Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays

Abstract

Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.

1. Introduction

Consider a well-known Wilson-Cowan type model [1, 2] with time-varying delays 𝑑𝑋_𝑃(𝑡)𝑑𝑡=−𝑋_𝑃(𝑡)+𝑘_𝑃−𝑟_𝑃𝑋_𝑃(𝑡𝑤)×𝐺¹_𝑃𝑋_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑋_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡),𝑑𝑋_𝑁(𝑡)𝑑𝑡=−𝑋_𝑁(𝑡)+𝑘_𝑁−𝑟_𝑁𝑋_𝑁(𝑡𝑤)×𝐺²_𝑃𝑋_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑋_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡), (1) where 𝑋_𝑃(𝑡),𝑋_𝑁(𝑡) represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant 𝑡, respectively. 𝑟_𝑃 and 𝑟_𝑁 are related to the duration of the refractory period, 𝑘_𝑃 and 𝑘_𝑁 are constants. 𝑤¹_𝑃, 𝑤¹_𝑁, 𝑤²_𝑃, and 𝑤²_𝑁 are the strengths of connections between the populations. 𝐼_𝑃(𝑡),𝐼_𝑁(𝑡) are the external inputs to the excitatory and the inhibitory populations. 𝐺(⋅) is the response function of neuronal activity and it is always assumed to be sigmoid type. 𝜏_𝑃(𝑡), 𝜏_𝑁(𝑡) correspond to the transmission time-varying delays.

It is interesting to revisit Wilson-Cowan system on the following points.(i)The Wilson-Cowan model has a realistic biological background which describes interactions between excitatory and inhibitory populations of neurons [1–3]. It has extensive application such as pattern analysis and image processing [4, 5].(ii)There exists rich dynamical behavior in Wilson-Cowan model. Theoretical results about stable limit cycles, equilibria, chaos, and oscillatory activity have been reported in [2, 3, 6–9]. Recently, Decker and Noonburg [8] reported new results about the existence of three periodic solutions when each neuron was stimulated by periodical inputs. However, under time-varying (periodic or almost periodic) inputs, Wilson-Cowan model can have more complex state space and coexistence of divergent solutions and local stable solutions which could not be easily estimated by its boundary. To see this, we can refer to Figure 1 for the phase portrait of solutions of the following Wilson-Cowan type model with 𝐺(𝑧)=tanh(𝑧) and almost periodic inputs [10–12]: 𝑑𝑋_𝑃(𝑡)𝑑𝑡=−𝑋_𝑃(𝑡)+1−𝑋_𝑃(𝑡𝑋)×𝐺_𝑃(𝑡)−𝑋_𝑁(𝑡)+5sin√2𝑡,𝑑𝑋_𝑁(𝑡)𝑑𝑡=−𝑋_𝑁(𝑡)+1−𝑋_𝑁(𝑡𝑋)×𝐺_𝑃(𝑡)−𝑋_𝑁(𝑡)+0.5cos𝑡sin𝑡. (2)(iii)Few works reported almost periodicity of Wilson-Cowan type model in the literature. Under almost periodic inputs, whether there exists a unique almost periodic solution of (1) which is stable? How to estimate its located boundary? Revealing these results can give a significant insight into the complex dynamical structure of Wilson-Cowan type model.

Figure 1 

Coexistence of divergent and local stable solutions of (2) with almost periodic inputs.

Throughout this paper, we always assume that 𝑘_𝑃, 𝑘_𝑁, 𝑟_𝑃, 𝑟_𝑁, 𝑤¹_𝑃, 𝑤²_𝑃, 𝑤¹_𝑁, and 𝑤²_𝑁 are positive constants, 𝜏_𝑃(𝑡), 𝜏_𝑁(𝑡), 𝐼_𝑃(𝑡), and 𝐼_𝑁(𝑡) are almost periodic functions [12], and set 𝜏^⊤_𝑃=sup_𝑡∈ℝ𝜏_𝑃(𝑡),𝜏^⊤_𝑁=sup_𝑡∈ℝ𝜏_𝑁(𝑡),𝐼^⊤_𝑃=sup_𝑡∈ℝ𝐼_𝑃(𝑡),𝐼^⊤_𝑁=sup_𝑡∈ℝ𝐼_𝑁(𝑡). (3)

Moreover, we need some basic assumptions in this paper.  (𝐻₁)𝐺(0)=0, sup_𝑣∈ℝ|𝐺(𝑣)|⩽𝐵_𝑠and there exists an 𝐿>0 such that |𝐺(𝑢)−𝐺(𝑣)|⩽𝐿|𝑢−𝑣|for∀𝑢,𝑣∈ℝ. (4) (𝐻₂) The quadratic equation (𝐾+𝑅𝑥)(𝐼+𝑊𝑥)=𝑥 has a positive solution 𝛿, where 𝑊=max𝐿𝑤¹_𝑃+𝑤¹_𝑁,𝐿𝑤²_𝑃+𝑤²_𝑁,𝐾=max𝑘_𝑃,𝑘_𝑁,𝑅=max𝑟_𝑃,𝑟_𝑁,𝐼=max𝐿𝐼^⊤_𝑃,𝐿𝐼^⊤_𝑁. (5) (𝐻₃)𝜏_𝑃(𝑡),𝜏_𝑁(𝑡) are bounded and continuously differentiable with 0≤𝜏_𝑃(𝑡)≤𝜏^⊤_𝑃, 0≤𝜏_𝑁(𝑡)≤𝜏^⊤_𝑁,1−𝜏̇_𝑃(𝑡)𝜏̇>0,1−_𝑁(𝑡)>0for𝑡∈𝑅. (6)

For for all 𝑢=(𝑢₁,𝑢₂)∈ℝ², we define the norm ‖𝑢‖=max{|𝑢₁|,|𝑢₂|}. Let 𝐵∶={𝜓∣𝜓=(𝜓₁,𝜓₁)}, where 𝜓 is an almost periodic function on ℝ². For all 𝜓∈𝐵, if we define induced nodule ‖𝜓‖_𝐵=sup_𝑡∈ℝ‖𝜓(𝑡)‖, then 𝐵 is a Banach space. The initial conditions of system (1) are of the form 𝑋_𝑃(𝑠)=𝜓_𝑃(𝑠),𝑋_𝑁(𝑠)=𝜓_𝑁(𝑠),𝑠∈[−𝜏,0], (7) where 𝜏=max{𝜏^⊤_𝑃,𝜏^⊤_𝑁} and 𝜓=(𝜓_𝑃,𝜓_𝑁)∈𝐵.

Definition 1 (see [12]). Let 𝑢(𝑡)∶ℝ→ℝ^𝑛 be continuous. 𝑢(𝑡) is said to be almost periodic on ℝ if, for any 𝜀>0, it is possible to find a real number 𝑙=𝑙(𝜀)>0, and for any interval with length 𝑙(𝜀), there exists a number 𝛿=𝛿(𝜀) in this interval such that |𝑢(𝑡+𝛿)−𝑢(𝑡)|<𝜀, for all 𝑡∈ℝ.
The remaining part of this paper is organized as follows. In Section 2, we will derive sufficient conditions for checking the existence of almost periodic solutions. In Section 3, we present delay-based exponential stability of the unique almost periodic solution of system (1). In Section 4, we will give an example to illustrate our results obtained in the preceding sections. Concluding remarks are given in Section 5.

2. Existence of Almost Periodic Solutions

Theorem 2. Suppose that (𝐻₁) and (𝐻₂) hold. If 𝐾𝑊+𝑅(𝐵_𝑠+𝛿𝑊)<1, then there exists a unique almost periodic solution of system (1) in the region 𝐵^∗=𝜓∣𝜓∈𝐵,𝜓_𝐵≤𝛿. (8)

Proof. For for all 𝜓=(𝜓_𝑃,𝜓_𝑁)∈𝐵^∗, we consider the almost periodic solution 𝑋^𝜓(𝑡)=(𝑋^𝜓_𝑃(𝑡),𝑋^𝜓_𝑁(𝑡)) of the following almost periodic differential equations: 𝑑𝑋_𝑃(𝑡)𝑑𝑡=−𝑋_𝑃(𝑡)+𝑘_𝑃−𝑟_𝑃𝜓_𝑃(𝑡𝑤)×𝐺¹_𝑃𝜓_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝜓_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡),𝑑𝑋_𝑁(𝑡)𝑑𝑡=−𝑋_𝑁(𝑡)+𝑘_𝑁−𝑟_𝑁𝜓_𝑁(𝑡𝑤)×𝐺²_𝑃𝜓_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝜓_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡). (9)
By almost periodicity of 𝜏_𝑃(𝑡), 𝜏_𝑁(𝑡), 𝐼_𝑃(𝑡), and 𝐼_𝑁(𝑡) and Theorem 3.4 in [12] or [10], (9) has a unique almost periodic solution 𝑋^𝜓_𝑃(𝑡)=^𝑡_−∞exp(−(𝑡−𝑠))𝑘_𝑃−𝑟_𝑃𝜓_𝑃(𝑠𝑤)×𝐺¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠𝑋)𝑑𝑠,_𝜓𝑁(𝑡)=^𝑡_−∞exp(−(𝑡−𝑠))𝑘_𝑁−𝑟_𝑁𝜓_𝑁(𝑠𝑤)×𝐺²_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤²_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑁(𝑠)𝑑𝑠. (10)
Define a mapping 𝐹∶𝐵^∗→𝐵 by setting 𝐹(𝜓)(𝑡)=(𝐹_𝑃(𝜓)(𝑡),𝐹_𝑁(𝜓)(𝑡))=𝑋^𝜓(𝑡), for all 𝜓∈𝐵^∗. Now, we prove that 𝐹 is a self-mapping from 𝐵^∗ to 𝐵^∗. From (10) and (𝐻₁), we obtain 𝐹_𝑃𝜓(𝑡)≤sup_𝑡∈ℝ^𝑡_−∞exp(−(𝑡−𝑠))𝑘_𝑃−𝑟_𝑃𝜓_𝑃(𝑠𝑤)×𝐺¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)𝑑𝑠≤sup_𝑡∈ℝ𝑘_𝑃+𝑟_𝑃𝜓_𝑃(𝑡𝑤)×𝐺¹_𝑃𝜓_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝜓_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡≤𝑘_𝑃)+𝑟_𝑃𝜓_𝐵×𝐿𝑤¹_𝑃𝜓_𝐵+𝑤¹_𝑁𝜓_𝐵+𝐼^⊤_𝑃≤𝑘_𝑃+𝑟_𝑃𝜓_𝐵×𝐿𝑤¹_𝑃+𝑤¹_𝑁𝜓_𝐵+𝐿𝐼^⊤_𝑃. (11)
By similar estimation, we can get 𝐹_𝑁𝜓≤𝑘_𝑁+𝑟_𝑁𝜓_𝐵×𝐿𝑤²_𝑃+𝑤²_𝑁𝜓_𝐵+𝐿𝐼^⊤_𝑁. (12)
Therefore, by the above estimations and (𝐻₂), we get 𝐹𝜓_𝐵=sup_𝑡∈ℝ𝐹_𝑃𝜓,𝐹_𝑁𝜓≤𝐾+𝑅𝜓_𝐵𝑊𝜓_𝐵≤()+𝐼𝐾+𝑅𝛿)(𝑊𝛿+𝐼=𝛿, (13) which implies that 𝐹(𝜓)∈𝐵^∗. So, the mapping 𝐹 is self-mapping from 𝐵^∗ to 𝐵^∗. Next, we prove that 𝐹 is a contraction mapping in the region 𝐵^∗. For all 𝜓, 𝜙∈𝐵^∗, by (10), we have 𝐹_𝑃𝜓(𝑡)−𝐹_𝑃𝜙(𝑡)≤sup_𝑡∈ℝ^𝑡_−∞exp(−(𝑡−𝑠))𝑘_𝑃−𝑟_𝑃𝜓_𝑃(𝑠𝑤)×𝐺¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠−)𝑑𝑠_𝑡−∞exp(−(𝑡−𝑠))𝑘_𝑃−𝑟_𝑃𝜙_𝑃(𝑠𝑤)×𝐺¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜙_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)𝑑𝑠, (14) which leads to 𝐹_𝑃𝜓(𝑡)−𝐹_𝑃𝜙(𝑡)≤sup_𝑡∈ℝ^𝑡_−∞(−(×𝑘exp𝑡−𝑠))_𝑃𝐿𝑤¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)−𝑤¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)+𝑤¹_𝑁𝜙_𝑁𝑠−𝜏_𝑁(𝑠)+𝑟_𝑃𝜓_𝑃(𝑠)𝐺𝑤¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)−𝜙_𝑃(𝑠)𝐺𝑤¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜙_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)𝑑𝑠≤𝑘_𝑃𝐿𝑤¹_𝑃sup_𝑡∈ℝ𝜓_𝑃(𝑡)−𝜙_𝑃(𝑡)+𝑤¹_𝑁sup_𝑡∈ℝ𝜓_𝑁(𝑡)−𝜙_𝑁(𝑡)+sup_𝑡∈ℝ^𝑡_−∞exp(−(𝑡−𝑠))𝑟_𝑃×𝜓_𝑃(𝑠)𝐺𝑤¹_𝑃𝜓_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜓_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)−𝜓_𝑃(𝑠)𝐺𝑤¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜙_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)+𝜓_𝑃(𝑠)𝐺𝑤¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜙_𝑁𝑠−𝜏_𝑁(𝑠)+𝐼_𝑃(𝑠)−𝜙_𝑃(𝑠)𝐺𝑤¹_𝑃𝜙_𝑃𝑠−𝜏_𝑃(𝑠)−𝑤¹_𝑁𝜙_𝑁(𝑠)+𝐼_𝑃(𝑠)|𝑑𝑠≤𝑘_𝑃𝐿𝑤¹_𝑃+𝑤¹_𝑁𝜓−𝜙_𝐵+𝑟_𝑃𝐵_𝑠𝜓−𝜙_𝐵+𝛿𝐿𝑤¹_𝑃+𝑤¹_𝑁𝜓−𝜙_𝐵. (15)
By similar argument, we can get 𝐹_𝑁𝜓(𝑡)−𝐹_𝑁𝜓(𝑡)≤𝑘_𝑁𝐿𝑤²_𝑃+𝑤²_𝑁𝜓−𝜙_𝐵+𝑟_𝑁𝐵_𝑠𝜓−𝜙_𝐵+𝛿𝐿𝑤²_𝑃+𝑤²_𝑁𝜓−𝜙_𝐵. (16)
From (15) and (16), we have 𝐹𝜓−𝐹𝜓_𝐵=sup_𝑡∈ℝ𝐹_𝑃𝜓−𝐹_𝑃𝜓,𝐹_𝑁𝜓−𝐹_𝑁𝜓𝐵≤𝐾𝑊𝜓−𝜙+𝑅_𝑠=𝜓−𝜙+𝛿𝑊𝜓−𝜙𝐾𝑊+𝑅𝐵_𝑠+𝑅𝛿𝑊𝜓−𝜙. (17)
Since 𝐾𝑊+𝑅𝐵_𝑠+𝑅𝛿𝑊∈(0,1), it is clear that the mapping 𝐹 is a contraction. Therefore the mapping 𝐹 possesses a unique fixed point 𝑋^∗∈𝐵^∗ such that 𝐹𝑋^∗=𝑋^∗. By (9), 𝑋^∗ is an almost periodic solution of system (1) in 𝐵^∗. The proof is complete.

Remark 3. Obviously, quadratic curve 𝒞(𝑣)∶=𝑅𝑊𝑣²+(𝐾𝑊+𝑅𝐼−1)𝑣+𝐾𝐼 satisfies with 𝒞(0)≥0. So, Δ∶=(𝐾𝑊+𝑅𝐼−1)²−4𝑅𝑊𝐾𝐼>0 and 𝐾𝑊+𝑅𝐼<1 guarantees the existence of 𝛿 in (𝐻₂) and 𝛿 lies in the following interval: 1−𝐾𝑊−𝑅𝐼−√Δ,√2𝑅𝑊1−𝐾𝑊−𝑅𝐼+Δ.2𝑅𝑊 (18)
By Theorem 2, we know that the unique almost periodic solution depends on 𝐾𝑊+𝑅(𝐵_𝑠+𝛿𝑊)<1. Choosing 𝛿=(1−(𝐾𝑊+𝑅𝐼))/2𝑅𝑊, we get a simple assumption as follows:(𝐻₂)𝐾𝑊+𝑅𝐼<1−2√𝐾𝑊𝑅𝐼,𝑅𝐵_𝑠<(1−(𝐾𝑊−𝑅𝐼))/2,and hence it leads to a parameter-based result.

Corollary 4. Suppose that (𝐻₁) and (𝐻₂) hold. Then there exists a unique almost periodic solution of system (1) in the region 𝐵^∗={𝜓∣𝜓∈𝐵,‖𝜓‖_𝐵≤(1−(𝐾𝑊+𝑅𝐼))/2𝑅𝑊}.

3. Delay-Based Stability of the Almost Periodic Solution

In this section, we establish locally exponential stability of the unique almost periodic solution of system (1) in the region 𝐵^∗, which is delay dependent.

Theorem 5. Suppose that (𝐻₁)–(𝐻₃) hold. If 𝐾𝑊+𝑅(𝐵_𝑠+𝛿𝑊)<1 and there exist constants ℓ₁>0, ℓ₂>0 such that 1−𝑟_𝑃𝐵_𝑠ℓ₁>sup_𝑡∈ℝ𝐶_𝑃1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡),1−𝑟_𝑁𝐵_𝑠ℓ₂>sup_𝑡∈ℝ𝐶_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡,) (19) where 𝐶_𝑃∶=𝐿[(𝑘_𝑃+𝑟_𝑃𝛿)ℓ₁𝑤¹_𝑃+(𝑘_𝑁+𝑟_𝑁𝛿)ℓ₂𝑤²_𝑃] and 𝐶_𝑁∶=𝐿[(𝑘_𝑃+𝑟_𝑃𝛿)ℓ₁𝑤¹_𝑁+(𝑘_𝑁+𝑟_𝑁𝛿)ℓ₂𝑤²_𝑁], 𝜐_𝑃⁻¹(𝑡) and 𝜐_𝑁⁻¹(𝑡) are the inverse functions of 𝜐_𝑃(𝑡)=𝑡−𝜏_𝑃(𝑡) and 𝜐_𝑁(𝑡)=𝑡−𝜏_𝑁(𝑡), then system (1) has exactly one almost periodic solution 𝑋^∗(𝑡) in the region 𝐵^∗ which is locally exponentially stable.

Proof. From Theorem 2, system (1) has a unique almost periodic solution 𝑋^∗(𝑡)∈𝐵^∗. Let 𝑋(𝑡)=(𝑋_𝑃(𝑡),𝑋_𝑁(𝑡)) be an arbitrary solution of system (1) with initial value 𝜓=(𝜓_𝑃,𝜓_𝑁)∈𝐵^∗. Set 𝑋(𝑡)=𝑋_𝑃(𝑡)−𝑋^∗_𝑃(𝑡), 𝑌(𝑡)=𝑋_𝑁(𝑡)−𝑋^∗_𝑁(𝑡). By system (1), we get 𝑑𝑋(𝑡)(𝑡)+𝑘𝑑𝑡=−𝑋_𝑃−𝑟_𝑃𝑋(𝑡)+𝑋^∗_𝑃(𝑡)×𝐺𝑤¹_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)𝐼_𝑃(𝑡−𝑘_𝑃)−𝑟_𝑃𝑋^∗_𝑃(𝑡𝑤)×𝐺¹_𝑃𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡,(𝑡))𝑑𝑌(𝑡)+𝑘𝑑𝑡=−𝑌_𝑁−𝑟_𝑁𝑌(𝑡)+𝑋^∗_𝑁(𝑡)×𝐺𝑤²_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡−𝑘_𝑁)−𝑟_𝑁𝑋^∗_𝑁(𝑡𝑤)×𝐺²_𝑃𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡). (20)
Construct the auxiliary functions 𝐹_𝑃(𝑢), 𝐹_𝑁(𝑢) defined on [0,+∞) as follows: 𝐹_𝑃(𝑢)∶=𝑢−1+𝑟_𝑃𝐵_𝑠ℓ₁+sup_𝑡∈ℝ𝐶_𝑃𝑒^{𝑢𝜏⊤𝑃}1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡),𝐹_𝑁(𝑢)∶=𝑢−1+𝑟_𝑁𝐵_𝑠ℓ₂+sup_𝑡∈ℝ𝐶_𝑁𝑒^{𝑢𝜏⊤𝑁}1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡). (21)
One can easily show that 𝐹_𝑃(𝑢), 𝐹_𝑁(𝑢) are well defined and continuous. Assumption (19) implies that 𝐹_𝑃(0)<0, 𝐹_𝑃(𝑢)→+∞ as 𝑢→+∞ and 𝐹_𝑁(0)<0, 𝐹_𝑁(𝑢)→+∞ as 𝑢→+∞. It follows that there exists a common 𝜆>0 such that 𝐹_𝑃(𝜆)<0 and 𝐹_𝑁(𝜆)<0.
Consider the Lyapunov functional 𝑉(𝑡)=ℓ₁|𝑋(𝑡)|+ℓ₂|𝑌(𝑡)|𝑒^𝜆𝑡+^𝑡_{𝑡−𝜏𝑃(𝑡)}𝐶_𝑃1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑠)|𝑋(𝑠)|𝑒^{𝜆(𝑠+𝜏⊤𝑃)}𝑑𝑠+^𝑡_{𝑡−𝜏𝑁(𝑡)}𝐶_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑠)|𝑌(𝑠)|𝑒^{𝜆(𝑠+𝜏⊤𝑁)}𝑑𝑠. (22)
Calculating the upper right derivative of 𝑉(𝑡) along system (1), one has 𝐷⁺𝑉(𝑡)≤𝜆ℓ₁|𝑋(𝑡)|+ℓ₂|𝑌(𝑡)|𝑒^𝜆𝑡+𝑒^𝜆𝑡ℓ₁[−|𝑋(𝑡+𝑘)|_𝑃−𝑟_𝑃𝑋(𝑡)+𝑋^∗_𝑃(𝑡)×𝐺𝑤¹_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡−𝑘_𝑃)−𝑟_𝑃𝑋^∗_𝑃(𝑡𝑤)×𝐺¹_𝑃𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡)+𝑒^𝜆𝑡ℓ₂[−|𝑌(𝑡+𝑘)|_𝑁−𝑟_𝑁𝑌(𝑡)+𝑋^∗_𝑁(𝑡)×𝐺𝑤²_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡−𝑘_𝑁)−𝑟_𝑁𝑋^∗_𝑁(𝑡𝑤)×𝐺²_𝑃𝑋^∗_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑋^∗_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡+𝐶_𝑃)1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡)|𝑋(𝑡)|𝑒^{𝜆(𝑡+𝜏⊤𝑃)}−𝐶_𝑃𝑋𝑡−𝜏_𝑃(𝑡)exp𝜆𝑡−𝜏_𝑃(𝑡)+𝜏^⊤_𝑃+𝐶_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡)|𝑌(𝑡)|𝑒^{𝜆(𝑡+𝜏⊤𝑁)}−𝐶_𝑁𝑌𝑡−𝜏_𝑁(𝑡)exp𝜆𝑡−𝜏_𝑁(𝑡)+𝜏^⊤_𝑁, (23) which leads to 𝐷⁺𝑉(𝑡)≤(𝜆−1)ℓ₁𝑒^𝜆𝑡|𝑋(𝑡)|+(𝜆−1)ℓ₂𝑒^𝜆𝑡|𝑌(𝑡)|+𝑒^𝜆𝑡ℓ₁𝑘_𝑃𝐿𝑤¹_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑤¹_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑟_𝑃|𝑋(𝑡)|𝐵_𝑠+𝑟_𝑃𝛿𝐿𝑤¹_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑤¹_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑒^𝜆𝑡ℓ₂𝑘_𝑁𝐿𝑤²_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑤²_𝑁𝑌𝑡−𝜏_𝑁(𝑡)+𝑟_𝑁|𝑌(𝑡)|𝐵_𝑠+𝑟_𝑁𝛿𝐿𝑤²_𝑃𝑋𝑡−𝜏_𝑃(𝑡)+𝑤²_𝑁𝑌𝑡−𝜏_𝑁(𝑡+𝐶_𝑃)1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡)|𝑋(𝑡)|𝑒^{𝜆(𝑡+𝜏⊤𝑃)}−𝐶_𝑃𝑋𝑡−𝜏_𝑃(𝑡)𝑒^𝜆𝑡+𝐶_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡)|𝑌(𝑡)|𝑒^{𝜆(𝑡+𝜏⊤𝑁)}−𝐶_𝑁𝑌𝑡−𝜏_𝑁(𝑡)𝑒^𝜆𝑡≤𝜆−1+𝑟_𝑃𝐵_𝑠ℓ₁+sup_𝑡∈ℝ𝐶_𝑃𝑒^{𝜆𝜏⊤𝑃}1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡)×𝑒^𝜆𝑡|𝑋(𝑡+)|𝜆−1+𝑟_𝑁𝐵_𝑠ℓ₂+sup_𝑡∈ℝ𝐶_𝑁𝑒^{𝜆𝜏⊤𝑁}1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡)×𝑒^𝜆𝑡|𝑌(𝑡)|≤−𝑐₁[|𝑋(𝑡)|+|𝑌(𝑡)|]𝑒^𝜆𝑡, (24) where 𝑐₁∶=−(1/2)max{𝐹_𝑃(𝜆),𝐹_𝑁(𝜆)}>0. We have from the above that 𝑉(𝑡)≤𝑉(𝑡₀) and minℓ₁,ℓ₂𝑒^𝜆𝑡(|𝑋(𝑡)|+|𝑌(𝑡)|)≤𝑉(𝑡)≤𝑉(0),𝑡≥0. (25)
Note that 𝑉(0)=ℓ₁+ℓ₂+𝜏^⊤_𝑃sup_{𝑠∈−𝜏⊤𝑃,0}𝐶_𝑃1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑠)𝑒^{𝜆𝜏⊤𝑃}+𝜏^⊤_𝑁sup_{𝑠∈−𝜏⊤𝑁,0}𝐶_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑠𝑒)^{𝜆𝜏⊤𝑁}𝜓−𝜓^∗, (26) where 𝜓^∗(𝑠)=𝑋^∗(𝑠), 𝑠∈[−𝜏,0]. Then there exists a positive constant 𝑀>1 such that 𝑋_𝑃(𝑡)−𝑋^∗_𝑃(𝑡)≤𝑀𝜓−𝜓^∗𝑒^−𝜆𝑡,𝑋_𝑁(𝑡)−𝑋^∗_𝑁(𝑡)≤𝑀𝜓−𝜓^∗𝑒^−𝜆𝑡. (27)
The proof is complete.

Set 𝛿=(1−(𝐾𝑊+𝑅𝐼))/2𝑅𝑊. It follows from Corollary 4 and Theorem 5 the following.

Corollary 6. Suppose that (𝐻₁), (𝐻₂), and (𝐻₃) hold. If there exist constants ℓ₁>0, ℓ₂>0 such that 1−𝑟_𝑃𝐵_𝑠ℓ₁>sup_𝑡∈ℝ𝐿𝛼_𝑃ℓ₁𝑤¹_𝑃+𝛼_𝑁ℓ₂𝑤²_𝑃1−𝜏̇_𝑃𝜐_𝑃⁻¹(𝑡),1−𝑟_𝑁𝐵_𝑠ℓ₂>sup_𝑡∈ℝ𝐿𝛼_𝑃ℓ₁𝑤¹_𝑁+𝛼_𝑁ℓ₂𝑤²_𝑁1−𝜏̇_𝑁𝜐_𝑁⁻¹(𝑡), (28) where 𝛼_𝑃∶=𝑘_𝑃+𝑟_𝑃1−(𝐾𝑊+𝑅𝐼),𝛼2𝑅𝑊_𝑁∶=𝑘_𝑁+𝑟_𝑁1−(𝐾𝑊+𝑅𝐼),2𝑅𝑊 (29)
𝜐_𝑃⁻¹(𝑡) and 𝜐_𝑁⁻¹(𝑡) are defined as Theorem 5, then system (1) has exactly one almost periodic solution 𝑋^∗(𝑡) in the region 𝐵^∗=𝜓∣𝜓∈𝐵,𝜓_𝐵≤1−(𝐾𝑊+𝑅𝐼)2𝑅𝑊, (30) which is locally exponentially stable.

4. An Example

In this section, we give an example to demonstrate the results obtained in previous sections. Consider a Wilson-Cowan type model with time-varying delays as follows: 𝑑𝑋_𝑃(𝑡)𝑑𝑡=−𝑋_𝑃(𝑡)+𝑘_𝑃−𝑟_𝑃𝑋_𝑃(𝑡𝑤)×𝐺¹_𝑃𝑋_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤¹_𝑁𝑋_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑃(𝑡),𝑑𝑋_𝑁(𝑡)𝑑𝑡=−𝑋_𝑁(𝑡)+𝑘_𝑁−𝑟_𝑁𝑋_𝑁(𝑡𝑤)×𝐺²_𝑃𝑋_𝑃𝑡−𝜏_𝑃(𝑡)−𝑤²_𝑁𝑋_𝑁𝑡−𝜏_𝑁(𝑡)+𝐼_𝑁(𝑡), (31) where 𝑘_𝑃=𝑘_𝑁=1, 𝑟_𝑃=𝑟_𝑁=0.01, 𝑤¹_𝑃=𝑤²_𝑃=𝑤¹_𝑁=𝑤²_𝑁=0.1, 𝜏_𝑃(𝑡)=𝜏_𝑁(𝑡)=0.1𝑡+10, 𝐼_𝑃(𝑡)=7sin√7𝑡, 𝐺(𝑣)=tanh(𝑣), and 𝐼_𝑁(𝑡)=7cos√2𝑡. It is easy to calculate that 𝜐𝐾=1,𝑊=0.2,𝑅=0.01,𝐼=7,_𝑃−1(𝑡)=𝜐_𝑁⁻¹(𝑡)=(𝑡+10),0.9𝐿=𝐵_𝑠=1,𝛿=182.5. (32)

It is easy to check that (𝐻₁) and (𝐻₂) hold. By Corollary 4, (31) has a unique almost periodic solution 𝑋^∗(𝑡) in region 𝐵^∗={𝜓∣𝜓∈𝐵,‖𝜓‖_𝐵≤182.5}. Setting ℓ₁=ℓ₂=0.5, we can check that (𝐻₃) and (28) hold and hence 𝑋^∗(𝑡) is exponentially stable in 𝐵^∗. Figure 2 shows the transient behavior of the unique almost periodic solution (𝑋_𝑃(𝑡),𝑋_𝑁(𝑡)) in 𝐵^∗. Phase portrait of attractivity of 𝑋_𝑃 and 𝑋_𝑁 is illustrated in Figure 3.

Figure 2 

Transient behavior of the almost periodic solution of (31) located in 𝐵∗.

Figure 3 

Phase portrait of attractivity of the unique almost periodic solution of (31) located in 𝐵∗.

5. Concluding Remarks

In this paper, we investigate Wilson-Cowan type model and obtain the existence of a unique almost periodic solution and its delay-based local stability in a convex subset. Our results are new and can reduce to periodic case, hence, complement existing periodic ones [7, 8]. We point out that there will exist multiple periodic (almost periodic) solution for system (1) under suitable parameter configuration. However, it is difficult to analyze its multistability of almost periodic solution by the existing method [10, 11, 13, 14]. We leave it for interested readers.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grants (11101187, 61273021), NCETFJ (JA11144), the Excellent Youth Foundation of Fujian Province (2012J06001), the Foundation of Fujian High Education (JA10184, JA11154), and the Foundation for Young Professors of Jimei University, China.