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 [

It is interesting to revisit Wilson-Cowan system on the following points.

The Wilson-Cowan model has a realistic biological background which describes interactions between excitatory and inhibitory populations of neurons [

There exists rich dynamical behavior in Wilson-Cowan model. Theoretical results about stable limit cycles, equilibria, chaos, and oscillatory activity have been reported in [

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 (

Coexistence of divergent and local stable solutions of (

Throughout this paper, we always assume that

Moreover, we need some basic assumptions in this paper.

The quadratic equation

For for all

Let

The remaining part of this paper is organized as follows. In Section

Suppose that

For for all

By almost periodicity of

Define a mapping

By similar estimation, we can get

Therefore, by the above estimations and

By similar argument, we can get

From (

Since

Obviously, quadratic curve

By Theorem

Suppose that

In this section, we establish locally exponential stability of the unique almost periodic solution of system (

Suppose that

From Theorem

Construct the auxiliary functions

One can easily show that

Consider the Lyapunov functional

Calculating the upper right derivative of

Note that

The proof is complete.

Set

Suppose that

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:

It is easy to check that

Transient behavior of the almost periodic solution of (

Phase portrait of attractivity of the unique almost periodic solution of (

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 [

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.