Almost Periodic Solutions for Wilson-Cowan Type Model with Time ...

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 683091, 7 pages http://dx.doi.org/10.1155/2013/683091

Research Article Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays Shasha Xie and Zhenkun Huang School of Science, Jimei University, Xiamen 361021, China Correspondence should be addressed to Zhenkun Huang; [email protected] Received 9 October 2012; Accepted 26 December 2012 Academic Editor: Junli Liu Copyright © 2013 S. Xie and Z. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 𝑑𝑋𝑃 (𝑡) = −𝑋𝑃 (𝑡) + [𝑘𝑃 − 𝑟𝑃 𝑋𝑃 (𝑡)] 𝑑𝑡

𝜏𝑃 (𝑡), 𝜏𝑁(𝑡) correspond to the transmission time-varying delays. It is interesting to revisit Wilson-Cowan system on the following points.

(1)

(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].

where 𝑋𝑃 (𝑡), 𝑋𝑁(𝑡) represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant 𝑡, respectively. 𝑟𝑃 and 𝑟𝑁 are related to the duration of the 1 refractory period, 𝑘𝑃 and 𝑘𝑁 are constants. 𝑤𝑃1 , 𝑤𝑁 , 𝑤𝑃2 , and 2 𝑤𝑁 are the strengths of connections between the populations. 𝐼𝑃 (𝑡), 𝐼𝑁(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.

(ii) There exists rich dynamical behavior in WilsonCowan 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

× 𝐺 [𝑤𝑃1 𝑋𝑃 (𝑡 − 𝜏𝑃 (𝑡)) 1 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)] , −𝑤𝑁

𝑑𝑋𝑁 (𝑡) = −𝑋𝑁 (𝑡) + [𝑘𝑁 − 𝑟𝑁𝑋𝑁 (𝑡)] 𝑑𝑡 × 𝐺 [𝑤𝑃2 𝑋𝑃 (𝑡 − 𝜏𝑃 (𝑡)) 2 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑁 (𝑡)] , −𝑤𝑁

2

Discrete Dynamics in Nature and Society (𝐻2 ) The quadratic equation (𝐾 + 𝑅𝑥)(𝐼 + 𝑊𝑥) = 𝑥 has a positive solution 𝛿, where

50 40 30 20

1 2 𝑊 = max{𝐿 (𝑤𝑃1 + 𝑤𝑁 ) , 𝐿 (𝑤𝑃2 + 𝑤𝑁 )} ,

XN

10 0

𝐾 = max{𝑘𝑃 , 𝑘𝑁} ,

−10

𝑅 = max{𝑟𝑃 , 𝑟𝑁} ,

(5)

⊤ }. 𝐼 = max {𝐿𝐼𝑃⊤ , 𝐿𝐼𝑁

−20 −30 −40 −50 −80 −60 −40 −20

0

20

40

60

80

100

(𝐻3 ) 𝜏𝑃 (𝑡), 𝜏𝑁(𝑡) are bounded and continuously differentiable with 0 ≤ 𝜏𝑃 (𝑡) ≤ 𝜏𝑃⊤ ,

XP

Figure 1: Coexistence of divergent and local stable solutions of (2) with almost periodic inputs.

Wilson-Cowan type model with 𝐺(𝑧) = tanh(𝑧) and almost periodic inputs [10–12]: 𝑑𝑋𝑃 (𝑡) = −𝑋𝑃 (𝑡) + [1 − 𝑋𝑃 (𝑡)] 𝑑𝑡 × 𝐺 [𝑋𝑃 (𝑡) − 𝑋𝑁 (𝑡) + 5 sin√2𝑡] , 𝑑𝑋𝑁 (𝑡) = −𝑋𝑁 (𝑡) + [1 − 𝑋𝑁 (𝑡)] 𝑑𝑡

(2)

× 𝐺 [𝑋𝑃 (𝑡) − 𝑋𝑁 (𝑡) + 0.5 cos 𝑡 sin 𝑡] .

Throughout this paper, we always assume that 𝑘𝑃 , 𝑘𝑁, 𝑟𝑃 , 1 2 𝑟𝑁, 𝑤𝑃1 , 𝑤𝑃2 , 𝑤𝑁 , and 𝑤𝑁 are positive constants, 𝜏𝑃 (𝑡), 𝜏𝑁(𝑡), 𝐼𝑃 (𝑡), and 𝐼𝑁(𝑡) are almost periodic functions [12], and set

𝑡∈R

󵄨 󵄨 𝐼𝑃⊤ = sup 󵄨󵄨󵄨𝐼𝑃 (𝑡)󵄨󵄨󵄨 , 𝑡∈R

⊤ = sup𝜏𝑁 (𝑡) , 𝜏𝑁 𝑡∈R

󵄨 󵄨 ⊤ 𝐼𝑁 = sup 󵄨󵄨󵄨𝐼𝑁 (𝑡)󵄨󵄨󵄨 .

̇ (𝑡) > 0 1 − 𝜏𝑁

(3)

1 − 𝜏𝑃̇ (𝑡) > 0, for 𝑡 ∈ 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 = sup𝑡∈R ‖𝜓(𝑡)‖, then 𝐵 is a Banach space. The initial conditions of system (1) are of the form 𝑋𝑃 (𝑠) = 𝜓𝑃 (𝑠) ,

(iii) Few works reported almost periodicity of WilsonCowan 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.

𝜏𝑃⊤ = sup𝜏𝑃 (𝑡) ,

⊤ 0 ≤ 𝜏𝑁 (𝑡) ≤ 𝜏𝑁 ,

𝑋𝑁 (𝑠) = 𝜓𝑁 (𝑠) ,

𝑠 ∈ [−𝜏, 0] , (7)

⊤ } and 𝜓 = (𝜓𝑃 , 𝜓𝑁) ∈ 𝐵. where 𝜏 = max{𝜏𝑃⊤ , 𝜏𝑁

Definition 1 (see [12]). Let 𝑢(𝑡) : R → R𝑛 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.

𝑡∈R

2. Existence of Almost Periodic Solutions Moreover, we need some basic assumptions in this paper. (𝐻1 ) 𝐺(0) = 0, sup𝑣∈R |𝐺(𝑣)| ⩽ 𝐵𝑠 and there exists an 𝐿 > 0 such that |𝐺 (𝑢) − 𝐺 (𝑣)| ⩽ 𝐿 |𝑢 − 𝑣|

for ∀𝑢, 𝑣 ∈ R.

(4)

Theorem 2. Suppose that (𝐻1 ) and (𝐻2 ) hold. If 𝐾𝑊 + 𝑅(𝐵𝑠 + 𝛿𝑊) < 1, then there exists a unique almost periodic solution of system (1) in the region 󵄩 󵄩 𝐵∗ = {𝜓 | 𝜓 ∈ 𝐵, 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 ≤ 𝛿} .

(8)

Discrete Dynamics in Nature and Society

3

Proof. For for all 𝜓 = (𝜓𝑃 , 𝜓𝑁) ∈ 𝐵∗ , we consider the almost 𝜓 𝜓 periodic solution 𝑋𝜓 (𝑡) = (𝑋𝑃 (𝑡), 𝑋𝑁(𝑡)) of the following almost periodic differential equations:

󵄨 󵄨 ≤ sup [𝑘𝑃 + 𝑟𝑃 󵄨󵄨󵄨𝜓𝑃 (𝑡)󵄨󵄨󵄨] 𝑡∈R

󵄨 × 𝐺 󵄨󵄨󵄨󵄨[𝑤𝑃1 𝜓𝑃 (𝑡 − 𝜏𝑃 (𝑡))

𝑑𝑋𝑃 (𝑡) = −𝑋𝑃 (𝑡) + [𝑘𝑃 − 𝑟𝑃 𝜓𝑃 (𝑡)] 𝑑𝑡

󵄨 1 −𝑤𝑁 𝜓𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)]󵄨󵄨󵄨󵄨 󵄩 󵄩 ≤ (𝑘𝑃 + 𝑟𝑃 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 )

× 𝐺 [𝑤𝑃1 𝜓𝑃 (𝑡 − 𝜏𝑃 (𝑡))

󵄩 󵄩 1 󵄩 󵄩󵄩𝜓󵄩󵄩󵄩 + 𝐼⊤ ) × 𝐿 (𝑤𝑃1 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 + 𝑤𝑁 󵄩 󵄩𝐵 𝑃

1 𝜓𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)] , −𝑤𝑁

𝑑𝑋𝑁 (𝑡) = −𝑋𝑁 (𝑡) + [𝑘𝑁 − 𝑟𝑁𝜓𝑁 (𝑡)] 𝑑𝑡

(9)

󵄩 1 󵄩 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 + 𝐿𝐼𝑃⊤ ) . × (𝐿 (𝑤𝑃1 + 𝑤𝑁

× 𝐺 [𝑤𝑃2 𝜓𝑃 (𝑡 − 𝜏𝑃 (𝑡))

(11)

2 𝜓𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑁 (𝑡)] . −𝑤𝑁

By similar estimation, we can get

By almost periodicity of 𝜏𝑃 (𝑡), 𝜏𝑁(𝑡), 𝐼𝑃 (𝑡), and 𝐼𝑁(𝑡) and Theorem 3.4 in [12] or [10], (9) has a unique almost periodic solution 𝜓

𝑋𝑃 (𝑡) 𝑡

−∞

×

󵄩 2 󵄩 ⊤ × (𝐿 (𝑤𝑃2 + 𝑤𝑁 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 + 𝐿𝐼𝑁 ).

(12)

𝑡

−∞

󵄩 󵄨 󵄨 󵄨 󵄨 󵄩󵄩 󵄩󵄩𝐹 (𝜓)󵄩󵄩󵄩𝐵 = sup {󵄨󵄨󵄨𝐹𝑃 (𝜓)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝐹𝑁 (𝜓)󵄨󵄨󵄨}

exp(− (𝑡 − 𝑠)) [𝑘𝑃 − 𝑟𝑃 𝜓𝑃 (𝑠)]

𝑡∈R

𝐺 [𝑤𝑃1 𝜓𝑃 (𝑠

− 𝜏𝑃 (𝑠))

1 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)] 𝑑𝑠, −𝑤𝑁 𝜓

󵄨󵄨 󵄩 󵄩 󵄨 󵄨󵄨𝐹𝑁 (𝜓)󵄨󵄨󵄨 ≤ (𝑘𝑁 + 𝑟𝑁󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 )

Therefore, by the above estimations and (𝐻2 ), we get

=∫

𝑋𝑁 (𝑡) = ∫

󵄩 󵄩 ≤ (𝑘𝑃 + 𝑟𝑃 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 )

󵄩 󵄩 󵄩 󵄩 ≤ (𝐾 + 𝑅󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 ) (𝑊󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 + 𝐼) ≤ (𝐾 + 𝑅𝛿) (𝑊𝛿 + 𝐼) = 𝛿,

(10)

exp(− (𝑡 − 𝑠)) [𝑘𝑁 − 𝑟𝑁𝜓𝑁 (𝑠)]

× 𝐺 [𝑤𝑃2 𝜓𝑃 (𝑠 − 𝜏𝑃 (𝑠)) 2 −𝑤𝑁 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑁 (𝑠)] 𝑑𝑠.

Define a mapping 𝐹 : 𝐵∗ → 𝐵 by setting 𝐹(𝜓)(𝑡) = (𝐹𝑃 (𝜓)(𝑡), 𝐹𝑁(𝜓)(𝑡)) = 𝑋𝜓 (𝑡), for all 𝜓 ∈ 𝐵∗ . Now, we prove that 𝐹 is a self-mapping from 𝐵∗ to 𝐵∗ . From (10) and (𝐻1 ), we obtain

which implies that 𝐹(𝜓) ∈ 𝐵∗ . So, the mapping 𝐹 is selfmapping 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 󵄨 󵄨󵄨 󵄨󵄨𝐹𝑃 (𝜓) (𝑡) − 𝐹𝑃 (𝜙) (𝑡)󵄨󵄨󵄨 󵄨󵄨 𝑡 󵄨 ≤ sup 󵄨󵄨󵄨∫ exp (− (𝑡 − 𝑠)) [𝑘𝑃 − 𝑟𝑃 𝜓𝑃 (𝑠)] 𝑡∈R 󵄨󵄨 −∞ × 𝐺 [𝑤𝑃1 𝜓𝑃 (𝑠 − 𝜏𝑃 (𝑠)) 1 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)] 𝑑𝑠 −𝑤𝑁

󵄨󵄨 󵄨 󵄨󵄨𝐹𝑃 (𝜓) (𝑡)󵄨󵄨󵄨 󵄨󵄨 𝑡 󵄨 ≤ sup 󵄨󵄨󵄨∫ exp (− (𝑡 − 𝑠)) [𝑘𝑃 − 𝑟𝑃 𝜓𝑃 (𝑠)] 𝑡∈R 󵄨󵄨 −∞ ×

(13)

𝐺 [𝑤𝑃1 𝜓𝑃 (𝑠

− 𝜏𝑃 (𝑠))

󵄨󵄨 󵄨 1 −𝑤𝑁 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)] 𝑑𝑠󵄨󵄨󵄨 󵄨󵄨

−∫

𝑡

−∞

exp (− (𝑡 − 𝑠)) [𝑘𝑃 − 𝑟𝑃 𝜙𝑃 (𝑠)] × 𝐺 [𝑤𝑃1 𝜙𝑃 (𝑠 − 𝜏𝑃 (𝑠)) 󵄨󵄨 󵄨 1 𝜙𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)] 𝑑𝑠󵄨󵄨󵄨 , −𝑤𝑁 󵄨󵄨 (14)

4

Discrete Dynamics in Nature and Society

which leads to

From (15) and (16), we have 󵄩 󵄩󵄩 󵄩󵄩𝐹 (𝜓) − 𝐹 (𝜓)󵄩󵄩󵄩𝐵 󵄨 󵄨 󵄨 󵄨 = sup {󵄨󵄨󵄨𝐹𝑃 (𝜓) − 𝐹𝑃 (𝜓)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝐹𝑁 (𝜓) − 𝐹𝑁 (𝜓)󵄨󵄨󵄨}

󵄨 󵄨󵄨 󵄨󵄨𝐹𝑃 (𝜓) (𝑡) − 𝐹𝑃 (𝜙) (𝑡)󵄨󵄨󵄨 ≤ sup ∫ 𝑡∈R

𝑡

−∞

exp (− (𝑡 − 𝑠))

𝑡∈R

󵄩 󵄩 ≤ 𝐾𝑊 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩

󵄨 1 × [𝑘𝑃 𝐿 󵄨󵄨󵄨󵄨𝑤𝑃1 𝜓𝑃 (𝑠 − 𝜏𝑃 (𝑠)) − 𝑤𝑁 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) −𝑤𝑃1 𝜙𝑃 (𝑠

−

1 𝜏𝑃 (𝑠))+𝑤𝑁 𝜙𝑁 (𝑠

󵄨 + 𝑟𝑃 󵄨󵄨󵄨󵄨𝜓𝑃 (𝑠) 𝐺 [𝑤𝑃1 𝜓𝑃 (𝑠 − 𝜏𝑃 (𝑠)) −

1 𝑤𝑁 𝜓𝑁 (𝑠

Since 𝐾𝑊 + 𝑅𝐵𝑠 + 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.

+𝐼𝑃 (𝑠)] − 𝜙𝑃 (𝑠) 𝐺 [𝑤𝑃1 𝜙𝑃 (𝑠 − 𝜏𝑃 (𝑠)) −

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:

− 𝜏𝑁 (𝑠))

󵄨 +𝐼𝑃 (𝑠) ] 󵄨󵄨󵄨󵄨 ] 𝑑𝑠 󵄨 󵄨 󵄨 󵄨 1 ≤ 𝑘𝑃 𝐿 (𝑤𝑃1 sup 󵄨󵄨󵄨𝜓𝑃 (𝑡) − 𝜙𝑃 (𝑡)󵄨󵄨󵄨 + 𝑤𝑁 sup 󵄨󵄨󵄨𝜓𝑁 (𝑡) − 𝜙𝑁 (𝑡)󵄨󵄨󵄨) 𝑡∈R

+ sup ∫ 𝑡∈R

𝑡

−∞

󵄩 󵄩 󵄩 󵄩 + 𝑅 (𝐵𝑠 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩 + 𝛿𝑊 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩) 󵄩 󵄩 = (𝐾𝑊 + 𝑅𝐵𝑠 + 𝑅𝛿𝑊) 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩 .

󵄨 − 𝜏𝑁 (𝑠))󵄨󵄨󵄨󵄨

− 𝜏𝑁 (𝑠))

1 𝜙𝑁 (𝑠 𝑤𝑁

𝑡∈R

󵄨 × [󵄨󵄨󵄨󵄨𝜓𝑃 (𝑠) 𝐺 (𝑤𝑃1 𝜓𝑃 (𝑠 − 𝜏𝑃 (𝑠))

1 − 𝐾𝑊 − 𝑅𝐼 − √Δ 1 − 𝐾𝑊 − 𝑅𝐼 + √Δ , ]. 2𝑅𝑊 2𝑅𝑊

(18)

̂2 ) 𝐾𝑊 + 𝑅𝐼 < 1 − 2√𝐾𝑊𝑅𝐼,𝑅𝐵𝑠 < (1 − (𝐾𝑊 − 𝑅𝐼))/2, (𝐻

1 𝜓𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)) −𝑤𝑁

and hence it leads to a parameter-based result.

− 𝜓𝑃 (𝑠) 𝐺 (𝑤𝑃1 𝜙𝑃 (𝑠 − 𝜏𝑃 (𝑠)) 1 − 𝑤𝑁 𝜙𝑁 (𝑠 − 𝜏𝑁 (𝑠))

󵄨 +𝐼𝑃 (𝑠))󵄨󵄨󵄨

󵄨 + 󵄨󵄨󵄨󵄨𝜓𝑃 (𝑠) 𝐺 (𝑤𝑃1 𝜙𝑃 (𝑠 − 𝜏𝑃 (𝑠))

1 𝜙𝑁 (𝑠 − 𝜏𝑁 (𝑠)) + 𝐼𝑃 (𝑠)) − 𝑤𝑁

− 𝜙𝑃 (𝑠) 𝐺 (𝑤𝑃1 𝜙𝑃 (𝑠 − 𝜏𝑃 (𝑠)) 󵄨 1 𝜙𝑁 (𝑠) + 𝐼𝑃 (𝑠))󵄨󵄨󵄨󵄨]|𝑑𝑠 − 𝑤𝑁 󵄩 1 󵄩 ≤ 𝑘𝑃 𝐿 (𝑤𝑃1 + 𝑤𝑁 ) 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵

̂2 ) hold. Then there Corollary 4. Suppose that (𝐻1 ) and (𝐻 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(𝐵𝑠 + 𝛿𝑊) < 1 and there exist constants ℓ1 > 0, ℓ2 > 0 such that (1 − 𝑟𝑃 𝐵𝑠 ) ℓ1 > sup

(15)

𝑡∈R

𝐶𝑃 , 1 − 𝜏𝑃̇ (𝜐𝑃−1 (𝑡))

𝐶𝑁 , −1 ̇ 1 − 𝜏 𝑡∈R 𝑁 (𝜐𝑁 (𝑡))

(19)

(1 − 𝑟𝑁𝐵𝑠 ) ℓ2 > sup

By similar argument, we can get 󵄨󵄨 󵄨 󵄨󵄨𝐹𝑁 (𝜓) (𝑡) − 𝐹𝑁 (𝜓) (𝑡)󵄨󵄨󵄨 󵄩 󵄩 2 󵄩 ≤ 𝑘𝑁𝐿 (𝑤𝑃2 + 𝑤𝑁 ) 󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵 󵄩 󵄩 󵄩 2 󵄩 + 𝑟𝑁 (𝐵𝑠 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵 + 𝛿𝐿 (𝑤𝑃2 + 𝑤𝑁 ) 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵 ) .

[

By Theorem 2, we know that the unique almost periodic solution depends on 𝐾𝑊 + 𝑅(𝐵𝑠 + 𝛿𝑊) < 1. Choosing 𝛿 = (1 − (𝐾𝑊 + 𝑅𝐼))/2𝑅𝑊, we get a simple assumption as follows:

exp (− (𝑡 − 𝑠)) 𝑟𝑃

󵄩 󵄩 󵄩 1 󵄩 + 𝑟𝑃 (𝐵𝑠 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵 + 𝛿𝐿 (𝑤𝑃1 + 𝑤𝑁 ) 󵄩󵄩󵄩𝜓 − 𝜙󵄩󵄩󵄩𝐵 ) .

(17)

(16)

where 𝐶𝑃 := 𝐿[(𝑘𝑃 + 𝑟𝑃 𝛿)ℓ1 𝑤𝑃1 + (𝑘𝑁 + 𝑟𝑁𝛿)ℓ2 𝑤𝑃2 ] and 𝐶𝑁 := 1 2 −1 + (𝑘𝑁 + 𝑟𝑁𝛿)ℓ2 𝑤𝑁 ], 𝜐𝑃−1 (𝑡) and 𝜐𝑁 (𝑡) are 𝐿[(𝑘𝑃 + 𝑟𝑃 𝛿)ℓ1 𝑤𝑁 the inverse functions of 𝜐𝑃 (𝑡) = 𝑡 − 𝜏𝑃 (𝑡) and 𝜐𝑁(𝑡) = 𝑡 − 𝜏𝑁(𝑡), then system (1) has exactly one almost periodic solution 𝑋∗ (𝑡) in the region 𝐵∗ which is locally exponentially stable.

Discrete Dynamics in Nature and Society

5

Proof. From Theorem 2, system (1) has a unique almost periodic solution 𝑋∗ (𝑡) ∈ 𝐵∗ . Let 𝑋(𝑡) = (𝑋𝑃 (𝑡), 𝑋𝑁(𝑡)) be an arbitrary solution of system (1) with initial value 𝜓 = (𝜓𝑃 , 𝜓𝑁) ∈ 𝐵∗ . Set 𝑋(𝑡) = 𝑋𝑃 (𝑡) − 𝑋𝑃∗ (𝑡), 𝑌(𝑡) = 𝑋𝑁(𝑡) − ∗ (𝑡). By system (1), we get 𝑋𝑁

Calculating the upper right derivative of 𝑉(𝑡) along system (1), one has 𝐷+ 𝑉 (𝑡) ≤ 𝜆 [ℓ1 |𝑋 (𝑡)| + ℓ2 |𝑌 (𝑡)|] 𝑒𝜆𝑡

𝑑𝑋 (𝑡) = −𝑋 (𝑡) 𝑑𝑡

+ 𝑒𝜆𝑡 ℓ1 [− |𝑋 (𝑡)| 󵄨 + 󵄨󵄨󵄨[𝑘𝑃 − 𝑟𝑃 (𝑋 (𝑡) + 𝑋𝑃∗ (𝑡))]

+ [𝑘𝑃 − 𝑟𝑃 (𝑋 (𝑡) + 𝑋𝑃∗ (𝑡))] ×

𝐺 [𝑤𝑃1 (𝑋 (𝑡

− 𝜏𝑃 (𝑡)) +

𝑋𝑃∗ (𝑡

× 𝐺 [𝑤𝑃1 (𝑋 (𝑡 − 𝜏𝑃 (𝑡)) + 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡)))

− 𝜏𝑃 (𝑡)))

1 ∗ (𝑌 (𝑡−𝜏𝑁 (𝑡)) + 𝑋𝑁 (𝑡−𝜏𝑁 (𝑡))) − 𝑤𝑁

1 ∗ (𝑌 (𝑡 − 𝜏𝑁 (𝑡)) + 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡))) 𝐼𝑃 (𝑡)] −𝑤𝑁

+𝐼𝑃 (𝑡)]

− [𝑘𝑃 − 𝑟𝑃 𝑋𝑃∗ (𝑡)]

− [𝑘𝑃 − 𝑟𝑃 𝑋𝑃∗ (𝑡)]

× 𝐺 [𝑤𝑃1 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡))

× 𝐺 [𝑤𝑃1 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡))

1 ∗ 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)] , −𝑤𝑁

󵄨 1 ∗ −𝑤𝑁 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)]󵄨󵄨󵄨󵄨]

𝑑𝑌 (𝑡) = −𝑌 (𝑡) 𝑑𝑡

+ 𝑒𝜆𝑡 ℓ2 [− |𝑌 (𝑡)| 󵄨 ∗ + 󵄨󵄨󵄨󵄨 [𝑘𝑁 − 𝑟𝑁 (𝑌 (𝑡) + 𝑋𝑁 (𝑡))]

∗ + [𝑘𝑁 − 𝑟𝑁 (𝑌 (𝑡) + 𝑋𝑁 (𝑡))]

× 𝐺 [𝑤𝑃2 (𝑋 (𝑡 − 𝜏𝑃 (𝑡)) + 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡)))

× 𝐺 [𝑤𝑃2 (𝑋 (𝑡 − 𝜏𝑃 (𝑡)) + 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡)))

2 ∗ −𝑤𝑁 (𝑌 (𝑡−𝜏𝑁 (𝑡))+𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)))+𝐼𝑁 (𝑡)]

2 ∗ (𝑌 (𝑡 − 𝜏𝑁 (𝑡)) + 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡))) − 𝑤𝑁

∗ − [𝑘𝑁 − 𝑟𝑁𝑋𝑁 (𝑡)]

+𝐼𝑁 (𝑡)] ∗ − [𝑘𝑁 − 𝑟𝑁𝑋𝑁 (𝑡)]

× 𝐺 [𝑤𝑃2 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡))

× 𝐺 [𝑤𝑃2 𝑋𝑃∗ (𝑡 − 𝜏𝑃 (𝑡))

2 ∗ 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑁 (𝑡)] . −𝑤𝑁

󵄨 2 ∗ 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑁 (𝑡)]󵄨󵄨󵄨󵄨] −𝑤𝑁

(20) Construct the auxiliary functions 𝐹𝑃 (𝑢), 𝐹𝑁(𝑢) defined on [0, +∞) as follows: ⊤

𝐶𝑃 𝑒𝑢𝜏𝑃 , ̇ (𝜐𝑃−1 (𝑡)) 𝑡∈R 1 − 𝜏𝑃 ⊤

(21)

One can easily show that 𝐹𝑃 (𝑢), 𝐹𝑁(𝑢) are well defined and continuous. Assumption (19) implies that 𝐹𝑃 (0) < 0, 𝐹𝑃 (𝑢) → +∞ as 𝑢 → +∞ and 𝐹𝑁(0) < 0, 𝐹𝑁(𝑢) → +∞ as 𝑢 → +∞. It follows that there exists a common 𝜆 > 0 such that 𝐹𝑃 (𝜆) < 0 and 𝐹𝑁(𝜆) < 0. Consider the Lyapunov functional

𝑡

𝑡−𝜏𝑃 (𝑡)

+∫

𝑡

𝑡−𝜏𝑁 (𝑡)

⊤ 𝐶𝑃 |𝑋 (𝑠)| 𝑒𝜆(𝑠+𝜏𝑃 ) 𝑑𝑠 1 − 𝜏𝑃̇ (𝜐𝑃−1 (𝑠)) ⊤ 𝐶𝑁 |𝑌 (𝑠)| 𝑒𝜆(𝑠+𝜏𝑁 ) 𝑑𝑠. −1 ̇ (𝜐𝑁 (𝑠)) 1 − 𝜏𝑁

+

⊤ 𝐶𝑁 |𝑌 (𝑡)| 𝑒𝜆(𝑡+𝜏𝑁 ) −1 ̇ 1 − 𝜏𝑁 (𝜐𝑁 (𝑡))

󵄨 󵄨 ⊤ − 𝐶𝑁 󵄨󵄨󵄨𝑌 (𝑡 − 𝜏𝑁 (𝑡))󵄨󵄨󵄨 exp [𝜆 (𝑡 − 𝜏𝑁 (𝑡) + 𝜏𝑁 )] ,

(23)

which leads to 𝐷+ 𝑉 (𝑡) ≤ [(𝜆 − 1) ℓ1 𝑒𝜆𝑡 |𝑋 (𝑡)| + (𝜆 − 1) ℓ2 𝑒𝜆𝑡 |𝑌 (𝑡)|] 󵄨 󵄨 + 𝑒𝜆𝑡 ℓ1 [𝑘𝑃 𝐿 (𝑤𝑃1 󵄨󵄨󵄨𝑋 (𝑡 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨

𝑉 (𝑡) = [ℓ1 |𝑋 (𝑡)| + ℓ2 |𝑌 (𝑡)|] 𝑒𝜆𝑡 +∫

⊤ 𝐶𝑃 |𝑋 (𝑡)| 𝑒𝜆(𝑡+𝜏𝑃 ) −1 1 − 𝜏𝑃̇ (𝜐𝑃 (𝑡))

󵄨 󵄨 − 𝐶𝑃 󵄨󵄨󵄨𝑋 (𝑡 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨 exp [𝜆 (𝑡 − 𝜏𝑃 (𝑡) + 𝜏𝑃⊤ )]

𝐹𝑃 (𝑢) := (𝑢 − 1 + 𝑟𝑃 𝐵𝑠 ) ℓ1 + sup

𝐶𝑁𝑒𝑢𝜏𝑁 . 𝐹𝑁 (𝑢) := (𝑢 − 1 + 𝑟𝑁𝐵𝑠 ) ℓ2 + sup −1 ̇ (𝜐𝑁 (𝑡)) 𝑡∈R 1 − 𝜏𝑁

+

(22)

󵄨 1 󵄨󵄨 + 𝑤𝑁 󵄨󵄨𝑌 (𝑡 − 𝜏𝑁 (𝑡))󵄨󵄨󵄨) + 𝑟𝑃 |𝑋 (𝑡)| 𝐵𝑠 󵄨 󵄨 + 𝑟𝑃 𝛿𝐿 (𝑤𝑃1 󵄨󵄨󵄨𝑋 (𝑡 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨 󵄨 1 󵄨󵄨 +𝑤𝑁 󵄨󵄨𝑌 (𝑡 − 𝜏𝑁 (𝑡))󵄨󵄨󵄨)]

6

Discrete Dynamics in Nature and Society 󵄨 󵄨 + 𝑒𝜆𝑡 ℓ2 [𝑘𝑁𝐿 (𝑤𝑃2 󵄨󵄨󵄨𝑋 (𝑡 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨

̂2 ), and (𝐻3 ) hold. If there Corollary 6. Suppose that (𝐻1 ), (𝐻 exist constants ℓ1 > 0, ℓ2 > 0 such that

󵄨 2 󵄨󵄨 + 𝑤𝑁 󵄨󵄨𝑌 (𝑡 − 𝜏𝑁 (𝑡))󵄨󵄨󵄨) + 𝑟𝑁 |𝑌 (𝑡)| 𝐵𝑠 +

󵄨 𝑟𝑁𝛿𝐿 (𝑤𝑃2 󵄨󵄨󵄨𝑋 (𝑡

(1 − 𝑟𝑃 𝐵𝑠 ) ℓ1 > sup

󵄨 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨

1 − 𝜏𝑃̇ (𝜐𝑃−1 (𝑡))

𝑡∈R

󵄨 2 󵄨󵄨 +𝑤𝑁 󵄨󵄨𝑌 (𝑡−𝜏𝑁 (𝑡))󵄨󵄨󵄨)] ⊤ 𝐶𝑃 + |𝑋 (𝑡)| 𝑒𝜆(𝑡+𝜏𝑃 ) 1 − 𝜏𝑃̇ (𝜐𝑃−1 (𝑡))

(1 − 𝑟𝑁𝐵𝑠 ) ℓ2 > sup

,

1 2 𝐿 (𝛼𝑃 ℓ1 𝑤𝑁 + 𝛼𝑁ℓ2 𝑤𝑁 )

𝑡∈R

−1 ̇ (𝜐𝑁 1 − 𝜏𝑁 (𝑡))

(28) ,

where

󵄨 󵄨 − 𝐶𝑃 󵄨󵄨󵄨𝑋 (𝑡 − 𝜏𝑃 (𝑡))󵄨󵄨󵄨 𝑒𝜆𝑡 +

𝐿 (𝛼𝑃 ℓ1 𝑤𝑃1 + 𝛼𝑁ℓ2 𝑤𝑃2 )

𝛼𝑃 := 𝑘𝑃 + 𝑟𝑃

⊤ 𝐶𝑁 |𝑌 (𝑡)| 𝑒𝜆(𝑡+𝜏𝑁 ) −1 ̇ (𝜐𝑁 (𝑡)) 1 − 𝜏𝑁

1 − (𝐾𝑊 + 𝑅𝐼) , 2𝑅𝑊

𝛼𝑁 := 𝑘𝑁 + 𝑟𝑁

󵄨 󵄨 − 𝐶𝑁 󵄨󵄨󵄨𝑌 (𝑡 − 𝜏𝑁 (𝑡))󵄨󵄨󵄨 𝑒𝜆𝑡

(29)

1 − (𝐾𝑊 + 𝑅𝐼) , 2𝑅𝑊

−1 𝜐𝑃−1 (𝑡) and 𝜐𝑁 (𝑡) are defined as Theorem 5, then system (1) has exactly one almost periodic solution 𝑋∗ (𝑡) in the region

⊤

𝐶𝑃 𝑒𝜆𝜏𝑃 ≤ [(𝜆 − 1 + 𝑟𝑃 𝐵𝑠 ) ℓ1 + sup ] ̇ (𝜐𝑃−1 (𝑡)) 𝑡∈R 1 − 𝜏𝑃

1 − (𝐾𝑊 + 𝑅𝐼) 󵄩 󵄩 𝐵∗ = {𝜓 | 𝜓 ∈ 𝐵, 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐵 ≤ }, 2𝑅𝑊

× 𝑒𝜆𝑡 |𝑋 (𝑡)|

(30)

which is locally exponentially stable. ⊤

𝐶𝑁𝑒𝜆𝜏𝑁 + [(𝜆 − 1 + 𝑟𝑁𝐵𝑠 ) ℓ2 + sup ] −1 ̇ (𝜐𝑁 (𝑡)) 𝑡∈R 1 − 𝜏𝑁

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:

× 𝑒𝜆𝑡 |𝑌 (𝑡)| ≤ −𝑐1 [|𝑋 (𝑡)| + |𝑌 (𝑡)|] 𝑒𝜆𝑡 , (24)

𝑑𝑋𝑃 (𝑡) = −𝑋𝑃 (𝑡) + [𝑘𝑃 − 𝑟𝑃 𝑋𝑃 (𝑡)] 𝑑𝑡 × 𝐺 [𝑤𝑃1 𝑋𝑃 (𝑡 − 𝜏𝑃 (𝑡))

where 𝑐1 := −(1/2) max{𝐹𝑃 (𝜆), 𝐹𝑁(𝜆)} > 0. We have from the above that 𝑉(𝑡) ≤ 𝑉(𝑡0 ) and min {ℓ1 , ℓ2 } 𝑒𝜆𝑡 (|𝑋 (𝑡)| + |𝑌 (𝑡)|) ≤ 𝑉 (𝑡) ≤ 𝑉 (0) ,

𝑡 ≥ 0. (25)

1 −𝑤𝑁 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑃 (𝑡)] ,

𝑑𝑋𝑁 (𝑡) = −𝑋𝑁 (𝑡) + [𝑘𝑁 − 𝑟𝑁𝑋𝑁 (𝑡)] 𝑑𝑡

Note that

(31)

× 𝐺 [𝑤𝑃2 𝑋𝑃 (𝑡 − 𝜏𝑃 (𝑡))

⊤ 𝐶𝑃 𝑉 (0) = [(ℓ1 + ℓ2 ) + 𝜏𝑃⊤ sup 𝑒𝜆𝜏𝑃 −1 ̇ (𝜐𝑃 (𝑠)) 𝑠∈[−𝜏𝑃⊤ ,0] 1 − 𝜏𝑃 [

2 −𝑤𝑁 𝑋𝑁 (𝑡 − 𝜏𝑁 (𝑡)) + 𝐼𝑁 (𝑡)] ,

⊤ 𝐶𝑁 󵄩 󵄩 ⊤ +𝜏𝑁 sup 𝑒𝜆𝜏𝑁 ] 󵄩󵄩󵄩𝜓 − 𝜓∗ 󵄩󵄩󵄩 , −1 ̇ 1 − 𝜏 (𝜐 (𝑠)) ⊤ 𝑁 𝑁 𝑠∈[−𝜏𝑁 ,0] ] (26)

where 𝜓∗ (𝑠) = 𝑋∗ (𝑠), 𝑠 ∈ [−𝜏, 0]. Then there exists a positive constant 𝑀 > 1 such that 󵄩 󵄨 󵄨󵄨 ∗ ∗ 󵄩 −𝜆𝑡 󵄨󵄨𝑋𝑃 (𝑡) − 𝑋𝑃 (𝑡)󵄨󵄨󵄨 ≤ 𝑀 󵄩󵄩󵄩𝜓 − 𝜓 󵄩󵄩󵄩 𝑒 , 󵄩 󵄨 󵄨󵄨 ∗ ∗ 󵄩 −𝜆𝑡 󵄨󵄨𝑋𝑁 (𝑡) − 𝑋𝑁 (𝑡)󵄨󵄨󵄨 ≤ 𝑀 󵄩󵄩󵄩𝜓 − 𝜓 󵄩󵄩󵄩 𝑒 .

(27)

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

1 = where 𝑘𝑃 = 𝑘𝑁 = 1, 𝑟𝑃 = 𝑟𝑁 = 0.01, 𝑤𝑃1 = 𝑤𝑃2 = 𝑤𝑁 2 𝑤𝑁 = 0.1, 𝜏𝑃 (𝑡) = 𝜏𝑁(𝑡) = 0.1𝑡 + 10, 𝐼𝑃 (𝑡) = 7 sin √7𝑡, 𝐺(𝑣) = tanh(𝑣), and 𝐼𝑁(𝑡) = 7 cos √2𝑡. It is easy to calculate that

𝐾 = 1,

𝑊 = 0.2,

𝑅 = 0.01,

−1 𝜐𝑃−1 (𝑡) = 𝜐𝑁 (𝑡) =

𝐿 = 𝐵𝑠 = 1,

(𝑡 + 10) , 0.9

𝐼 = 7, (32)

𝛿 = 182.5.

̂2 ) hold. By Corollary 4, (31) It is easy to check that (𝐻1 ) and (𝐻 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 (𝑋𝑃 (𝑡), 𝑋𝑁(𝑡)) in 𝐵∗ . Phase portrait of attractivity of 𝑋𝑃 and 𝑋𝑁 is illustrated in Figure 3.

Discrete Dynamics in Nature and Society

7

1.5

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.

1 0.5

References

0 −0.5 −1

0

10

20

30

40

50

60

70

80

Time (𝑡)

𝑋𝑃 (𝑡) 𝑋𝑁 (𝑡)

Figure 2: Transient behavior of the almost periodic solution of (31) located in 𝐵∗ . 3 2

XN (t)

1 0 −1 −2 −3 −3

−2

−1

0 XP (t)

1

2

3

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),

[1] H. R. Wilson and J. D. Cowan, “Excitatory and inhibitory interactions in localized populations of model neurons,” Biophysical Journal, vol. 12, no. 1, pp. 1–24, 1972. [2] H. R. Wilson and J. D. Cowan, “A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,” Kybernetik, vol. 13, no. 2, pp. 55–80, 1973. [3] C. van Vreeswijk and H. Sompolinsky, “Chaos in neuronal networks with balanced excitatory and inhibitory activity,” Science, vol. 274, no. 5293, pp. 1724–1726, 1996. [4] S. E. Folias and G. B. Ermentrout, “New patterns of activity in a pair of interacting excitatory-inhibitory neural fields,” Physical Review Letters, vol. 107, no. 22, Article ID 228103, 2011. [5] K. Mantere, J. Parkkinen, T. Jaaskelainen, and M. M. Gupta, “Wilson-Cowan neural-network model in image processing,” Journal of Mathematical Imaging and Vision, vol. 2, no. 2-3, pp. 251–259, 1992. [6] L. H. A. Monteiro, M. A. Bussab, and J. G. C. Berlinck, “Analytical results on a Wilson-Cowan neuronal network modified model,” Journal of Theoretical Biology, vol. 219, no. 1, pp. 83–91, 2002. [7] B. Pollina, D. Benardete, and V. W. Noonburg, “A periodically forced Wilson-Cowan system,” SIAM Journal on Applied Mathematics, vol. 63, no. 5, pp. 1585–1603, 2003. [8] R. Decker and V. W. Noonburg, “A periodically forced WilsonCowan system with multiple attractors,” SIAM Journal on Mathematical Analysis, vol. 44, no. 2, pp. 887–905, 2012. [9] P. L. William, “Equilibrium and stability of wilson and cowan’s time coarse graining model,” in Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS ’12), Melbourne, Australia, July 2012. [10] Z. Huang, S. Mohamad, X. Wang, and C. Feng, “Convergence analysis of general neural networks under almost periodic stimuli,” International Journal of Circuit Theory and Applications, vol. 37, no. 6, pp. 723–750, 2009. [11] Y. Liu and Z. You, “Multi-stability and almost periodic solutions of a class of recurrent neural networks,” Chaos, Solitons and Fractals, vol. 33, no. 2, pp. 554–563, 2007. [12] C. Y. He, Almost Periodic Differential Equation, Higher Education, Beijing, China, 1992. [13] H. Zhenkun, F. Chunhua, and S. Mohamad, “Multistability analysis for a general class of delayed Cohen-Grossberg neural networks,” Information Sciences, vol. 187, pp. 233–244, 2012. [14] H. Zhenkun and Y. N. Raffoul, “Biperiodicity in neutral-type delayed difference neural networks,” Advances in Difference Equations, vol. 2012, article 5, 2012.

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