Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling

Abstract

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results.

1. Introduction

In recent years, a number of different classes of neural networks with or without delays, including Hopfield networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associate memory neural networks have been active research topic as [1], and substantial efforts have been made in neural network models, for example, Huang et al. [2] studied the global exponential stability and the existence of periodic solution of a class of cellular neural networks with delays, Guo and Huang [3] investigated the Hopf bifurcation natures of a ring of neurons with delays, Yan [4] analyzed the stability and bifurcation of a delayed tri-neuron network model, Hajihosseini et al. [5] made a discussion on the Hopf bifurcation of a delayed recurrent neural network in the frequency domain, and Liao et al. [6] did a theoretical and empirical investigation of a two-neuron system with distributed delays in the frequency domain. Agranovich et al. [7] considered the impulsive control of a hysteresis cellular neural network model. For more information, one can see [8–24]. In 1986 and 1987, Babcock and Westervelt [25, 26] had investigated the stability and dynamics of the following simple neural network model of two neurons with inertial coupling: 𝑑𝑥₁𝑑𝑡=𝑥₃,𝑑𝑥₂𝑑𝑡=𝑥₄,𝑑𝑥₃𝑑𝑡=−2𝜉𝑥₃−𝑥₁+𝐴₂tanh𝑥₂,𝑑𝑥₄𝑑𝑡=−2𝜉𝑥₄−𝑥₂+𝐴₁tanh𝑥₁, (1.1) where 𝑥_𝑖(𝑖=1,2) is the input voltage of the 𝑖th neuron, 𝑥_𝑗(𝑗=3,4) denotes the output of the 𝑗th neuron, 𝜉>0 is the damping factor, and 𝐴_𝑖(𝑖=1,2) is the overall gain of the neuron which determines the strength of the nonlinearity. For a more detailed interpretation of the parameters, one can see [25, 26]. In 1997, Lin and Li [27] made a detail discussion on the bifurcation direction of periodic solution for system (1.1).

From applications point of view, considering that there is a time delay (we assume that it is 𝜏) in the response of the output voltages to changes in the input, that is, there exists a feedback delay of the input voltage of the 𝑖th neuron to the growth of the output of the 𝑗th neuron, then we modify system (1.1) as follows: 𝑑𝑥₁𝑑𝑡=𝑥₃,𝑑𝑥₂𝑑𝑡=𝑥₄,𝑑𝑥₃𝑑𝑡=−2𝜉𝑥₃−𝑥₁(𝑡−𝜏)+𝐴₂tanh𝑥₂(𝑡−𝜏),𝑑𝑥₄𝑑𝑡=−2𝜉𝑥₄−𝑥₂(𝑡−𝜏)+𝐴₁tanh𝑥₁(𝑡−𝜏). (1.2) It is well known that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation is very critical. When delays are incorporated into the network models, stability and Hopf bifurcation analysis become more difficult. To obtain a deep and clear understanding of dynamics of neural network model of two neurons with inertial coupling, we will make a discussion on system (1.2), that is, we study the stability, the local Hopf bifurcation for system (1.2).

The remainder of this paper is organized as follows. In Section 2, we investigate the stability of the equilibrium and the occurrence of local Hopf bifurcations. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results.

2. Stability of the Equilibrium and Local Hopf Bifurcations

In this section, we shall study the stability of the equilibrium and the existence of local Hopf bifurcations. For simplification, we only consider the zero equilibrium. One can check that if the following condition: (𝐻1)𝐴₁𝐴₂<1 (2.1) holds, then (1.2) has a unique equilibrium 𝐸(0,0,0,0). The linearization of (1.2) at 𝐸(0,0,0,0) is given by 𝑑𝑥₁𝑑𝑡=𝑥₃,𝑑𝑥₂𝑑𝑡=𝑥₄,𝑑𝑥₃𝑑𝑡=−2𝜉𝑥₃−𝑥₁(𝑡−𝜏)+𝐴₂𝑥₂(𝑡−𝜏),𝑑𝑥₄𝑑𝑡=−2𝜉𝑥₄−𝑥₂(𝑡−𝜏)+𝐴₁𝑥₁(𝑡−𝜏), (2.2) whose characteristic equation takes the form of 𝑒det𝜆0−100𝜆0−1_−𝜆𝜏−𝐴₂𝑒^−𝜆𝜏𝜆+2𝜉0−𝐴₁𝑒^−𝜆𝜏𝑒^−𝜆𝜏0𝜆+2𝜉=0, (2.3) that is, 𝜆⁴+4𝜉𝜆³+4𝜉²𝜆²+2𝜆²+4𝜉𝜆𝑒^−𝜆𝜏+1−𝐴₁𝐴₂𝑒^−2𝜆𝜏=0. (2.4) Multiplying 𝑒^𝜆𝜏 on both sides of (2.4), it is easy to obtain 𝜆⁴+4𝜉𝜆³+4𝜉²𝜆²𝑒^𝜆𝜏+2𝜆²+4𝜉𝜆+1−𝐴₁𝐴₂𝑒^−𝜆𝜏=0. (2.5) In order to investigate the distribution of roots of the transcendental equation (2.5), the following lemma is helpful.

Lemma 2.1 (see [28]). For the following transcendental equation: 𝑃𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜏1+𝑝+⋯_1(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜏𝑚}=0, (2.6) as (𝜏₁,𝜏₂,𝜏₃,…,𝜏_𝑚) vary, the sum of orders of the zeros of 𝑃(𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}) in the open right half-plane can change, and only a zero appears on or crosses the imaginary axis.

For 𝜏=0, (2.5) becomes 𝜆⁴+4𝜉𝜆³+4𝜉²+2𝜆²+4𝜉𝜆+1−𝐴₁𝐴₂=0. (2.7) In view of Routh-Hurwitz criteria, we know that all roots of (2.7) have a negative real part if the following condition: (𝐻2)4𝜉²+𝐴₁𝐴₂>0 (2.8) is fulfilled.

For 𝜔>0,𝑖𝜔 is a root of (2.5) if and only if 𝜔⁴−4𝜉𝜔³𝑖−4𝜉²𝜔²(cos𝜔𝜏+𝑖sin𝜔𝜏)−2𝜔²+4𝜉𝜔𝑖+1−𝐴₁𝐴₂(cos𝜔𝜏−𝑖sin𝜔𝜏)=0. (2.9) Separating the real and imaginary parts gives 𝜔⁴−4𝜉²𝜔²+1−𝐴₁𝐴₂cos𝜔𝜏+4𝜉𝜔³sin𝜔𝜏=2𝜔²,𝜔⁴−4𝜉²𝜔²−1+𝐴₁𝐴₂sin𝜔𝜏−4𝜉𝜔³cos𝜔𝜏=−4𝜉𝜔. (2.10) Then, we obtain sin𝜔𝜏=4𝜉𝜔⁵+16𝜉³𝜔³−1−𝐴₁𝐴₂𝜔⁴+4𝜉²𝜔²²−1−𝐴₁𝐴₂²,cos𝜔𝜏=2𝜔⁶+8𝜉²𝜔⁴−1−𝐴₁𝐴₂𝜔⁴+4𝜉²𝜔²²−1−𝐴₁𝐴₂². (2.11)(2.12) In view of sin²𝜔𝜏+cos²𝜔𝜏=1, then, we have 4𝜉𝜔⁵+16𝜉³𝜔³−1−𝐴₁𝐴₂²+2𝜔⁶+8𝜉²𝜔⁴−1−𝐴₁𝐴₂²=𝜔⁴+4𝜉²𝜔²²−1−𝐴₁𝐴₂²², (2.13) which is equivalent to 𝜔¹⁶+𝑙₁𝜔¹⁴+𝑙₂𝜔¹²+𝑙₃𝜔¹⁰+𝑙₄𝜔⁸+𝑙₅𝜔⁶+𝑙₆𝜔⁵+𝑙₇𝜔⁴+𝑙₈𝜔³+𝑙₉=0, (2.14) where 𝑙₁=16𝜉²,𝑙₂=96𝜉⁴−4,𝑙₃=256𝜉⁶−48𝜉²,  𝑙₄=256𝜉⁸−64𝜉⁴−128𝜉³−2(1−𝐴₁𝐴₂)²,𝑙₅=2(1−𝐴₁𝐴₂)−16(1−𝐴₁𝐴₂)²𝜉²−256𝜉⁶,  𝑙₆=2𝜉(1−𝐴₁𝐴₂),𝑙₇=16𝜉²(1−𝐴₁𝐴₂)−32𝜉⁴(1−𝐴₁𝐴₂)²,  𝑙₈=32𝜉³(1−𝐴₁𝐴₂),𝑙₉=−[(1−𝐴₁𝐴₂)⁴+(1−𝐴₁𝐴₂)²].

Denote ℎ(𝜔)=𝜔¹⁶+𝑙₁𝜔¹⁴+𝑙₂𝜔¹²+𝑙₃𝜔¹⁰+𝑙₄𝜔⁸+𝑙₅𝜔⁶+𝑙₆𝜔⁵+𝑙₇𝜔⁴+𝑙₈𝜔³+𝑙₉. (2.15) Since 𝑙₉<0 and lim_𝜔→+∞ℎ(𝜔)=+∞, then we can conclude that (2.14) has at least one positive root. Without loss of generality, we assume that (2.14) has sixteen positive roots, denoted by 𝜔_𝑘(𝑘=1,2,3,…,16). Then, by (2.12), we have 𝜏_𝑘^(𝑗)=1𝜔_𝑘arccos2𝜔⁶+8𝜉²𝜔⁴−1−𝐴₁𝐴₂𝜔⁴+4𝜉²𝜔²²−1−𝐴₁𝐴₂²+2𝑗𝜋, (2.16) where 𝑘=1,2,3,…,16;𝑗=0,1,…, then ±𝑖𝜔_𝑘 are a pair of purely imaginary roots of (2.4) with 𝜏_𝑘^(𝑗). Define 𝜏₀=𝜏_𝑘⁽⁰⁾₀=min_{{}𝑘∈1,2,3,…,16}𝜏_𝑘⁽⁰⁾. (2.17) The above analysis leads to the following result.

Lemma 2.2. If (𝐻1) and (𝐻2) hold, then all roots of (2.4) have a negative real part when 𝜏∈[0,𝜏₀) and (2.4) admits a pair of purely imaginary roots ±𝜔_𝑘 when 𝜏=𝜏_𝑘^(𝑗)(𝑘=1,2,3,…,16;𝑗=0,1,2,…).

Let 𝜆(𝜏)=𝛼(𝜏)+𝑖𝜔(𝜏) be a root of (2.5) near 𝜏=𝜏_𝑘^(𝑗), 𝛼(𝜏_𝑘^(𝑗))=0, and 𝜔(𝜏_𝑘^(𝑗))=𝜔_𝑘. Due to functional differential equation theory, for every 𝜏_𝑘^(𝑗),𝑘=1,2,3,…,16;𝑗=0,1,2,…, there exists 𝜀>0 such that 𝜆(𝜏) is continuously differentiable in 𝜏 for |𝜏−𝜏_𝑘^(𝑗)|<𝜀. Substituting 𝜆(𝜏) into the left hand side of (2.5) and taking derivative with respect to 𝜏, we have 𝑑𝜆𝑑𝜏⁻¹=4𝜆³+12𝜉𝜆²+8𝜉²𝜆𝑒^𝜆𝜏+4𝜆+4𝜉𝜆1−𝐴₁𝐴₂𝑒^−𝜆𝜏−𝜆𝜆⁴+4𝜉𝜆³+4𝜉²𝜆²𝑒^𝜆𝜏−𝜏𝜆. (2.18) Noting that 4𝜆³+12𝜉𝜆²+8𝜉²𝜆𝑒^𝜆𝜏+4𝜆+4𝜉_{𝜏=𝜏𝑘(𝑗)}=𝐾₁+𝐾₂𝜆𝑖,1−𝐴₁𝐴₂𝑒^−𝜆𝜏−𝜆𝜆⁴+4𝜉𝜆³+4𝜉²𝜆²𝑒^𝜆𝜏_{𝜏=𝜏𝑘(𝑗)}=𝑃₁+𝑃₂𝑖, (2.19) where 𝐾₁=4𝜉−12𝜉𝜔²_𝑘cos𝜔_𝑘𝜏_𝑘^(𝑗)−(18𝜉²𝜔_𝑘−4𝜔³_𝑘)sin𝜔_𝑘𝜏_𝑘^(𝑗),  𝐾₂=4𝜔_𝑘+(8𝜉²𝜔_𝑘−4𝜔³_𝑘)cos𝜔_𝑘𝜏_𝑘^(𝑗)−12𝜉𝜔²_𝑘sin𝜔_𝑘𝜏_𝑘^(𝑗).  𝑃₁=4𝜉𝜔⁴_𝑘cos𝜔_𝑘𝜏_𝑘^(𝑗)+[(1−𝐴₁𝐴₂)𝜔_𝑘−𝜔⁵_𝑘−4𝜉²𝜔³_𝑘]sin𝜔_𝑘𝜏_𝑘^(𝑗) 𝑃₂=[(1−𝐴₁𝐴₂)𝜔_𝑘−𝜔⁵_𝑘−4𝜉²𝜔³_𝑘]cos𝜔_𝑘𝜏_𝑘^(𝑗)−4𝜉𝜔⁴_𝑘sin𝜔_𝑘𝜏_𝑘^(𝑗),

we derive 𝑑(Re𝜆(𝜏))𝑑𝜏⁻¹_{𝜏=𝜏𝑘}^(𝑗)=Re4𝜆³+12𝜉𝜆²+8𝜉²𝜆𝑒^𝜆𝜏+4𝜆+4𝜉𝜆1−𝐴₁𝐴₂𝑒^−𝜆𝜏−𝜆𝜆⁴+4𝜉𝜆³+4𝜉²𝜆²𝑒^𝜆𝜏_{𝜏=𝜏𝑘(𝑗)}𝐾=Re¹+𝐾₂𝑖𝑃₁+𝑃₂𝑖=𝐾₁𝑃₁−𝐾₂𝑃₂𝑃²₁+𝑃²₂. (2.20) We assume that the following condition holds: (𝐻3)𝐾₁𝑃₁≠𝐾₂𝑃₂. (2.21) According to above analysis and the results of Kuang [29] and Hale [30], we have the following.

Theorem 2.3. If (𝐻1) and (𝐻2) hold, then the equilibrium 𝐸(0,0,0,0) of system (1.2) is asymptotically stable for 𝜏∈[0,𝜏₀). Under the conditions (𝐻1) and (𝐻2), if the condition (𝐻3) holds, then system (1.2) undergoes a Hopf bifurcation at the equilibrium 𝐸(0,0,0,0) when 𝜏=𝜏_𝑘^(𝑗), 𝑘=1,2,3,…,16; 𝑗=0,1,2,….

3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained conditions for Hopf bifurcation to occur when 𝜏=𝜏_𝑘^(𝑗),𝑘=1,2,3,…,16;𝑗=0,1,2,…. In this section, we shall obtain the explicit formulae for determining the direction, stability, and periods of these periodic solutions bifurcating from the equilibrium 𝐸(0,0,0,0) at these critical value of 𝜏, by using techniques from normal form and center manifold theory [31]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcation at the equilibrium 𝐸(0,0,0,0) for 𝜏=𝜏_𝑘^(𝑗),𝑘=1,2,3,…,16;𝑗=0,1,2,…, and then ±𝑖𝜔_𝑘 are corresponding purely imaginary roots of the characteristic equation at the equilibrium 𝐸(0,0,0,0).

For convenience, let 𝑥_𝑖(𝑡)=𝑥_𝑖(𝜏𝑡)(𝑖=1,2,3,4) and 𝜏=𝜏_𝑘^(𝑗)+𝜇, where 𝜏_𝑘^(𝑗) is defined by (2.16) and 𝜇∈𝑅, drop the bar for the simplification of notations, then system (2.2) can be written as an FDE in 𝐶=𝐶([−1,0]),𝑅⁴) as 𝑢̇(𝑡)=𝐿_𝜇𝑢_𝑡+𝐹𝜇,𝑢_𝑡, (3.1) where 𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡))^𝑇∈𝐶 and 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃)=(𝑥₁(𝑡+𝜃),𝑥₂(𝑡+𝜃),𝑥₃(𝑡+𝜃),𝑥₄(𝑡+𝜃))^𝑇∈𝐶, and 𝐿_𝜇∶𝐶→𝑅,𝐹∶𝑅×𝐶→𝑅 are given by 𝐿_𝜇𝜙=𝜏_𝑘^(𝑗)𝜙+𝜇0010000100−2𝜉0000−2𝜉¹(0)𝜙₂(0)𝜙₃(0)𝜙₄(0)+𝜏_𝑘^(𝑗)+𝜇00000000−1𝐴₂𝐴00₁𝜙−100¹()𝜙−1₂()𝜙−1₃()𝜙−1₄(),𝑓=𝜏−1𝜇,𝜙_𝑘^(𝑗)00𝐴+𝜇₂𝜙³₂()𝐴−1+h.o.t.₁𝜙³₁(),−1+h.o.t. (3.2) respectively, where 𝜙(𝜃)=(𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃),𝜙₄(𝜃))^𝑇∈𝐶.

From the discussion in Section 2, we know that if 𝜇=0, then system (3.1) undergoes a Hopf bifurcation at the equilibrium 𝐸(0, 0, 0, 0) and the associated characteristic equation of system (3.1) has a pair of simple imaginary roots ±𝜔_𝑘𝜏_𝑘^(𝑗).

By the representation theorem, there is a matrix function with bounded variation components 𝜂(𝜃,𝜇),𝜃∈[−1,0] such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂𝜃,𝜇𝜙(𝜃)for𝜙∈𝐶. (3.3) In fact, we can choose 𝜂𝜃,𝜇=𝜏_𝑘^(𝑗)𝛿(𝜃)−𝜏+𝜇0010000100−2𝜉0000−2𝜉_𝑘(𝑗)+𝜇00000000−1𝐴₂𝐴00₁𝛿(),−100𝜃+1 (3.4) where 𝛿 is the Dirac delta function.

For 𝜙∈𝐶([−1,0],𝑅⁴), define 𝐴𝜇(𝜃)𝜙=𝑑𝜙𝑑𝜃,−1≤𝜃<0,₀₋₁𝜙(𝑠)𝑓𝑑𝜂𝑠,𝜇,𝜃=0,𝑅𝜙=0,−1≤𝜃<0,𝜇,𝜙,𝜃=0. (3.5) Then, (3.1) is equivalent to the following abstract differential equation: 𝑢̇_𝑡=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡, (3.6) where 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃),𝜃∈[−1,0]. For 𝜓∈𝐶([0,1],(𝑅⁴)^∗), define 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)(],𝑑𝑠,𝑠∈0,1₀₋₁𝑑𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0. (3.7) For 𝜙∈𝐶([−1,0],𝑅⁴) and 𝜓∈𝐶([0,1],(𝑅⁴)^∗), define the following bilinear form: 𝜓,𝜙=𝜓(0)𝜙(0)−⁰₋₁^𝜃_𝜉=0𝜓^𝑇(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (3.8) where 𝜂(𝜃)=𝜂(𝜃,0), and the 𝐴=𝐴(0) and 𝐴^∗ are adjoint operators. By the discussions in Section 2, we know that ±𝑖𝜔_𝑘𝜏_𝑘^(𝑗) are eigenvalues of 𝐴(0), and they are also eigenvalues of 𝐴^∗ corresponding to 𝑖𝜔_𝑘𝜏_𝑘^(𝑗) and −𝑖𝜔_𝑘𝜏_𝑘^(𝑗), respectively. By direct computation, we can obtain 𝑞(𝜃)=1,𝛼,𝛽,𝛾^𝑇𝑒^{𝑖𝜔𝑘𝜏𝑘(𝑗)𝜃},𝑞^∗(𝑠)=𝐷1,𝛼^∗,𝛽^∗,𝛾^∗𝑒^{𝑖𝜔𝑘𝜏𝑘(𝑗)𝑠}, (3.9) where 𝛼=𝑖𝜔_𝑘𝑖𝜔_𝑘+2𝜉+𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}𝐴₂𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)},𝛽=−𝑖𝜔_𝑘,𝛾=𝑖𝜔_𝑘𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}−𝜔²_𝑘𝑖𝜔_𝑘+2𝜉𝐴₂𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)},𝛼^∗=𝑖𝜔_𝑘2𝜉−𝑖𝜔_𝑘−𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}𝐴₁𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)},𝛽^∗=12𝜉−𝑖𝜔_𝑘,𝛾^∗=𝜔²_𝑘𝑖𝜔_𝑘+2𝜉−𝑖𝜔_𝑘𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}𝐴₂𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)},1𝐷=1+𝛼𝛼^∗+𝛽𝛽^∗+𝛾𝛾^∗+𝜏_𝑘^(𝑗)𝛾𝐴₁−𝛼^∗−𝛽𝐴₂𝛼^∗+1𝑒^{𝑖𝜔𝑘𝜏𝑘(𝑗)}. (3.10) Furthermore, ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩1 and ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩0.

Next, we use the same notations as those in Hassard et al. [31] and we first compute the coordinates to describe the center manifold 𝐶₀ at 𝜇=0. Let 𝑢_𝑡 be the solution of (3.1), when 𝜇=0.

Define 𝑧(𝑡)=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜃)=𝑢_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃), (3.11) on the center manifold 𝐶₀, and we have 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃), (3.12) where 𝑊(𝑧(𝑡),𝑧(𝑡),𝜃)=𝑊(𝑧,𝑧)=𝑊₂₀𝑧²2+𝑊₁₁𝑧𝑧+𝑊₀₂𝑧²2+⋯, (3.13) and 𝑧 and 𝑧 are local coordinates for center manifold 𝐶₀ in the direction of 𝑞^∗ and 𝑞^∗. Noting that 𝑊 is also real if 𝑢_𝑡 is real, we consider only real solutions. For solutions 𝑢_𝑡∈𝐶₀ of (3.1), we have 𝑧̇(𝑡)=𝑖𝜔_𝑘𝜏_𝑘^(𝑗)𝑧+𝑞^∗(𝜃)𝑓0,𝑊(𝑧,𝑧,𝜃)+2Re𝑧𝑞(𝜃)=^def𝑖𝜔_𝑘𝜏_𝑘^(𝑗)𝑧+𝑞^∗(0)𝑓₀. (3.14) That is, 𝑧̇(𝑡)=𝑖𝜔_𝑘𝜏_𝑘^(𝑗)𝑧+𝑔(𝑧,𝑧), (3.15) where 𝑔(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (3.16) Hence, we have 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑞^∗(0)𝑓0,𝑢_𝑡=𝜏_𝑘^(𝑗)𝐷1,𝛼^∗,𝛽^∗,𝛾^∗00𝐴₂𝑥³_2𝑡()𝐴−1+h.o.t.₁𝑥³_1𝑡()=−1+h.o.t.𝐷𝜏_𝑘^(𝑗)3𝛽^∗𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}+3𝛾^∗𝛼²𝛼𝑒^{−2𝑖𝜔𝑘𝜏𝑘(𝑗)}𝑧²𝑧+h.o.t, (3.17) and we obtain 𝑔₂₀=𝑔₁₁=𝑔₀₂𝑔=0,₂₁=2𝐷𝜏_𝑘^(𝑗)3𝛽^∗𝑒^{−𝑖𝜔𝑘𝜏𝑘(𝑗)}+3𝛾^∗𝛼²𝛼𝑒^{−2𝑖𝜔𝑘𝜏𝑘(𝑗)}. (3.18)(3.19) Thus, we derive the following values: 𝑐₁(0)=𝑖2𝜔_𝑘𝜏_𝑘^(𝑗)𝑔₂₀𝑔₁₁−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝑐=−Re₁(0)Re𝜆^′𝜏_𝑘^(𝑗),𝛽₂=2Re𝑐₁(0),𝑇₂𝑐=−Im₁(0)+𝜇₂Im𝜆^′𝜏_𝑘^(𝑗)𝜔_𝑘𝜏_𝑘^(𝑗), (3.20) which determine the quantities of bifurcation periodic solutions of (3.1) on the center manifold at the critical value 𝜏=𝜏_𝑘^(𝑗),(𝑘=1,2,3,…,16;𝑗=0,1,2,3,…). Summarizing the results obtained above leads to the following theorem.

Theorem 3.1. The periodic solution is forward (backward) if 𝜇₂>0(𝜇₂<0). The bifurcating periodic solutions on the center manifold are orbitally asymptotically stable with asymptotical phase (unstable) if 𝛽₂<0(𝛽₂>0). The periods of the bifurcating periodic solutions increase (decrease) if𝑇₂>0(𝑇₂<0).

4. Numerical Examples

In this section, we present some numerical results of system (1.2) to verify the analytical predictions obtained in the previous section. Let us consider the following system: 𝑑𝑥₁𝑑𝑡=𝑥₃,𝑑𝑥₂𝑑𝑡=𝑥₄,𝑑𝑥₃𝑑𝑡=−0.8𝑥₃−𝑥₁(𝑡−𝜏)+0.2tanh𝑥₂(𝑡−𝜏),𝑑𝑥₄𝑑𝑡=−0.8𝑥₄−𝑥₂(𝑡−𝜏)+0.4tanh𝑥₁(𝑡−𝜏), (4.1) which has an equilibrium 𝐸₀(0,0,0,0) and satisfies the conditions indicated in Theorem 2.3. The equilibrium 𝐸₀(0,0,0,0) is asymptotically stable for 𝜏=0. Using the software MATLAB (here take 𝑗=0 for example), we derive 𝜔₀≈4.1208,𝜏₀≈0.6801,𝜆^′(𝜏₀)≈0.3307−3.1524𝑖,𝑔₂₁≈−1.4203−4.5518𝑖. Thus by algorithm (3.20) derived in Section 3, we have 𝑐₁(0)≈−0.7102−2.1609𝑖,  𝜇₂≈−2.1476, 𝛽₂≈−1.4202,  𝑇₂≈3.1867. Furthermore, it follows that 𝜇₂>0 and 𝛽₂<0. Thus, the equilibrium 𝐸₀(0,0,0,0) is stable when 𝜏<𝜏₀≈0.6801. Figures 1(a)–1(j) show that the equilibrium 𝐸₀(0,0,0,0) is asymptotically stable when 𝜏=0.65<𝜏₀≈0.6801. It is observed from Figures 1(a)–1(j) that the input voltage of the 𝑖(𝑖=1,2)th neuron and the output of the 𝑗(𝑗=3,4)th neuron converge to their steady states in finite time. If we gradually increase the value of 𝜏 and keep other parameters fixed, when 𝜏 passes through the critical value 𝜏₀≈0.6801, the equilibrium 𝐸₀(0,0,0,0) loses its stability and a Hopf bifurcation occurs, that is, the input voltage of the 𝑖(𝑖=1,2)th neuron the output of the 𝑗(𝑗=3,4)th neuron will keep an oscillary mode near the equilibrium 𝐸₀(0,0,0,0). Due to 𝜇₂>0 and 𝛽₂<0, the direction of the Hopf bifurcation is 𝜏>𝜏₀≈0.6801, and these bifurcating periodic solutions from 𝐸₀(0,0,0,0) at 𝜏₀≈0.6801 are stable. Figures 1(j)–2(d) suggest that Hopf bifurcation occurs from the equilibrium 𝐸₀(0,0,0,0) when 𝜏=0.8>𝜏₀≈0.6801.

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Figure 1 

The time histories and phase trajectories of system (4.1) with 𝜏=0.65<𝜏0≈0.6801 and the initial value (0.05, 0.05, 0.05, 0.025). The equilibrium 𝐸0(0,0,0,0) is asymptotically stable.

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Figure 2 

The time histories and phase trajectories of system (4.1) with 𝜏=0.8>𝜏0≈0.6801 and the initial value (0.05, 0.05, 0.05, 0.025). Hopf bifurcation occurs from the equilibrium 𝐸0(0,0,0,0).

5. Conclusions

In this paper, we have studied the bifurcation natures of a delayed neural network model of two neurons with inertial coupling. Regarding delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the equilibrium 𝐸₀(0,0,0,0), we proposed the conditions to define the parameters for the occurrence of Hopf bifurcation and the oscillatory solutions of the models equations. It is shown that if conditions (𝐻1) and (𝐻2) hold, the equilibrium 𝐸₀(0,0,0,0) of system (1.2) is asymptotically stable for all 𝜏∈[0,𝜏₀). Under conditions (𝐻1) and (𝐻2), if condition (𝐻3) is satisfied, as the delay 𝜏 increases and crosses a threshold value 𝜏_𝑘^(𝑗), the equilibrium loses its stability and the delayed network model of two neurons with inertial coupling enters into a Hopf bifurcation. In addition, using the normal form method and center manifold theorem, explicit formulaes for determining the properties of periodic solutions are worked out. Simulations are included to verify the theoretical findings. The obtained findings are useful in applications of network control.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 60902044), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100) and Doctoral Foundation of Guizhou University of Finance and Economics (2010), Governor Foundation of Guizhou Province (2012), and the Science and Technology Program of Hunan Province (no. 2010FJ6021).