ([tau] = 5). Bifurcation diagram with respect to time delay [tau] is also illustrated in Figure 3.
Example 1. Consider
[mathematical expression not reproducible]. (22)
Example 2. Consider
[mathematical expression not reproducible]. (23)
4.2. Verification of Global Stability Criteria. Here, we consider the model (5) when n = 5 and the parameter values are so chosen ([[alpha].sub.2] = 4.5) which satisfy the global stability criteria independent of delay. Considering [tau] = 15, we observe stable behavior of the system in Figure 4. But considering the same time delay ([tau] = 15), the numerical simulation of Example 3 violating the conditions of global stability criteria shows the chaotic behavior in Figure 5.
Example 3. Consider
[mathematical expression not reproducible]. (24)
4.3. Bifurcation Results
4.3.1. Bifurcation with Respect to Total Time Delay a. We consider model (14) with total time delay [sigma](= [[tau].sub.1] + [[tau].sub.2] + [[tau].sub.3]) in the following example.
Example 4. Consider
[mathematical expression not reproducible]. (25)
Substituting the system parameters into (19) and (20) for [eta] = 0, we obtain [[omega].sub.0] = 1.3, [[sigma].sub.0] = 0.36, and [lambda]'([[sigma].sub.0]) = 0.1 - 0.13i and the system has a positive equilibrium [E.sup.*] (0.64,2.15,3.14). When [sigma] = 0, the positive equilibrium [E.sup.*] is asymptotically stable and it remains stable when [sigma] < [[sigma].sub.0] as is illustrated by the computer simulations (see Figure 6(a)) and system (25) is unstable for all [sigma] > [[sigma].sub.0] (see Figure 6(b)). Bifurcation diagram with respect to time delay a showing critical value is illustrated in Figure 7(a).
4.3.2. Bifurcation with Respect to Neuronal Gain Parameter [[beta].sub.1]. We consider model (12) with the same time delay in the following example.
Example 5. Consider
[mathematical expression not reproducible]. (26)
Bifurcation diagram with respect to [[beta].sub.1] is demonstrated in Figure 7(b). Gradual increase of slope [[beta].sub.1] of the activation function causes periodic oscillation and then chaos through period doubling of the system (26).
In this paper, we studied the global stability of artificial neural network model of n-neurons with time delay and obtained the criteria of involving the synaptic weight and decay parameters, independent of delay. The system shows chaotic behavior without delay and introduction of delay plays a vital role in controlling chaos in the system. Also bifurcation analysis of the model with respect to time delay shows transition from stable to unstable (i.e., resting to rhythmic) behavior increasing the value of delay. We have also illustrated the effect of changing slope of the sigmoidal activation function (gain parameter in the dynamics of the neural network. Neural gain can be thought of as an amplifier of neural communication. When gain is increased, excited neurons become more active and inhibited neurons become less active . The numerical simulation shows that the system dynamics undergo changes from stable to periodic and then to chaotic behavior through period doubling as the gain parameter is increased. A similar kind of behavior can be observed also for other neuronal gain parameters. The network with very small gain and small delay may not be very useful for implementation in models of oscillation generators and associative memories.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Amitava Kundu and Pritha Das
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India
Correspondence should be addressed to Pritha Das; email@example.com
Received 29 May 2014; Accepted 15 September 2014; Published 8 October 2014
Academic Editor: Matt Aitkenhead
Caption: FIGURE 1: Structure of the five neural networks investigated.
Caption: FIGURE 2: Time series (a) of the system (22) without delay showing chaotic behavior; (b) control of chaos of the system (23) can be seen in solution trajectory with showing periodic behavior when delay t = 5.
Caption: FIGURE 3: Bifurcation diagram with respect to time delay t showing control of chaos of system (22).
Caption: FIGURE 4: Time series and corresponding phase portrait showing stable behavior satisfying global stability conditions when delay [tau] = 15.
Caption: FIGURE 5: Time series and corresponding phase portrait showing chaotic behavior when delay [tau] = 15 violating global stability conditions.
Caption: FIGURE 6: Solution trajectory showing (a) asymptotic stable behavior when [sigma] = 0.2 < 0.36 = [[sigma].sub.0]; (b) periodic behavior with [sigma] = 0.8 > 0.36 = [[sigma].sub.0].
Caption: FIGURE 7: Bifurcation diagram (a) of the system (25) with respect to time delay [sigma] showing critical value [[sigma].sub.0] = 0.36; (b) of the system (26) with respect to [[beta].sub.1] when [tau] = 12.
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|Title Annotation:||Research Article|
|Author:||Kundu, Amitava; Das, Pritha|
|Publication:||Advances in Artificial Neural Systems|
|Date:||Jan 1, 2014|
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