Abstract
Conditions for the global asymptotic stability of delayed artificial neural network model of n (≥3) neurons have been derived. For bifurcation analysis with respect to delay we have considered the model with three neurons and used suitable transformation on multiple time delays to reduce it to a system with single delay. Bifurcation analysis is discussed with respect to single delay. Numerical simulations are presented to verify the analytical results. Using numerical simulation, the role of delay and neuronal gain parameter in changing the dynamics of the neural network model has been discussed.
1. Introduction
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. [6] 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 [12]. The first chaos control was proposed by Ott et al. (the OGY method) [13]. Since this pioneer work of OGY, various methods such as the occasional proportional feedback (OPF) method [14], continuous feedback control [15], and pinning method [16] 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 [21].
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 -neural network model of time delayed connections between the neurons using system of delay differential equations: where is the response functions and and are the threshold and the slope (neuronal gain parameter) of the response function of the neuron , respectively. The value of this sigmoidal type of response function is always nonnegative, bounded by 0 and 1. denotes the weight of the synaptic connection from the first neuron to theth neuron, is the decay or degradation rate parameter, and is the signal transmission delay and nonnegative constant.
In the following, we assume that each of the relations between the output of the cell and the state of the cell possesses the following properties.(H1) is bounded function in ;(H2) 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 satisfies the hypotheses (H1) and (H2) above. Then all solutions of (1) remain bounded for .
Proof. It is easy to verify that all solutions of (1) satisfy differential inequalities of the form where , ; and Using (3), one can easily prove that solutions of (1) remain bounded on . This completes the proof.
By coordinate translation , , and assuming , (1) can be written as where Clearly, is an equilibrium of (5). To prove the global asymptotic stability of 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) and (ii) , then the trivial solution of (5) is globally asymptotically stable for any delay.
Proof. We consider Lyapunov functional defined by
As and satisfy the propositions on and , we have the following expressions:
where ,
where , and there exist , such that , , .
Therefore
Similarly, .
Thus
So if and , then when ; 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 only with three distinct delays. Hence our mathematical model will take the form as For simplicity, the following set of new state variables are introduced: Also, let .
Now, system (12) takes the following equivalent form with single delay as Linearizing the system about the steady state , we get where , , and .
The characteristic equation associated with the linearized system (14) is where and so , , and .
Let be a root of (16); then Eliminating the harmonic terms in (18), we have The existence of unique is given by So for different values of , we will get different values of . For simplicity, we will denote the minimum value of by and let be the positive and simple root of (18). By differentiating with respect to at in (16), we will get Next, based on the above results, a theorem is established.
Theorem 3. is asymptotically stable for , and it is impossible that it remains stable for all . Hence, there exists a , such that, for , is asymptotically stable and, for , is unstable and as increases through , bifurcates into small amplitude periodic solutions of Hopf type [22].
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 . Similar results can be obtained also for . We used Matlab 7.10.0.499 for simulation of numerical examples.
4.1. Chaos Control
We consider the system 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 as system parameter keeping other system parameter values as in system (22). Figure 2(b) shows periodic behavior in presence of delay (). Bifurcation diagram with respect to time delay is also illustrated in Figure 3.
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Example 1. Consider
Example 2. Consider
4.2. Verification of Global Stability Criteria
Here, we consider the model (5) when and the parameter values are so chosen () which satisfy the global stability criteria independent of delay. Considering , we observe stable behavior of the system in Figure 4. But considering the same time delay (), the numerical simulation of Example 3 violating the conditions of global stability criteria shows the chaotic behavior in Figure 5.
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Example 3. Consider
4.3. Bifurcation Results
4.3.1. Bifurcation with Respect to Total Time Delay
We consider model (14) with total time delay ) in the following example.
Example 4. Consider Substituting the system parameters into (19) and (20) for , we obtain , , and and the system has a positive equilibrium . When , the positive equilibrium is asymptotically stable and it remains stable when as is illustrated by the computer simulations (see Figure 6(a)) and system (25) is unstable for all (see Figure 6(b)). Bifurcation diagram with respect to time delay showing critical value is illustrated in Figure 7(a).
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4.3.2. Bifurcation with Respect to Neuronal Gain Parameter
We consider model (12) with the same time delay in the following example.
Example 5. Consider Bifurcation diagram with respect to is demonstrated in Figure 7(b). Gradual increase of slope of the activation function causes periodic oscillation and then chaos through period doubling of the system (26).
5. Conclusion
In this paper, we studied the global stability of artificial neural network model of -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 [21]. 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.