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Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associative memory. In this paper, we establish a delayed neural system with external stimulations. The complex dynamical behaviors induced by external simulations are investigated employing theoretical analysis and numerical simulation. Firstly, we illustrate number of equilibria by the saddle-node bifurcation of nontrivial equilibria. It implies that the neural system has one/three equilibria for the external stimulation. Then, analyzing characteristic equation to find Hopf bifurcation, we obtain the equilibrium’s stability and illustrate periodic activity induced by the external stimulations and time delay. The neural system exhibits a periodic activity with the increased delay. Further, the external stimulations can induce and suppress the periodic activity. The system dynamics can be transformed from quiescent state (i.e., the stable equilibrium) to periodic activity and then quiescent state with stimulation increasing. Finally, inspired by ubiquitous rhythm in living organisms, we introduce periodic stimulations with low frequency as rhythm activity from sensory organs and other regions. The neural system subjected by the periodic stimulations exhibits some interesting activities, such as periodic spiking, subthreshold oscillation, and bursting-like activity. Moreover, the subthreshold oscillation can switch its position with delay increasing. The neural system may employ time delay to realize Winner-Take-All functionality.

Imitating the properties of a biological nerve system to build artificial neural network model plays an important role in the fields of neural network applications. Complexity and dynamical analysis in neural systems are important requirements for the application of optimization problem and associative memory. In fact, many neural network systems, such as Hopfield neural system, Cohen–Grossberg neural system, and cellular neural system, have been applied in many research fields such as associative memory, secure communication, and signal processing. Actually, associative memory storage in neural networks is defined by the stable equilibrium points, periodic orbits, and even chaotic attractors. Neural system receives external stimulations from sensory organs and brain regions and then produces different types of firing behaviors, such as periodic activity, spiking, bursting, and chaos to transfer and integrate neural information [

In this paper, we investigate a simplified delay neural system with external stimulations. The Wilson-Cowan (W-C) neural model with time delay [

The W–C neural system with time delay and external stimulations is described by the following delayed differential equation:

The outline of this paper is as follows: in Section

We start with equilibrium analysis by employing the dynamical bifurcation theory. It is obvious that system (

The number and value of the nontrivial equilibrium just depend on synaptic weights

By

The corresponding characteristic equation is

In fact, the critical value of equilibrium’s number corresponds to a static bifurcation [

It should be noted that

The number of equilibria can be obtained by dynamic nullclines, as shown in Figure

Intersection points of dynamic nullclines illustrate the saddle-node bifurcation of nontrivial equilibrium, where (a) one nontrivial equilibrium for

In this section, we will analyze the equilibrium’s stability and find periodic activity induced by both external stimulations and time delay. To this end, using

The characteristic equation of system (

It follows that the equilibrium of system (

Using the Routh-Hurwitz criterion, we obtain a necessary and sufficient condition to assure the equilibrium of system (

With delay

Separating equation (

Eliminating

By

In general, based on conditions

Then, equation (

Define

The Hopf bifurcation happens when the system eigenvalues cross the imaginary axis with nonzero velocity. Differentiating

By the Hopf bifurcation theory, we obtain the following conclusion with condition (11). If

For example, we choose system parameters as

Partial eigenvalues of the system equilibrium

Time histories of system (

The real parts of system eigenvalues with

On the other hand, external stimulations can induce and suppress the periodic activity in system (

Time histories of system (

In the section above, we have studied the equilibrium stability and find a periodic activity, where system (

It follows from equilibrium analysis in Section

Firstly, we choose system parameters as

Time histories of system (

For case 2, system parameters are fixed as

Time histories of system (20) with delay varying (a)

Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associative memory. In this paper, we considered a delayed neural system with content/periodic external stimulations. The results show that content stimulations can induce and suppress a periodic activity. The neural system exhibits a periodic activity with delay increasing. Further, the system dynamics can be changed from quiescent state to periodic activity and then enter into the quiescent state with stimulation increasing. Additionally, in view of the ubiquitous rhythm in living organisms, we introduce the periodic stimulations with low frequency as the rhythm activity. The results show that the neural system subjected by periodic stimulations exhibits some interesting activities, such as the periodic spiking, subthreshold oscillation, and bursting-like ones. Further, with delay increasing slightly, the subthreshold oscillation can change its position from top to down. The neural system having multiple equilibria may employ time delay to realize Winner-Take-All functionality.

All data, models, and code generated or used during the study are provided within the article.

The authors declared that they have no conflicts of interest regarding this work.

This work was supported by the National Science Foundation of China (grant nos. 11672177 and 11672185).