Stability and Hopf-bifurcation Analysis of Delayed Bam Neural Network under Dynamic Thresholds

In this paper the dynamics of a three neuron model with self-connection and distributed delay under dynamical threshold is investigated. With the help of topological degree theory and Homotopy invariance principle existence and uniqueness of equilibrium point are established. The conditions for which the Hopf-bifurcation occurs at the equilibrium are obtained for the weak kernel of the distributed delay. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and central manifold theorem. Lastly global bifurcation aspect of such periodic solutions is studied. Some numerical simulations for justifying the theoretical analysis are also presented.


Introduction
In many areas of science, for example biology, population dynamics, neuroscience, economics network with connection delays arise [1][2][3].However in neural networks delays occur in the signal transmission between neurons or electronic model neurons due to finite propagation velocity of action potentials (axonal delay) non negligible time of a signal from a neuron to reach the receiving site of a postsynaptic neuron (synaptic delay) and some finite switching speed.
In some artificial neural network information is stored as stable equilibrium points of the system.Retrieval occurs when the system is initialized within the basin of attraction of one of the equilibria and the network is allowed to stabilize in its steady state [1,4].Delay may render such networks more versatile [5,6].Nevertheless uncontrolled delays may degrade network performance by rendering the equilibria unstable, and makes the retrieval of the corresponding information impossible [7].Thus delay is an important controlled parameter in neural system.Delays may be discrete or continuous in nature.We have considered here a three neuron BAM network with distributed delay.
In 1980, Hopfield proposed a simplified neuron network model in the development of memory in human [1].In this model each neuron is represented by a linear circut consisting of a resistor and a capacitor and is connected to the other neuron via nonlinear sigmoidal activation function.Since then hopfield model have been widely developed and studied both in theory and application including both continuous and discrete time delay.
Since the complexity found in simple models can often be carried over to large scale networks in some way thereby yielding much better understanding of the later from a careful study of the former,most network has focussed on the neural networks where all connection terms have same time delay [8].
X. Liao, S. Guo, C. Li [23] considered a simple delayed three neuron network model and obtained sufficient delay dependent criteria to ensure global assymptotic stability of the euilibrium.They also paid attention to the double Hopf-bifurcation associated with resonance.Das Gupta, Majee, and Roy [24] studied stability,bifurcation and global existence of a Hopf-bifurcating periodic solution for a tri neuron Hopfield type general model.They derived sufficient condition for linear stability, instability and occurrence of Hopf-bifurcation with respect to delay parameter about the trivial euilibrium.They also studied asymptotic stability, orbital stability of Hopf-bifurcating periodic solution for a three neuron network with distributed delay [25].Baldi and Atiya [26] investigated the effect of delays on the dynamics of a n-neuron ring network and discussed its oscillating properties.Recently Wei and Velarde [27] studied stability and other properties of delay induced Hopf-bifurcation for Baldi and Atiya model with three neurons.There are many other research works [28][29][30] on three neuron network.
Bidirectional associate memorial (BAM) neural networks are a type of network with the neurons arrayed in two layers.Netwroks with such a bidirectional structure have practical applications in storing paired patterns or memories and possess the ability of searching the desired patterns via both directions: forward and backward [31][32][33][34][35].A BAM neural netwok can be described by the following system of ordinary differential equations.
Realizing the ubiquitous existences of delay in neural networks Gopalsamy and He [31] incorporated time delays into the model and considered the following system of delay differential equations A diagonally dominant and delay independent criterion for global stability of above model was established [31].Recently Mohammad [35] addressed the exponential stability of above model.More recently Wang and Zou [36] considered a special case of above model where all delays in each layer are identical, and performed local stability and Hopf bifurcation analysis of that.
Liao et al. [8,9] proposed the following two neuron system with distributed delays and no self-connection: and found that Hopf-bifurcation occurred for the weak kernel.However as far as we know, there is few works dealing with self connected delayed neural network systems.
In this paper we have investigated local asymptotic stability; existence, uniqueness of equilibrium, existence and directional stability of Hopf bifurcating periodic solution for a BAM three-neuron network with distributed delay.In this network neurons 1, 2 and neurons 1, 3 are coupled.Each neuron possesses non-linear self feedback under dynamic threshold (Fig. 1).

The paper is structured as follows:
In Section 2 the model is presented and a sufficient condition for existence and uniqueness of equilibrium is given using degree theory.In Section 3 local stability analysis of trivial equilibrium has been done.In Section 4 existence of Hopf-bifurcating periodic solution about origin is studied.In Section 5 direction,period and stability of that bifurcating periodic solution is studied.In Section 6, existence of global Hopfbifurcation has been studied.To verify the theoretical analysis, numerical simulations are demonstrated in Section 7. Finally a conclusion has been drawn in Section 8.
In this paper our aim is to consider a three neuron system with distributed delay and having self connections under dynamic threshold.Such a model can be expressed in following form.
In this model x i (i = 1, 2, 3) denotes the mean soma potential of the neuron i.The nonnegative constant β ′ , corresponds to the strength of neurons to itself.b ′ corresponds to the strength of neuron 2 and 3 on neuron 1. a ′ represents the strength of neuron 1 on neuron 2 and 3. γ = 0 is the measure of the inhibitory influence of the past history.c i (i = 1, 2, 3) > 0 denotes the neuronal threshold.The term x i in the argument of function f represents local positive feedback.In biological literature, such a feedback is known as reverbation, while in the literature of artificial neural network it is known as excitation from other neurons.The weight function F (s) is a non-negative bounded function defined on [0, ∞) to reflect the influence of the past states on the current dynamics.F (s) is called the delay kernel.
Let us assume that: (H1) f ∈ C 4 (R), f (0) = 0, and µf (µ) > 0 for µ = 0; The general form of delay kernel F (s) is as follows: where α is a parameter denoting the rate of decay of the effects of past memories and it is a positive real number.It is also known as exponentially fading memory.n = 0 represents weak kernel whereas n = 1 represents strong kernel.
In this paper we study the effect of weak kernel only, that is In this section we are interested in the existence and uniqueness of the equilibrium of the system (1).The initial condition associated with (1) is of the form We shall use here topological degree and homotopy invariance principle to establish the existence and uniqueness of the equilibrium of the system (1). where

Theorem 1. If (H2) holds and F (s) is of form (3) and there exists positive constants
then system (1) has a unique equilibrium x * .
Proof.The system (1) can be rewritten as Here ) is an equilibrium of system (5) if and only if the following condition holds: where Obviously the solution of h(x) = 0 are equilibrium of system (5).We define a homotopic mapping where and Then it follows that By ( 5) we have Let Then ξ 0 > 0 by (4) and a 0 is a positive constant by (H2).Let It follows from ( 13) that for any By (12) we can obtain that for any λ ∈ (0, 1] Hence it is easy to prove deg(g, U (0), 0) = 1 where g(x) = x is differentiable and strictly monotonic increasing.Therefore by Homotopy invariance principle By topological degree theory we can conclude that equation h(x) = 0 has at least a solution in U (0).That is to say system (5) has at least one equilibrium x * implying system (1) has at least one equilibrium.Now we consider uniqueness of equilibrium x * of system (5).Suppose y * = (y * 1 , y * 2 , y * 3 ) is also an equilibrium of system ( 5).Then we have By ( 6) and ( 16) According to (H2) In view of (4) as . Hence x * = y * .Therefore the system (1) has a unique equilibrium.
Corollary 1.If in (1) we assume f (x) ≡ tanh(x) and F (s) = αe −αs then from Theorem 1 follows that corresponding system has unique equilibrium if

Local stability analysis of trivial equilibrium
In this section we focus on investigating the local stability of equilibrium and existence of Hopf-bifurcation for system (1).
For convenience we set γ = 1 and Then system (1) is equivalent to the following model: From ( 4) it is clear that if γ = 1, then the corresponding system has a unique steady state and as c 1 = c 2 = c 3 = 0 it is obvious that (0, 0, 0) is the unique steady state of (20).The linearization of system (20) about (0, 0, 0) takes the form: where . Now the associated characteristic equation of the linearized system (20) after substituting F (s) = αe −αs , α > 0 is where m 1 , m 2 , m 3 . . .m 6 are given by Now the trivial equilibrium (0, 0, 0) is locally asymptotically stable iff all the roots of equation ( 23) have negative real parts.

Existence of Hopf bifurcating periodic solution
Let λ = iω, ω > 0 be a root of equation (23).Substituting λ = iω in ( 23) and then separating real and imaginary parts we get Eliminating ω from equations in (24) and then substituting values of m 1 , m 2 , m 3 , . . ., m 6 after simplification it is obtained as (25) implies that characteristic equation ( 24) has a purely imaginary root iω if To study the existence of Hopf-bifurcating periodic solution first let us assume 0 < 2ab < 3α.
Therefore characteristic equation ( 23) reduces to Applying Descarte's rule of sign and relation between roots and coefficients of a polynomial it can be concluded that apart from two purely imaginary roots other roots of ( 23) can not have real negative part.Therefore in this case there is no possibility of Hopf-bifurcation.
Now applying Descarte's rule of sign and relation between roots and coefficients of a polynomial it can be concluded that ( 23) has two purely imaginary roots and other four roots have real negative part.
Let at α = α 2 , ±iω 2 is a pair of purely imaginary roots of characteristic equation ( 23) then from (24) we get As in this case value of ω 2 is indeterminate, characteristic equation ( 23) can have no purely imaginary root ±iω 2 at α = α 2 .So there is no possibility of Hopf-bifuircation.Hence we have the following theorem:

Direction, period and stability of Hopf-bifurcating periodic solution
In the previous section we obtained the condition for Hopf Bifurcation to occure at the critical value α = α 1 .In this section we are interested to determine the stability and direction of the periodic solutions bifurcating from the equilibrium (0, 0, 0) following the idea of the Normal form and the centre manifold theory [39].
The nonlinear system (20) can be expanded into first second and higher order terms near origin and then we have the following matrix form. where and a bilinear inner product where L T , K T , ψT are the transpose of L, K, ψ respectively.Since A and A * are adjoint operators, if ±iω 0 are eigen values of A, then they are also eigen values of A * .Let q(θ) be the eigen vector of A 0 associated with the eigenvalue iω 0 =⇒ A(0)q(θ) = iω 0 q(θ).
Now we first compute the coordinates describing the center manifold C 0 at µ = 0. Let y t be the solution of ( 22) at µ = 0.
Thus we can calculate the following quantities Then from conclusion of [39] we obtain the following result: The direction of the Hopf-bifurcating periodic solution is determined by When µ 2 > 0 (< 0), the Hopf-bifurcation is super critical (subcritical) and the bifurcating periodic solution exits for α > α 0 (< α 0 ).
(ii) The period of the bifurcating periodic solution can be estimated by The period increases (decreases) if T 2 > 0 (< 0).
(iii) The stability of the bifurcating periodic solution is determined by When b 2 > 0 (< 0) the bifurcating periodic solution is unstable (stable).

Global Hopf-bifurcation
In this section we shall consider the global existence of Hopf-bifurcating periodic solution that is continuation of the bifurcating periodic solutions as the bifurcation parameter α increases and varies over the interval (α 1 , ∞) assuming that the Hopf-bifurcation is supercritical.This phenomenon will be proved by using the technique of Alexander and Auchmuty [41].
Let P 1 denote the space of all x : R → R 6 which are periodic with period 2π.The space P 1 is a Banach space with the norm Let Λ denote the open interval (0, ∞).Let L(P 1 ) denote the set of all continuous linear maps of P 1 into itself with the induced norm topology.Let F : P 1 × Λ → P 1 be continuous and let us consider the problem of finding the solutions (y, α, ω) ∈ P 1 × Λ × (0, ∞) of the equation If y(t) is a solution of equation (58), and if x(t) = y(ωt), then x(t) is a solution of where x is periodic with period T = 2π/ω.Now let Then ( 22) can be written as (60) The system of integro differential equations ( 22) is equivalent to system (60) of ordinary differential equations [42].Now if we denote the system (60) by then (61) can be rewritten as where A(α) is variational matrix of (60) about trivial equilibrium (0, 0, 0, 0, 0, 0) and R(Z, α) = F 1 (Z, α) − A(α)Z.Now theorem which is used to prove global existence of Hopf-bifurcating prriodic solution is stated under (proof is in [41]).Theorem 5. Let F be a Frechet differentiable map of P 1 × Λ into P 1 .There is a global bifurcation of 2π-periodic solutions of equation (58) from a solution (y * , α 0 , ω 0 ) provided (i) A(α) ∈ L(P 1 ) for α ∈ Λ.The mapping α → A(α) is continuous and 0 is not in the spectrum of A(α 0 ).
(ii) The number of linearly independent solutions in P 1 of ω 0 dW dt = A(α)W is finite and congruent to 2 mod 4.
(iii) There are positive δ and ε such that if λ(α) is in the spectrum of A(α) and Reλ(α We will verify the above sufficient conditions of global Hopf-bifurcation for the system (62).We consider the linearized system and let us suppose that it has a periodic solution of period 2π; let it be The coefficients d k are solutions of the linear system ω 0 ikd k = A(α)d k , k = 0, ±1, ±2, . ... Nontrivial periodic solutions of period 2π exist iff ikω 0 is an eigen value of A(α).That is ikω 0 is a solution of equation (24).From previous discussion it is clear that non-trivial periodic solutions exist only for k = ±1 at α = α 1 , and hence there is only one periodic solution of period 2π for the linearized system (63).It has been shown that Also A(α) is continuous in α.Thus all the conditions of Theorem 5 are satisfied.Therefore there is a global bifurcation of 2π periodic solutions from the point (0, α 1 , ω 1 ).
Thus the global Hopf-bifurcation has been established in a product space of the phase space,parameter space and the frequency space.

Numerical simulation
In this section using the Matlab software we carry out the numerical simulation on a particular form of system (1).Let f (x) ≡ tanh(x);   Here origin is locally asymptotically stable.A periodic solution exists near origin.A periodic solution exists near origin.

Conclusion
In this paper we have analyzed a BAM neural network model composed of three neurons with distributed delay.It is a generalization of the model studied in [8,9].To our knowledge, the stability analysis and bifurcation of a bidirectional associate memorial network with self-connection have not been investigated in literature.This paper is an attempt to do this.As the distributed delay can become a discrete delay when the delay kernel is a delta function at a certain time, a neural network model with distributed delay is more general than that with discrete delay.In this paper in Theorem 1 sufficient condition for existence of unique equilibrium has been studied.In Theorem 2 we have obtained the criteria under which the trivial equilibrium remains locally asymptotically stable.In Theorem 3 condition for existence of Hopf-bifurcating periodic solution about origin has been obtained.In Theorem 4 we have studied the direction, period and stability of such Hopf-bifurcating periodic solution.
Then the global stability of that Hopf-bifurcating periodic solution has been studied.