Theory and Applications of Periodic Solutions and Almost Periodic SolutionsView this Special Issue
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Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays
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.
Consider a well-known Wilson-Cowan type model [1, 2] with time-varying delays where represent the proportion of excitatory and inhibitory neurons firing per unit time at the instant , respectively. and are related to the duration of the refractory period, and are constants. , , , and are the strengths of connections between the populations. are the external inputs to the excitatory and the inhibitory populations. is the response function of neuronal activity and it is always assumed to be sigmoid type. , 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  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 and almost periodic inputs [10–12]: (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.
Throughout this paper, we always assume that , , , , , , , and are positive constants, , , , and are almost periodic functions , and set
Moreover, we need some basic assumptions in this paper. , and there exists an such that The quadratic equation has a positive solution , where are bounded and continuously differentiable with ,
For for all , we define the norm . Let , where is an almost periodic function on . For all , if we define induced nodule , then is a Banach space. The initial conditions of system (1) are of the form where and .
Definition 1 (see ). Let be continuous. is said to be almost periodic on if, for any , it is possible to find a real number , and for any interval with length , there exists a number in this interval such that , for all .
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 and hold. If , then there exists a unique almost periodic solution of system (1) in the region
Proof. For for all , we consider the almost periodic solution of the following almost periodic differential equations:
By almost periodicity of , , , and and Theorem 3.4 in  or , (9) has a unique almost periodic solution
Define a mapping by setting , for all . Now, we prove that is a self-mapping from to . From (10) and , we obtain
By similar estimation, we can get
Therefore, by the above estimations and , we get which implies that . So, the mapping is self-mapping from to . Next, we prove that is a contraction mapping in the region . For all , , by (10), we have which leads to
By similar argument, we can get
From (15) and (16), we have
Since , it is clear that the mapping is a contraction. Therefore the mapping possesses a unique fixed point such that . By (9), is an almost periodic solution of system (1) in . The proof is complete.
Remark 3. Obviously, quadratic curve satisfies with . So, and guarantees the existence of in and lies in the following interval:
By Theorem 2, we know that the unique almost periodic solution depends on . Choosing , we get a simple assumption as follows:,,and hence it leads to a parameter-based result.
Corollary 4. Suppose that and hold. Then there exists a unique almost periodic solution of system (1) in the region .
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 , which is delay dependent.
Theorem 5. Suppose that – hold. If and there exist constants , such that where and , and are the inverse functions of and , then system (1) has exactly one almost periodic solution in the region which is locally exponentially stable.
Proof. From Theorem 2, system (1) has a unique almost periodic solution . Let be an arbitrary solution of system (1) with initial value . Set , . By system (1), we get
Construct the auxiliary functions , defined on as follows:
One can easily show that , are well defined and continuous. Assumption (19) implies that , as and , as . It follows that there exists a common such that and .
Consider the Lyapunov functional
Calculating the upper right derivative of along system (1), one has which leads to where . We have from the above that and
Note that where , . Then there exists a positive constant such that
The proof is complete.
Corollary 6. Suppose that , , and hold. If there exist constants , such that
and are defined as Theorem 5, then system (1) has exactly one almost periodic solution in the region 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: where , , , , , , and . It is easy to calculate that
It is easy to check that and hold. By Corollary 4, (31) has a unique almost periodic solution in region . Setting , we can check that and (28) hold and hence is exponentially stable in . Figure 2 shows the transient behavior of the unique almost periodic solution in . Phase portrait of attractivity of and is illustrated in Figure 3.
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.
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.
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Copyright © 2013 Shasha Xie and Zhenkun 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.