Dynamical Properties of Discrete-Time Background Neural Networks with Uniform Firing Rate

The dynamics of a discrete-time background network with uniform firing rate and background input is investigated.The conditions for stability are firstly derived. An invariant set is then obtained so that the nondivergence of the network can be guaranteed. In the invariant set, it is proved that all trajectories of the network starting from any nonnegative value will converge to a fixed point under some conditions. In addition, bifurcation and chaos are discussed. It is shown that the network can engender bifurcation and chaos with the increase of background input. The computations of Lyapunov exponents confirm the chaotic behaviors.


Introduction
A bright red light may trigger a sudden motor action in a driver crossing an intersection: stepping on the brakes at once.The same red light may be entirely inconsequential if it appears inside a movie theater [1].Clearly, the context information determines whether a particular stimulus will trigger a motor response.To study the neural correlate of this class of phenomena, the background neural networks were proposed in [1].Simulations showed that the proposed networks can indeed explain the working of this phenomenon, and then it is very important to study the dynamical properties of the background networks.
Convergence is one of the most important dynamical properties of neural networks [2][3][4][5].However, the convergence analysis for the background networks is not easy.In [1], local stability is discussed.When all neurons are firing at equal rates and the total synaptic input to all neurons is the same, the convergence of the background network is studied in [6].In this paper, we apply the forward Euler scheme to discretize the background network with uniform firing rate and study the dynamical properties of the discretetime version.To obtain the conditions for stability and convergence, we present two lemmas and three theorems.Since the background networks proposed in [1] originate from the study of the activities of human brain and chaos is essential to normal brain functioning at many levels of activity [7][8][9][10], it is very significant to study the bifurcation and chaos of the networks.However, chaotic behavior in background networks has not been reported so far.This paper will explore the chaotic behavior of the discrete-time background network with uniform firing rate.Bifurcation diagrams and Lyapunov exponents are presented to illustrate the existence of chaotic behavior.
The rest of this paper is organized as follows.In Section 2, preliminaries will be presented.The conditions for nondivergence of the discrete-time background network will be obtained in Section 3. In Section 4, we will study the stability and convergence of the network.The bifurcation and chaos will be discussed in Section 5. Finally, conclusions are drawn in Section 6.

Preliminaries
Consider the following background network presented in [1] in which the firing rates for all neurons are equal: where  is the uniform firing rate of all neurons,  > 0 is a time constant,  > 0 is a saturation constant, V > 0 is the strength of a synaptic connection,  is the total number of neurons, ℎ ≥ 0 represents the background input whose value is independent of the network's activity, and the total synaptic input  > 0 to each neuron is equal.Applying the forward Euler scheme to the network (1), we obtain the discrete-time system as follows: for  ≥ 0, where  is the sampling time.That is, for  ≥ 0, where  = / > 0 is called step size.The following lemmas are useful.

Invariant Set
Definition 3. A compact set  is called an invariant set of (3), if for any (0) ∈ , the trajectory of (3) starting from (0) will remain in  for all  ≥ 0.
An invariant set provides a method to guarantee nondivergence of trajectories.Denote that Clearly,  is a bounded constant.
Proof.Let  be the maximal real root of the following equation: Since V > 0, it follows from the Vieta theorem that  > 0.

Stability and Convergence Analysis
In this section, we will study the stability and convergence of the network (3).Definition 5. A point  * ∈  is called a fixed point of (3), if and only if Clearly, there may exist three fixed points.Thus, the study of dynamics belongs to a multistability problem [11].
According to the Lyapunov indirect method [12], a fixed point of a discrete-time nonlinear system is stable if the absolute of each eigenvalue of the Jacobian matrix of the system at this point is less than 1.For the network (3), the eigenvalues of Jacobian matrix at each fixed point will be computed.Theorem 6.Let  * be a fixed point of the network (3).If and both hold, then  * is stable. Proof.Let We have It follows from (11) that If inequality ( 12) holds, we have |/|  * | < 1.Thus, the fixed point  * is stable.The proof is completed.
Theorem 6 gives the stability conditions for a fixed point of (3).Specially, when the learning rate 0 <  < 1, we will further study the convergence of the trajectories of (3).Theorem 7. Suppose that 0 <  < 1.If  ≤ /Vℎ and 0 ≤ (0) ≤ , then the trajectories of (3) starting from (0) will converge to a fixed point, where  is the maximal real root of the following cubic equation: Proof.It follows from Definition 5 that a fixed point of (3) is the root of (16).Since V > 0, according to the Vieta theorem, there must exist a positive real root.In the following, we will prove the convergence of the trajectories of (3) in all three different cases, where Δ represents the discriminant of (16).
For convenience, the maximal real root of ( 16) is denoted by  uniformly in all the cases.Clearly,  > 0.
Case 1. Suppose that Δ > 0. Clearly, there exist one real root and two imaginary roots; that is,  is the unique fixed point of (3).Firstly, we will prove that That is, By Lemma 1 and (3) and (18), it holds that Since  ≤ /Vℎ, it follows from Lemma 2 that max From Lemma 1 and ( 19) and (20), we have 0 ≤ ( + 1) ≤ .
Theorem 7 shows that the trajectories starting from [0, ] will converge.What will happen if (0) > ?We have the following theorem.Theorem 8. Suppose that 0 <  < 1.If  ≤ /Vℎ and (0) ≥ 0, then the trajectories of (3) starting from (0) will converge to a fixed point, where  is the maximal real root of (16).
If (0) ≥  > 0, it holds from (9) that there must exist an integer  ≥ 0 such that 0 ≤ () ≤ .By Theorem 7, it follows that the trajectory starting from () will converge to a fixed point of (3).The proof is completed.Theorem 8 gives the conditions guaranteeing the convergence of (3).If 0 <  < 1 and  ≤ /Vℎ, then the trajectory starting from any nonnegative real number will converge to a fixed point of (3).
Next, we will give an example to further illustrate the previous analysis.Condition (a) (where ℎ = 8).It follows from (16) that Δ < 0. It is easy to check that there are three fixed points of the network (3).By Theorem 6, it holds that  1 = 3.5401 and  = 82.7021are stable while  2 = 17.0781 is unstable.The unstable fixed point acts as a threshold that separates two stable fixed points.Since  < /Vℎ = 454.7647, it follows from Theorem 7 that the trajectories starting from [0,  2 ) should converge to  1 = 3.5401, while the trajectories starting from [ 2 , ] should converge to  = 82.7021.Figure 1 illustrates these results.
The only difference between the previous two conditions is the background input ℎ.Figures 1 and 2 show that the low-rate disappears with an increase in background inputs, switching the network between regimes with one or two stable uniform rates.

Bifurcation and Chaos
With the increase in background input ℎ, the network (3) may exhibit more complicated dynamical properties.To illustrate this, we will draw the bifurcation diagram of the network (3) and calculate Lyapunov exponents of the network.Condition (a) (where  = 0.9).Theorem 4 guarantees the nondivergence of the network (3).Since Δ > 0, then there exists only one fixed point.With the increase of the background input ℎ, the changes of the dynamical behavior of network (3) are illustrated in Figure 3.It can be seen that ℎ ≃ 4.7 is a bifurcation point of network (3), where the fixed point becomes unstable and a 2-periodic orbit shows up.As ℎ ≃ 176, the 2-periodic orbit becomes unstable and a 4period orbit appears.The 4-period orbits are always stable until ℎ = 600.
Condition (b) (where  = 0.99).The bifurcation diagram of network (3) is shown in Figure 4. To further show the route to chaos with the increase of ℎ, another bifurcation diagram is given in Figure 5.The only difference in Figures 4 and 5 is the value range of ℎ. Figure 4 shows that ℎ ≃ 4.2 is a bifurcation point.From Figure 5, we can see that ℎ ≃ 36 is also a bifurcation point, where the 2-period orbit becomes unstable and a 3-period orbit shows up, and then 5-period orbit appears, and so forth; as ℎ approaches 103, the network enters chaotic state.
To further illustrate the chaotic property, the Lyapunov exponents are computed according to Dror and Tsodyks [13].For each value of ℎ, the Lyapunov exponent of the network (3) is plotted in Figure 6.As ℎ ≃ 103, the Lyapunov exponent is positive and the network becomes chaotic.
From Examples 9 and 10, we can see that with the change of background input, the discrete-time background neural network with uniform firing rate may switch among several dynamical states: single fixed point, periodic orbits, and chaos.

Conclusions
The discrete-time background network obtained by Euler method is investigated in this paper.The nondivergence,    stability, and convergence of the network are studied, respectively.As shown in this paper, though the nonlinear system is one dimensional, it is not easy to study its convergence.Furthermore, as the background input ℎ varies, it is shown that the network may exhibit complex dynamical behaviors such as bifurcation and chaos.This means that the dynamics of the discrete-time network can produce a much richer set of patterns than those discovered in continuous-time network.Since the background networks originate from the study of the activities of brain and chaotic activities are ubiquitous in the human brain, the chaos analysis of the background networks is significant.