Stability analysis of delayed neural networks with slope-bounded activation functions

: This paper deals with the global asymptotic stability problem of delayed neural networks with unbounded activation functions and network parameter uncertainties. New stability criteria for global asymptotic stability of the delayed neural networks are derived by employing suitable Lyapunov functionals. These results reported in this paper can be regarded as generalizations of some existing stability results. The effectiveness and usefulness of the obtained results can be verified by comparing our results with the previously published results.


Introduction
In the past decades, there has been a steady increase in the interest on the applications of dynamical neural networks in solving various classes of engineering problems such as image and signal processing, associative memory design, combinatorial optimization, and pattern recognition. In fact, most of these practical applications need the process information to be in the form of stable states. Hence, in order to apply neural networks to solve these practical problems, the stability property of the equilibrium point for the designed neural networks will be crucially essential.
For a biological neural network or artificial neural network, time delays are sometimes unavoidable to be taken into account. For instance, in electronic circuits of neural networks, time delays will PUBLIC INTEREST STATEMENT In this paper, a more general class of activation functions are presented and they are not required to be bounded, differentiable, and monotonically increasing. Different from existing results, the slope of this class of activation functions exist both upper and lower bounds, and they may be positive, negative, or zero. In addition, more information of the states, activation functions, and upper bounds of the delay derivative of the time varying delays are taken into consideration. A new Lyapunov functional is constructed and utilized to derive sufficient conditions to guarantee the global asymptotic stability of the neural network.
This paper should be of interest to a broad readership including those interested in neural networks.
I confirm that this manuscript has not been published elsewhere and is not under consideration by another publication. always occur during the signal procession and transmission, which may lead to oscillation and deteriorate the stability performance. In addition, due to the existence of external disturbances and parameter fluctuations, uncertainties are other critical issues that may destabilize the neural networks. Therefore, robust stability analysis of neural networks in the presence of time delays and uncertainties will be of theoretical and practical importance.
Motivated by the above discussions, in this paper, we will focus on investigating better sufficient conditions ensuring the existence, uniqueness and global asymptotic stability for delayed neural networks. The obtained results can be regarded as generalizations of the previously published corresponding results by the following improvements. (1) A more general class of activation functions is presented and they are not required to be bounded, differentiable, and monotonically increasing. Different from existing results in (Arik, 2014a(Arik, , 2014bFaydasicok & Arik, 2013), the slope of this class of activation functions exist both upper and lower bounds, and they may be positive, negative, or zero. (2) More information of the states, activation functions, and upper bounds of the delay derivative of the time varying delays are taken into consideration. A new Lyapunov functional is constructed and utilized to derive sufficient conditions to guarantee the global asymptotic stability of the neural network.

Problem description and preliminaries
In this paper, we will study the robust stability of the following delayed neural network model: where n the number of the neurons, x i (t) denotes the state of the neuron i and c i is the charging rate for the neuron i. a ij and b ij denote the strengths of connectivity between neurons j and i at time t and t − ij (t), respectively. u i is the constant input to the i-th neuron. f j x j (t) denotes the j-th neuron activation function. The delay parameters are time-varying and denoted by τ ij (t).
The uncertainties in the network parameters A = (a ij ), B = (b ij ) and C = diag (c i > 0) can be formulated as follows: (1) In this section, we will present sufficient conditions for robust asymptotic stability of equilibrium point of delayed neural network (1). We first shift the equilibrium point x* of system (1) to the origin.
The transformation z i (.) = x i (.) − x * i is used to put system (1) in the following form: Note that it can easily be verified that the function g i satisfies the assumptions on f i, that is, Since z(t) → 0 implies x → x*, it is thus necessary and sufficient to prove the stability of the transformed model (3) instead of considering stability of the original one (1).

Comparisons and examples
We now consider the following example to compare our results with those previous results given above: Example 1 Assume that the networks parameters of the delayed neural network (1) are given as follows: Now let us apply the result of Theorem 1 to this example. In this case, we first note that In the case of applying the result of Theorem 11 in Xie et al. (2014) for this example, we obtain from which it can be calculated that Ψ > 0 if and only if c m > 6.16, meaning that the sufficient condition for robust stability is obtained when c m > 6.16. In order to compare the result of Theorem 12 in Xie et al. (2014) for this example, we first get , the matrix Θ is obtained in the form of The choice c m > 4.52 implies that Θ > 0 which guarantees the global robust stability of system (1). When checking the constraint condition of Theorem 2 in Ozcan and Arik (2014) for the network parameters of this example, we obtain ζ i = 4c m -14, ζ i = 4c m -13, ζ i = 2c m -10, ζ i = 2c m -11. Hence, for the network parameter of this example, we note that the conditions of Theorem 2 in Ozcan and Arik (2014) are satisfied if c m > 5.5. In the case of applying the result of Theorem 2 in (Faydasicok and Arik (2013), we obtain ∈ = c m − 4in which the robust stability conditions are satisfied if c m > 4.

Conclusion
This paper has studied the robust stability problem of neural networks with time-varying delays. An appropriate Lyapunov functional is constructed to derive sufficient conditions to ensure their globally asymptotic stability. The obtained results cannot only be used to testify the dynamic behaviors of the equilibrium point, but also generalize the existing results.