EVANS FUNCTIONS AND BIFURCATIONS OF STANDING WAVE SOLUTIONS IN DELAYED SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the following nonlinear singularly perturbed system of integral differential equations ∂u ∂t + f(u) + w

and the scalar integral differential equation singular perturbation parameter ε in mathematical models. In this paper, we will show by using rigorous mathematical analysis that when the ratio of sodium activation and potassium activation reaches a certain value, which is closely related to the threshold above which an action potential is generated, bifurcations of standing wave solutions occur. Consider the following nonlinear singular perturbed system of integral differential equations arising from delayed synaptically coupled neuronal networks and ∂u ∂t + f (u) where u = u(x, t) represents the membrane potential of a neuron at position x and time t in a delayed synaptically coupled neuronal network, w = w(x, t) represents the leaking current. The kernel functions K and W are defined on R. They represent synaptic couplings between neurons. The gain function is the Heaviside step function: H(u − θ) = 0 for all u < θ, H(0) = 1 2 , and H(u − θ) = 1 for all u > θ. The probability density functions ξ and η are defined on (0, ∞). The function ξ represents a statistical distribution of action potential speeds. Additionally, ξ may have a compact support [c 1 , c 2 ], where c 1 and c 2 are positive constants, denoting the lower bound and upper bound of biologically possible speeds, respectively. Moreover, a, b, α, β, ε, θ and Θ are nonnegative or positive constants, representing various biological mechanisms. The functions f = f (u) and g = g (u) are smooth functions. In addition, either f (u) + g(u) = m(u − n) + k(u − l) is a linear function, where k > 0 and m > 0 are positive constants, l and n are real constants; or f (u) + g(u) = u(u − 1)(Du − 1) is a cubic polynomial function, where D > 0 is a positive constant. Biologically, to model the sodium current and the potassium current, a nonlinear function is much better than a linear function because sodium channels and potassium channels are voltage gated channels (in another word, sodium conductance and potassium conductance should be functions of voltage). On the other hand, mathematically, a linear function is much better than a nonlinear function because a linear function is easy to handle and in some sense it is a good approximation of the nonlinear function. We may interpret the constant m as the sodium conductance and the constant n as the sodium reversal potential. Similarly, we may interpret the constant k as the potassium conductance and the constant l as the potassium reversal potential. See [2,3,4,5,7,8,9,10,13,14,15,16,17,18,19,20,21,22,23,24] for the same or very similar model equations.
is a standing wave solution of system (1.1)-(1.2), then ( u(x, t), w(x, t) is also a standing wave solution of (1.1)-(1.2), for any real constant h. There may exist both simple standing wave solutions (which cross only one threshold) and complicated standing wave solutions (which cross both thresholds). Our results show that there exist two complicated standing wave solutions and under additional conditions, there are four simple standing wave solutions. We will highlight complicated standing wave solutions. We call them standing wave solutions because they are special traveling wave solutions where the wave speed is equal to zero. We may also call them stationary solutions or steady state solutions.
Previously, Amari [1], Guo and Chow [6], Pinto and Ermentrout [9] and Zhang [19] have studied the existence, stability/instability of standing wave solutions of some integral differential equations arising from synaptically coupled neuronal networks. However, the existence, stability/instability and bifurcations of standing wave solutions of system (1.1)-(1.2) have been open for a long time. An interesting feature on the stability/instability analysis and the bifurcation analysis is that the eigenvalue problems derived from linearization of the nonlinear system are nonlinear in λ (this is the eigenvalue parameter). This difficulty arises because the system involves two kinds of delays and any of the two delays may cause such difficulty. We are able to overcome the difficulty to find the eigenvalues of the eigenvalue problems by the construction of Evans functions and by studying their properties.
Suppose that either f (u)+g(u) = m(u−n)+k(u−l), for some positive constants k > 0 and m > 0 and for some real constants l and n; or f (u)+g(u) = u(u−1)(Du− 1), for some positive constant D > 0. Suppose that for all x ∈ R, and are not essential. Actually, if they are not equal to one, then we can always rescale the constants a, b, α and β so that the integrals are equal to one. For the existence and stability/instability analysis of traveling wave solutions, we must make the additional assumption for two positive constants C > 0 and ρ > 0. For example, K(x) = ρ 2 exp(−ρ|x|) and W (x) = √ ρ π exp(−ρx 2 ), where ρ > 0 is a parameter, satisfy this condition. For the existence and stability/instability analysis of standing wave solutions, we do not need such assumptions. For non-symmetric kernel functions K and W , the results obtained in this paper may not be true.
Let us find the constant solutions of the system.
and ϕ 2 is a constant solution, such that ϕ 2 > θ and ϕ 2 > Θ, then m( Hence The constant solution ϕ 1 does not exist if θ = Θ. All of them are stable constant solutions.

The Main Goal
In this paper, we will use rigorous mathematical analysis to establish the existence, stability/instability and bifurcations of the standing wave solutions to the nonlinear singularly perturbed system of integral differential equations with ε > 0. First of all, we will obtain explicit standing wave solutions for the nonlinear system. Then, we derive some eigenvalue problems. Then by constructing and making use of some complex analytic functions, called Evans functions, we will accomplish the stability/instability and the bifurcations of the standing wave solutions. Evans functions are complex analytic functions defined in some right half complex plane. Moreover, λ 0 is an eigenvalue of the eigenvalue problem if and only if λ 0 is a zero of the Evans function. First of all, we establish the existence of the standing wave solutions to the nonlinear singularly perturbed system of integral differential equations with ε > 0 and f (u) + g(u) = m(u−n)+k(u−l), where k > 0 and m > 0 are positive constants, l and n are real constants. As mentioned before, the linear function is a good approximation of the nonlinear function.

Existence of the Standing
(2.10) (I) There exist two complicated standing wave solutions to the nonlinear system of integral differential equations (1.1)-(1.2), where Define the positive constants The standing wave solutions satisfy the following boundary conditions ) .
(II) Suppose that the constants satisfy the additional conditions There exist four simple standing wave solutions to the system (1.1)-(1.2) of integral differential equations, where i = 3, 4, 5, 6 and These standing wave solutions satisfy the following boundary conditions ) .
Remark 2.1. If the kernel functions K and W are positive functions, then the conditions (9)- (14) and (17) hold automatically. If one synaptic coupling represents a pure excitation and another denotes a lateral inhibition, then these conditions can be easily satisfied. If one synaptic coupling represents a pure excitation and another denotes a lateral excitation, then these conditions can also be easily satisfied.
Similarly, there exists a real number Z 0 ′′ , such that For simplicity, we assume that the model parameters and the kernel functions are chosen appropriately, such that Z 0

Proof.
Define Standing wave solutions satisfy ∂u ∂t = 0 and ∂w ∂t = 0. Substituting a solution of the ] .
(I) Suppose that the first standing wave solution satisfies the conditions ϕ < θ on (−∞, 0), ϕ(0) = θ and ϕ > θ on (0, ∞); ϕ < Θ on (−∞, Z 0 ), ϕ(Z 0 ) = Θ and ϕ > Θ on (Z 0 , ∞), for some nonnegative constant Z 0 ≥ 0. Then the right hand side of the first equation in the last system becomes Hence we obtain the first standing wave solution ( to the nonlinear system (1.1)-(1.2) of integral differential equations, where ϕ 1 = ϕ 1 (x) is given by The derivative of ϕ 1 = ϕ 1 (x) is given by In particular, we have However, this is only a formal solution. We have to show that it is compatible, namely, we have to verify that the standing wave solution is below and above the threshold θ on (−∞, 0) and (0, ∞), respectively; and it is below and above the threshold Θ on (−∞, Z 0 ) and (Z 0 , ∞), respectively. The following inequalities are equivalent to one another (below, the symbol "<=>" means that either we always take "<" or we always take "=" or we always take ">"): Moreover, the following inequalities are equivalent to each other: Define the following auxiliary functions on R: Then we have Hence, both A(x) and B(x) are increasing functions on R. Therefore, we find that The existence of the second standing wave solution can be proved very similarly. Indeed, suppose that the second standing wave solution satisfies the conditions: ϕ 2 > θ on (−∞, 0), ϕ 2 (0) = θ and ϕ 2 < θ on (0, ∞); ϕ 2 > Θ on (−∞, −Z 0 ), ϕ 2 (−Z 0 ) = Θ and ϕ 2 < Θ on (−Z 0 , ∞), for the same constant Z 0 as in the analysis of ϕ 1 . Then the differential equations become Therefore, we obtain the second standing wave solution ( The derivative of ϕ 2 = ϕ 2 (x) is given by In particular, we have It is easy to check that this standing wave solution also satisfies the prescribed conditions. (II) To obtain the expressions of the standing wave solutions ( , we may simply let b = 0, Z 0 = 0 and β = 0 in the proof of Theorem 2.1, because ( are below the big threshold Θ. To obtain the representations of the standing wave solu- are above the small threshold θ. All other details are the same as those in (I) and they are omitted. The proof of Theorem 2.1 is finished.

The Scalar Integral Differential Equation (1.3)
The standing wave solutions satisfy the following boundary conditions (II) Suppose that the constants satisfy the additional conditions There exist four additional standing wave solutions

L. Zhang
These standing wave solutions satisfy the following boundary conditions Proof. The proof follows from Theorem 2.1.

Remark 2.4.
It is not difficulty to show that for all x ∈ R. Therefore, we obtain These relationships will play important roles in the construction of Evans functions and to establish the stability/instability of the standing wave solutions.
where γ = 1 for the first standing wave solution and γ = −1 for the second standing wave solution.
Both standing wave solutions of equation (1.3) enjoy the estimates where γ = 1 for the first standing wave solution and γ = −1 for the second standing wave solution.

Stability/Instability of the Standing Wave Solutions
In this section, our goal is to establish the stability/instability of the standing wave solutions. First of all, we derive some eigenvalue problems. Then we construct complex analytic functions (namely, the Evans functions) corresponding to the eigenvalue problems. Then we study properties of the Evans functions. Finally, we finish the stability/instability analysis. We will focus on the mathematical analysis of the eigenvalue problems and the Evans functions of the first two standing wave solutions because they cross both thresholds θ and Θ. For the next two standing wave solutions ( do not cross the big threshold Θ. For the last two standing wave solutions ( do not cross the small threshold θ.

Derivation of the Eigenvalue Problems
Recall that all standing wave solutions satisfy Linearizing the nonlinear system of integral differential equations about the standing wave solutions, we obtain the linear system of integral differential equations where γ = 1 for the first standing wave solution ( ϕ 1 , ψ 1 ) and γ = −1 for the second standing wave solution ( is a solution of this system, where λ is a complex number and ψ 1 (x) and ψ 2 (x) are bounded complex functions. Then By canceling out the exponential function exp(λt), we obtain the eigenvalue problems The eigenvalue problems may be written as where L(λ) : [L ∞ (R)] 2 → [L ∞ (R)] 2 is a family of linear operators. As we can easily see, due to the presence of any of the delays, these eigenvalue problems are nonlinear in λ. This new feature makes the eigenvalue problems more difficult to solve. We will construct Evans functions to investigate the eigenvalues of this family of linear operators.
Definition 3.1. If there exists a complex number λ 0 and there exists a nontrivial bounded continuous function ψ 0 defined on R, such that L(λ 0 )ψ 0 = λ 0 ψ 0 , then λ 0 is called an eigenvalue and ψ 0 is called an eigenfunction of the eigenvalue problem.

Derivation of the Evans Functions
Define a real constant γ: γ = 1 for the first standing wave solution and γ = −1 for the second standing wave solution. Note that for the standing wave solutions, we have Now we are ready to solve the eigenvalue problems and construct the Evans functions to study the stability/instability of the standing wave solutions. Define the positive constants Define two complex analytic functions Letting x = 0 and x = γZ 0 , respectively, in the eigenvalue problems, we have For the standing wave solutions ( , we only need the first system to construct the Evans function E 2 (λ, ε), where b = 0, Z 0 = 0 and β = 0. For the standing wave solutions ( ϕ 5 , ψ 5 ) and ( ϕ 6 , ψ 6 ) , we only need the second system to construct the Evans function E 3 (λ, ε), where we must let Z 0 = 0 and K(0) |ϕ ′ (0)| = 0.
(VI) Define the Evans function E = E 1 (λ) for the first two standing wave solutions φ 1 and φ 2 of the scalar integral differential equation (1.3) by for all λ ∈ Ω 0 .
(VII) Define the Evans function E = E 2 (λ) for the next two standing wave solutions φ 3 and φ 4 of the scalar integral differential equation (1.3) by for all λ ∈ Ω 0 .
(VIII) Define the Evans function E = E 3 (λ) for the last two standing wave solutions φ 5 and φ 6 of the scalar integral differential equation (1.3) by for all λ ∈ Ω 0 .
Similarly, the definitions of the Evans functions E i (λ) for the last four standing wave solutions of (1.3) make sense only if ) .
(III) Clearly, the conclusion is correct. The proof of Theorem 3.1 is finished.

Remark 3.2.
We believe that the algebraic multiplicity of any eigenvalue λ 0 of the eigenvalue problem L(λ)ψ = λψ is equal to the order of λ 0 as a zero of the corresponding Evans function E = E 1 (λ, ε). However, it may be very complicated to prove.
Let us find the derivatives of the Evans functions. For the nonlinear singularly perturbed system (1.1)-(1.2), by using Definition 3.2, we find that Moreover, for the scalar integral differential equation (1.3), we have In particular, for λ = 0, we have Let us also find the partial derivatives of the Evans functions E 1 (λ, ε) with respect to ε: For the next four standing wave solutions of the nonlinear system (1.1)-(1.2), we find that For the next four standing wave solutions of the scalar integral differential equation of the system (1.1)-(1.2) by using the conditions Therefore, we find that Theorem 3.2. (I) λ = 0 is a simple eigenvalue of the eigenvalue problem L(λ)ψ = λψ except for ε = ε 1 , that is (II) For any ε ∈ (0, ε 1 ), there exist two positive numbers λ 0 (ε) and λ 1 (ε), such that λ 1 (ε) > λ 0 (ε) > 0, and (III) In the unbounded domain Ω, the Evans functions Proof. (I) Differentiating the standing wave equations with respect to x, we get ] .
From this system, we see that λ = 0 is an eigenvalue and is an eigenfunction of the eigenvalue problem L(λ)ψ = λψ.
(III) For all ε ∈ (ε 1 , ∞) and for all λ ∈ Ω, if Reλ ≥ 0 and λ ̸ = 0, then by using Lemma 3.1, we have the following estimates Therefore, we obtain the desired estimate The proof of Theorem 3.2 is finished.
Moreover, for all complex number λ with λ ̸ = 0 and Reλ ≥ 0, we have Proof. The first half is easy to prove. Let us establish the estimates in the second half. We have the following estimates Therefore, we obtain The proof of Corollary 3 is finished.

Theorem 3.3. There hold the following results
For all ε < ε 1 , there hold For all ε > ε 1 , there hold 2) corresponding to the initial data for all t > 0, where the constant h satisfies the condition , then we say the standing wave solution (ϕ, ψ) is a stable. Otherwise, we say it is unstable.
The Linearized Stability Criterion. (I) If there exists no nonzero eigenvalue to the eigenvalue problem L(λ)ψ = λψ in { λ ∈ C : Reλ ≥ 0 } and if λ = 0 is a simple eigenvalue of the eigenvalue problem, then the standing wave solution is stable.
(II) If there exists a positive eigenvalue to the eigenvalue problem L(λ)ψ = λψ, then the standing wave solution is unstable.
(III) If λ = 0 is not a simple eigenvalue of the eigenvalue problem L(λ)ψ = λψ, then the standing wave solution is unstable.
The proof of the linearized stability criterion may be found in [22]. Proof. For each of the six standing wave solutions (

Bifurcations of the Standing Wave Solutions
, there exists no nonzero eigenvalue to the eigenvalue problem L(λ)ψ = λψ in the region { λ ∈ C : Reλ ≥ 0 } . Moreover, the neutral eigenvalue λ = 0 is simple. By using the linearized stability criterion, we find that the standing wave solutions of the system (1.1)-(1.2) are exponentially stable. The proof of Theorem 4.2 is finished.
It is worth of mentioning that for the scalar integral differential equation, the standing wave solutions are always stable. However, for the nonlinear singularly perturbed system of integral differential equations, even though the parameter ε plays no role in the existence analysis of the standing wave solutions, it does play a very important role in the stability/instability analysis and the bifurcation analysis. The main idea to establish the stability/instability and the bifurcations is to construct Evans functions corresponding to several associated eigenvalue problems to find the eigenvalues.

Open Problems
Under different conditions on the model parameters and functions, there may exist standing pulse solutions rather than standing front solutions. But these problems have not been investigated rigorously.
The results for the nonlinear system are surprisingly interesting in mathematical neuroscience.