Oscillation and global asymptotic stability of a neuronic equation with two delays

In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay.


Introduction
Delay differential equations have been used to describe the dynamics of a single neuron to take into account the processing time.Pakdaman et al [15] considered a neuron that has a delayed self-connection with weight a > 0 and delay τ .Implementing a decay rate λ in the model, they found that the neuron activation at time t; say x(t), follows the delay differential equation dx(t) dt = −λx(t) + K + af (x(t − τ )), (1.1) where K is the constant input received by the neuron and the neuron transfer function f is defined by f (x) = 1 1+e −x .In [4] the delay differential equation has been proposed to describe the behavior of the activation level x(t) of a single neuron which is capable of self-activation modulated by a dynamic threshold C with a single delay τ .In the absence of the threshold effect, equation (1.2) has the form Letting y(t) = x(t) − bx(t − τ ) in (1.3).Then y satisfies the equation dy(t) dt = −y(t) + a tanh(y(t)) − ab tanh(y(t − τ )), (1.4) the stability and/or bifurcation analysis of (1.4) have been studied in [4,16] and [13] but with more general activation function.It is also proved by [13,14] that (1.4) is not capable of producing chaos violating the existence of chaos conjectured by [16].
Gopalsamy and Leung [4] proved that the unique equilibrium of (1.4) and hence of (1.3) is globally asymptotically stable if a(1 − b) < 1 and a(1 + b) < 1 when a > 0 and b ≥ 0, which agrees with the findings of [13,16].El-Morshedy and Gopalsamy [2] improves the above condition by allowing the equality signs to be nonstrict.In fact, Theorem 3.1 in [2] is the best known absolute (delay-independent) global asymptotic stability criteria for (1.2).
Liao et al [10] considered a single neuron model with general activation function; namely, dx(t) dt = −x(t) + af (x(t) − bx(t − τ ) + C), They discussed the local stability as well as the existence of Hopf bifurcation under the assumption that f has a continuous third derivative.
Based on [8]; Györi and Hartung [5] investigate the stability character of the single As one may observe; all the above models contain only one delay.It has been demonstrated by [3,7] that models of single neuron can contain many delays.In this where a, b, c ∈ R, λ > 0 and with each solution of (1.5) an initial function is associated where l = max{τ, σ}.For generality reasons we will not assume that f is a tanh-like function only.Instead, we assume that f is continuous on R, f (0) = 0 and satisfies some or all of the following conditions It can be seen that the substitution In Section 2, we investigate the oscillatory character of (1.5).We say that a solution x(t) of (1.5) is nonoscillatory if it is eventually positive or eventually negative, otherwise x(t) is called oscillatory.Equation (1.5) is called oscillatory if all its solutions are oscillatory.If equation (1.5) has at least one nonoscillatory solution, then it is called nonoscillatory.The oscillation theory of the delay differential equations can be found in [3,6].In contrast with the stability of these equations, there are no absolute (delayindependent) oscillation criteria for first order delay differential equations.Although the oscillatory properties of models arising from many fields as mathematical biology is now completely characterized (see [3,6] for more details), the oscillation of equations of the form (1.5) has not yet received the deserved attention.It seems that [2] is the only work on this type of equations.
The asymptotic behavior of the trivial solution of (1.5) will be considered in Section 3. Theorem 3.1 in [2] will be extended to (1.5) and interesting delay dependent global asymptotic stability criteria are obtained which are new even for the special case (1.2).

The Oscillatory Behavior
Suppose that x(t) is a solution of (1.5).Define a function M as follows: ≥ 0 for all t ∈ [0, t 0 ).
Suppose that (ii) holds.Define a function F by Then Consequently, F (y) will be positive for all sufficiently small positive values of y.Since lim y→∞ F (y) = −∞, then there exists a positive value c such that F (c) = 0. Set x(t) = c, it follows that x(t) satisfies (1.5); i.e., (1.5) is nonoscillatory.
In view of the idea used in the proof of Theorem 2.1(ii), a more general form from it can be obtained by replacing (ii) by the following phrase: The equation −λy + af (1 − b + c)y) = 0 has at least one nontrivial root.
In the next oscillation results we will make use of the following theorem which is adapted from [6, Corollary 3.4.1]concerning the oscillation of the equation Then each of the following two conditions is sufficient for the oscillation of equation (2.4): We refer here to the fact that Theorem 2.2 is a consequence of [6, Theorem 3. Theorem 2.3 Assume that (H1), (H2), (2.3) hold, and either one of the conditions ) is satisfied.Then (1.5) is oscillatory.
Proof.To the contrary let us assume that (1.
Suppose that µ = 0. Then Taking into account that The last inequality leads to the existence of a constant ν > 0 such that dx(t) dt < −ν for all sufficiently large t.Integrating the last inequality from a suitable large t (say Integrating the above inequality from t − τ to t we obtain, That is Notice that (2.3) and (2.11) imply that c ≤ 0. This restriction is very important.In fact when c > 0, the conditions a > λ and (2.3) imply that ab ≤ 0 which leads to the nonoscillation of (1.5) according to Theorem 2.1.
Lemma 2.1 Assume that a = 0 and (H4) holds.If x is any solution of (1.5), then there exists t 0 ≥ 0 such that Proof.From (1.5), we have Since |f (x)| < 1 for all x ∈ R, Therefore the function N, where N(t) = e λt (x(t) − |a| λ ), is nonincreasing on [0, ∞).This implies that N(t) must be eventually of one sign.We claim that N(t) is eventually negative.Suppose not.Then there exists T ≥ 0 such that N(t) > 0 for all t ≥ T .But N(t) is nonincreasing, then it has a nonnegative finite limit as t → ∞.Also we have  (2.14).Thus N(t) must be eventually negative as claimed; i.e., there exists t 0 ≥ 0 such that e λt (x(t) − |a| λ ) < 0, t ≥ t 0 .This inequality holds only if x(t) < |a| λ for all t ≥ t 0 .The left inequality of (2.12) can be proved similarly. ) is satisfied where r = min{ Proof.To the contrary let us assume that (1.5) has a solution x(t) such that x(t) > 0, for all t ≥ t 0 ≥ 0. Then as in the previous proofs, This inequality and (H3) yield But (H1) implies 1 > r.
Then M(t) > r, for all t ≥ t 1 . (2.17) Defining the functions p, q as in the proof of Theorem 2.
or bc ≥ 0 and a(1 is satisfied.Then all solutions of (1.5) satisfy that Proof.We will prove the theorem when (3.1) holds.The proof when (3.2) holds is similar and will be omitted to avoid repetition.First we assume that x is a solution of (1.5).It follows from Lemma 2.1 that x is bounded.Therefore there exist L, S ∈ R such that Thus for any ε > 0 there exists t 0 ≥ 0 such that In view of the continuity of x one can choose two sequences  From the previous two inequalities we get which is impossible in view of (3.1) and the fact that S > L. This contradiction implies that L = S.
When a < 0, c < 0 and b > 0, using similar arguments as before, we find Also Let n → ∞ in (3.15), (3.16).Then we obtain respectively that and Since ε is arbitrary small, we get which is impossible.Therefore S ≤ 0 and hence L < 0. From (3.19) we get which is also impossible.Then L = S.
Consider now the last possible case; that is a > 0, c > 0 and b < 0,.The above Since the trivial solution is the unique equilibrium, due to the second inequality of (3.1), we get L = S = 0.
Lemma 3.1 If all solutions of (2.2) are bounded and λ > a > 0 or a ≤ 0, then the zero solution of (1.5) is globally exponentially stable.
Remark 3.1 It should be noted that there are many interesting linear stability criteria that can be applied here (see, e.g., [1,17] and the references cited therein) but of course it is not possible to apply all these results due to space limitation.
We conclude our results with the following consequences of Theorems 3.4-3.6(with c = 0, l = τ ) on the single delay model (1.3).As far as the authors know these results are new.
4.2] but we use it here since (a) and (b) are practically easier to apply than the original condition provided in [6, Theorem 3.4.2].