# Dynamics of a class of neutral three neurons network with delay

- Ming Liu
^{1}, - Chunrui Zhang
^{1}Email author and - Xiaofeng Xu
^{1}

**2013**:338

https://doi.org/10.1186/1687-1847-2013-338

© Liu et al.; licensee Springer. 2013

**Received: **31 January 2013

**Accepted: **28 October 2013

**Published: **22 November 2013

## Abstract

In this paper, a class of neutral neural networks with delays is investigated. The linear stability of the model is studied. It is found that a Hopf bifurcation also occurs when some delays pass through a sequence of critical values. The direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are determined by using the normal form method and center manifold theory. The existence of a permanent oscillation is established using Chafee’s criterion. Numerical simulations are performed to support the analytical results.

## Keywords

## 1 Introduction

*etc.*can be written in neurons network models. In 2008, Li and Yuan considered a Hopfield-type network of three identical neurons coupled in any possible way in [1]:

*j*to

*i*are connected or not; $a,b,{a}_{2},{b}_{2}\in R$ denote the strength in self-connection and neighboring-connection, respectively; ${\tau}_{s},{\tau}_{n}\ge 0$ are the corresponding time delays. Furthermore,

*f*,

*g*, ${f}_{2}$, ${g}_{2}$ are assumed to be adequately smooth and to satisfy the following conditions:

Then we derive the stability of this system and conditions of existence of the bifurcation with ${a}_{12}={a}_{13}={a}_{23}={a}_{32}=1$, ${a}_{21}={a}_{31}=0$; ${b}_{12}={b}_{13}={b}_{23}={b}_{32}=1$, ${b}_{21}={b}_{31}=0$, ${\tau}_{s}={\tau}_{n}=\tau $. The remainder of this paper is organized as follows. In Section 2 we introduce the stability of the equilibrium point and the conditions of existence of a local Hopf bifurcation. We are devoted to establishing the direction and stability of the Hopf bifurcation in Section 3. In Section 4 we discuss the existence of a permanent oscillation. Finally, we carry out some numerical simulation to support the analysis result in Section 5.

## 2 Stability and Hopf bifurcation analysis

Under the given hypotheses (H_{1}) and (H_{2}), it is easy to check that $x={(0,0,0)}^{T}$ is an equilibrium point of system (2.1). By using a similar method to that in [3], we have the following results on stability to system (2.1).

**Theorem 1**

*Let*$|{a}_{2}+{b}_{2}|<1$

*and*$|{a}_{2}-{b}_{2}|<1$.

- (1)
*If*$(a,b)\in {D}_{1}$,*then the zero solution of system*(2.1)*is absolutely stable*. - (2)
*If*$(a,b)\in {D}_{2}$,*then the zero solution of system*(2.1)*is conditionally stable*,*i*.*e*., $\tau \in [0,{\tau}_{0})$,*the zero solution of system*(2.1)*is asymptotically stable*; $\tau >{\tau}_{0}$,*the zero solution of system*(2.1)*is unstable*,*with*$\begin{array}{c}{\tau}_{1j}=\sqrt{\frac{1-{a}_{2}^{2}}{{a}^{2}-1}}[arccos\frac{a{a}_{2}+1}{a+{a}_{2}}+2\pi j],\hfill \\ {\tau}_{2j}=\sqrt{\frac{1-{({a}_{2}+{b}_{2})}^{2}}{{(a+b)}^{2}-1}}[arccos\frac{(a+b)({a}_{2}+{b}_{2})+1}{a+b+{a}_{2}+{b}_{2}}+2\pi j],\hfill \\ {\tau}_{3j}=\sqrt{\frac{1-{({a}_{2}-{b}_{2})}^{2}}{{(a-b)}^{2}-1}}[arccos\frac{(a-b)({a}_{2}-{b}_{2})+1}{a-b+{a}_{2}-{b}_{2}}+2\pi j],\hfill \\ {\tau}_{0}=min[{\tau}_{10},{\tau}_{20},{\tau}_{30}],\hfill \end{array}$- (a)
*if*$a<-1$,*system*(2.1)*undergoes a Hopf bifurcation at the origin when*$\tau ={\tau}_{1j}$, $j=0,1,2,\dots $ , - (b)
*if*$a+b<-1$,*system*(2.1)*undergoes a Hopf bifurcation at the origin when*$\tau ={\tau}_{2j}$, $j=0,1,2,\dots $ , - (c)
*if*$a-b<-1$,*system*(2.1)*undergoes a Hopf bifurcation at the origin when*$\tau ={\tau}_{3j}$, $j=0,1,2,\dots $

- (a)

*Proof*From hypotheses (H

_{1}) and (H

_{2}), the characteristic equation associated with the linearization of system (2.1) is

Separately analyzing the roots of ${\mathrm{\Delta}}_{i}(\lambda )=0$ ($i=1,2,3$), by using the method in [3], we have the following results.

If $-1\le a<1$, then all roots of ${\mathrm{\Delta}}_{1}(\lambda )=0$ have negative real parts for all $\tau \ge 0$. If $a<-1$, then ${\mathrm{\Delta}}_{1}(\lambda )=0$ has a pair of purely imaginary roots when $\tau ={\tau}_{1j}$.

If $-1\le a+b<1$, then all roots of ${\mathrm{\Delta}}_{2}(\lambda )=0$ have negative real parts for all $\tau \ge 0$. If $a+b<-1$, then ${\mathrm{\Delta}}_{2}(\lambda )=0$ has a pair of purely imaginary roots when $\tau ={\tau}_{2j}$.

If $-1\le a-b<1$, then all roots of ${\mathrm{\Delta}}_{3}(\lambda )=0$ have negative real parts for all $\tau \ge 0$. If $a-b<-1$, then ${\mathrm{\Delta}}_{3}(\lambda )=0$ has a pair of purely imaginary roots when $\tau ={\tau}_{3j}$.

Additionally, all roots of ${\mathrm{\Delta}}_{i}(\lambda )=0$ ($i=1,2,3$) have negative real parts when $\tau =0$ and $Re\frac{d\lambda}{d\tau}{|}_{\tau ={\tau}_{ij}}>0$ ($i=1,2,3$) is satisfied.

Summarizing the conclusions above, the proof is completed. □

## 3 Properties of Hopf bifurcation

*τ*as a bifurcation parameter. In this section, we shall investigate the direction of the Hopf bifurcation and stability of bifurcating periodic solutions by taking ${f}_{2}(u)={g}_{2}(u)=u$. Rewrite Eq. (2.1) as the following system:

Comparing with the previous characteristic equation, we find $\gamma =\lambda \tau $. For convenience, we denote $\gamma =({\tau}_{j}+\nu )\lambda $, where ${\tau}_{j}={\tau}_{sj}$ ($s=1,2,3$; $j=0,1,2,\dots $) and $\nu \in R$. According to Theorem 1, we know that system (3.2) undergoes a Hopf bifurcation at the origin when $\nu =0$.

*i.e.*,

Consequently, we have the following theorem on the bifurcating periodic solution.

**Theorem 2**

*For system*(3.1),

*assume*$0<{a}_{2}<\frac{\sqrt{2}}{2}$, $a<-\frac{1}{{a}_{2}}$,

*then*

- (1)
*if*${f}^{\u2034}(0)>0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{1j}$*is supercritical and the bifurcating periodic solutions are asymptotically stable*; - (2)
*if*${f}^{\u2034}(0)<0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{1j}$*is subcritical and the bifurcating periodic solutions are unstable*.

*Proof*When $\tau ={\tau}_{1j}$, by calculation, we easily obtain the following results:

This completes the proof of Theorem 2. □

Similarly, we can prove Theorem 3 and Theorem 4, we omit the proof here.

**Theorem 3**

*For system*(3.1),

*assume*$0<{a}_{2}+{b}_{2}<\frac{\sqrt{2}}{2}$, $a+b<-\frac{1}{{a}_{2}+{b}_{2}}$,

*then*

- (1)
*if*$a{f}^{\u2034}(0)+b{g}^{\u2034}(0)>0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{2j}$*is subcritical and the bifurcating periodic solutions are unstable*; - (2)
*if*$a{f}^{\u2034}(0)+b{g}^{\u2034}(0)<0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{2j}$*is supercritical and the bifurcating periodic solutions are asymptotically stable*.

**Theorem 4**

*For system*(3.1),

*assume*$0<{a}_{2}-{b}_{2}<\frac{\sqrt{2}}{2}$, $a-b<-\frac{1}{{a}_{2}-{b}_{2}}$,

*then*

- (1)
*if*$a{f}^{\u2034}(0)-b{g}^{\u2034}(0)>0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{3j}$*is subcritical and the bifurcating periodic solutions are unstable*; - (2)
*if*$a{f}^{\u2034}(0)-b{g}^{\u2034}(0)<0$,*the direction of the Hopf bifurcation at*$\tau ={\tau}_{3j}$*is supercritical and the bifurcating periodic solutions are asymptotically stable*.

## 4 Permanent oscillation

Based on Chafee’s criterion, if system (2.1) has a unique equilibrium point which is unstable, and the solutions of system (2.1) are globally bounded, this system generates a limit cycle, namely a permanent oscillation [5].

*f*,

*g*, ${f}_{2}$, ${g}_{2}$ are nonlinear bounded functions and satisfy Lipschitz condition,

We have the following lemmas.

**Lemma 5** *If* $|\alpha |{L}_{1}+|\beta |{L}_{2}<1$ *holds*, *system* (2.1) *has a unique equilibrium point*.

*Proof*Suppose that ${X}^{\ast}$ is the equilibrium point of the system, then we have

where $|{c}_{i}|\le {L}_{1}$, $|{d}_{j}|\le {L}_{2}$ ($i=1,2,3$; $j=2,3$). Under the given condition, Φ is an invertible matrix. Then $u=v$, namely $H(X)$ is injective on ${R}^{3}$. Noting that *f* and *g* are bounded continuous functions, it is easily to obtain that $\parallel H(u)\parallel \to \mathrm{\infty}$, when $\parallel u\parallel \to \mathrm{\infty}$. So $H(X)$ is a homeomorphism on ${R}^{3}$ and system (2.1) has a unique equilibrium point. □

**Lemma 6** *The solutions of system* (2.1) *are globally bounded*.

*Proof*Since

*f*,

*g*, ${f}_{2}$ and ${g}_{2}$ are bounded continuous functions, there is $M>0$ such that

with $i=1,2,3$. This proves the lemma. □

**Lemma 7**

*The equilibrium point*$(0,0,0)$

*of system*(2.1)

*is unstable when one of the following conditions are satisfied*:

- (1)
${\alpha}_{2}>0$, ${\alpha}_{2}>\alpha \tau $,

*and*$\tau +(1-\frac{\alpha}{{\alpha}_{2}}\tau )<{\alpha}_{2}{e}^{-(1-\frac{\alpha}{{\alpha}_{2}}\tau )}$, - (2)
${\alpha}_{2}+{\beta}_{2}>0$, ${\alpha}_{2}+{\beta}_{2}>(\alpha +\beta )\tau $,

*and*$\tau +(1-\frac{\alpha +\beta}{{\alpha}_{2}+{\beta}_{2}}\tau )<({\alpha}_{2}+{\beta}_{2}){e}^{-(1-\frac{\alpha +\beta}{{\alpha}_{2}+{\beta}_{2}}\tau )}$, - (3)
${\alpha}_{2}-{\beta}_{2}>0$, ${\alpha}_{2}-{\beta}_{2}>(\alpha -\beta )\tau $,

*and*$\tau +(1-\frac{\alpha -\beta}{{\alpha}_{2}-{\beta}_{2}}\tau )<({\alpha}_{2}-{\beta}_{2}){e}^{-(1-\frac{\alpha -\beta}{{\alpha}_{2}-{\beta}_{2}}\tau )}$.

*Proof*Based on analysis in [2], we know the roots of the following equation:

are the characteristic roots of the linearized system of (2.1). When condition (1) holds, Eq. (4.1) has at least a positive real root, and the equilibrium point $(0,0,0)$ of system (2.1) is unstable.

Using the same method, we can obtain conditions (2) and (3). The proof of the lemma is completed. □

Up to now, we have prepared sufficiently to state the following results.

**Theorem 8**

*System*(2.1)

*generates a permanent oscillation*,

*when*$|\alpha |{L}_{1}+|\beta |{L}_{2}<1$

*holds and one of the following conditions are satisfied*:

- (1)
${\alpha}_{2}>0$, ${\alpha}_{2}>\alpha \tau $,

*and*$\tau +(1-\frac{\alpha}{{\alpha}_{2}}\tau )<{\alpha}_{2}{e}^{-(1-\frac{\alpha}{{\alpha}_{2}}\tau )}$, - (2)
${\alpha}_{2}+{\beta}_{2}>0$, ${\alpha}_{2}+{\beta}_{2}>(\alpha +\beta )\tau $,

*and*$\tau +(1-\frac{\alpha +\beta}{{\alpha}_{2}+{\beta}_{2}}\tau )<({\alpha}_{2}+{\beta}_{2}){e}^{-(1-\frac{\alpha +\beta}{{\alpha}_{2}+{\beta}_{2}}\tau )}$, - (3)
${\alpha}_{2}-{\beta}_{2}>0$, ${\alpha}_{2}-{\beta}_{2}>(\alpha -\beta )\tau $,

*and*$\tau +(1-\frac{\alpha -\beta}{{\alpha}_{2}-{\beta}_{2}}\tau )<({\alpha}_{2}-{\beta}_{2}){e}^{-(1-\frac{\alpha -\beta}{{\alpha}_{2}-{\beta}_{2}}\tau )}$.

## 5 Numerical simulation

In the section, we carry out some numerical simulations for system (2.1).

*i.e.*, $\tau \in [0,{\tau}_{20}=0.48)$, the zero solution of system (2.1) is asymptotically stable; $\tau >{\tau}_{20}=0.48$, the zero solution of system (2.1) is unstable, and system (2.1) undergoes a Hopf bifurcation at the origin when $\tau ={\tau}_{20}$. Furthermore, the direction of the Hopf bifurcation at $\tau ={\tau}_{2j}$ is subcritical and the bifurcating periodic solutions are unstable. The simulation results as shown in Figures 3 and 4.

## 6 Conclusion

For a neutral model including three cells with time delay, we have given the general condition for the stability and shown the delay-independent and delay-dependent local stability regions. We have also obtained the condition to determine the direction of Hopf bifurcations, the stability of bifurcating periodic solutions and a permanent oscillation.

As we know, the extension of local periodic solutions for large time delay would appear when some conditions are satisfied. Further study of the patterns is undergoing.

## Declarations

### Acknowledgements

The authors are very grateful to the referees for their valuable suggestions. This work is supported by the Fundamental Research Funds for the Central Universities (No. DL12BB29) and the National Natural Science Foundation of China (Grant No. 41304093).

## Authors’ Affiliations

## References

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