EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the model equation in synaptically coupled neuronal networks ∂u ∂t +m(u− n) = (α− au) ∫∞ 0 ξ(c) [∫ R K(x− y)H ( u ( y, t− 1 c |x− y| ) − θ ) dy ] dc +(β − bu) ∫∞ 0 η(τ) [∫ R W (x− y)H(u(y, t− τ)− θ)dy ] dτ. In this model equation, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The kernel functions K ≥ 0 and W ≥ 0 represent synaptic couplings between neurons in synaptically coupled neuronal networks. The Heaviside step function H = H(u − θ) represents the gain function and it is defined by H(u − θ) = 0 for all u < θ, H(0) = 1 2 and H(u−θ) = 1 for all u > θ. The functions ξ and η represent probability density functions. The function f(u) ≡ m(u−n) represents the sodium current, where m > 0 is a positive constant and n is a real constant. The constants a ≥ 0, b ≥ 0, α ≥ 0, β ≥ 0 and θ > 0 represent biological mechanisms. This model equation is motivated by previous models in synaptically coupled neuronal networks. We will couple together intermediate value theorem, mean value theorem and many techniques in dynamical systems to prove the existence and uniqueness of a traveling wave front of this model equation. One of the most interesting and difficult parts is the proof of the existence and uniqueness of the wave speed. We will introduce several auxiliary functions and speed index functions to prove the existence and uniqueness of the front and the wave speed.

In this model equation, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The kernel functions K ≥ 0 and W ≥ 0 represent synaptic couplings between neurons in synaptically coupled neuronal networks. The Heaviside step function H = H(u − θ) represents the gain function and it is defined by H(u − θ) = 0 for all u < θ, H(0) = 1 2 and H(u−θ) = 1 for all u > θ. The functions ξ and η represent probability density functions. The function f (u) ≡ m(u−n) represents the sodium current, where m > 0 is a positive constant and n is a real constant. The constants a ≥ 0, b ≥ 0, α ≥ 0, β ≥ 0 and θ > 0 represent biological mechanisms. This model equation is motivated by previous models in synaptically coupled neuronal networks.
We will couple together intermediate value theorem, mean value theorem and many techniques in dynamical systems to prove the existence and uniqueness of a traveling wave front of this model equation. One of the most interesting and difficult parts is the proof of the existence and uniqueness of the wave speed. We will introduce several auxiliary functions and speed index functions to prove the existence and uniqueness of the front and the wave speed.

The Mathematical Model Equations
Consider the model equation in synaptically coupled neuronal networks dτ. c |x − y| denotes the spatial temporal delay. The parameter τ > 0 represents a constant delay. The sodium channels are voltage gated channels, in other words, sodium conductance is a function of the membrane potential. The sodium current is derived by using Ohm's law and should be a nonlinear smooth function of u, just like the nonlinear function f (u) = u(u−1)(u−a) in the Hodgkin-Huxley equations or in the Fitzhugh-Nagumo equations. The linear function f (u) ≡ m(u − n) appeared in this paper stands for a good approximation of the sodium current, where m > 0 is a positive constant and n is a real constant. We may interpret the constant m as the sodium conductance and the constant n as the sodium reversal potential. The constants a ≥ 0, b ≥ 0, α ≥ 0, β ≥ 0 and θ > 0 represent biological mechanisms in synaptically coupled neuronal networks. The integrals represent nonlocal spatial temporal interactions between neurons. This model equation is somehow similar to the equation Here are some examples of synaptic couplings where δ represents the Dirac delta impulse function, M ≥ 1 and N ≥ 1 are positive integers, 0 < ρ < ∞, 0 < ρ 1 < ρ 2 < · · · < ρ M < ∞, 0 < σ < ∞, 0 < σ 1 < σ 2 < · · · < σ N < ∞ are positive constants. Here are some examples of probability density functions where p ≥ 1 and q ≥ 1 are positive integers, 0 < c 1 < c 2 < · · · < c p < ∞ and 0 < τ 1 < τ 2 < · · · < τ q < ∞ are positive constants. The model equation (1.1) contains many important equations in synaptically coupled neuronal networks as particular examples.
(IV) If a = 0 and α = 0, then (1.1) becomes the equation with another kind of delay (VI) If a = 0, b = 0, β = 0, m = 1, n = 0 and ξ(c) = δ(c − ∞), then (1.1) reduces to the equation These model equations occupy a central position in synaptically coupled neuronal networks. For each of these equations, under appropriate assumptions on the model equations, there exists a traveling wave front with a positive wave speed. Please see [1]- [30].

The Main Goal
In this paper, we will couple together intermediate value theorem, mean value theorem and many techniques in dynamical systems to prove the existence and uniqueness of a traveling wave front of the model equation (1.1). We will make use of several auxiliary functions and speed index functions and make use of rigorous mathematical analysis to prove the existence and uniqueness of the traveling wave front, under appropriate conditions on the constants and functions of the model equation (1.1). The proof of the existence and uniqueness of the wave speed is the most interesting and difficult part in this paper.

The Mathematical Assumptions
Suppose that the probability density functions ξ ≥ 0 and η ≥ 0 are defined on Suppose that the synaptic couplings K ≥ 0 and W ≥ 0 are at least piecewise smooth functions defined on R. Suppose that for two positive constants C > 0 and ρ > 0. Define the auxiliary function in
(I) For the model equation we have β = 0, b = 0 and the auxiliary functions become (II) For the model equation we have α = 0, a = 0 and the auxiliary functions become

The Speed Index Function
Let us define the speed index function in (0, c 0 ) by It is easy to see that ϕ = ϕ(µ) is a continuous function of µ even if K and W are piecewise continuous on R. Also it is easy to find that The speed function is a very important concept in mathematical neuroscience and will play very important role in proving the existence and uniqueness of the wave speed of the traveling wave front of (1.1). As before, the speed index function may be simplified for special cases.
(I) For the model equation we have β = 0, b = 0 and Now the speed index function becomes (II) For the model equation we have α = 0, a = 0 and Now the speed index function reduces to

The Main Result
The wave speed µ 0 is the unique solution of the speed equation

The Mathematical Analysis -the Proof of the Main Result
The main goal of this section is to prove the existence and uniqueness of the traveling wave front and the existence and uniqueness of the wave speed.
Suppose that u(x, t) = U (x + µt) is the traveling wave front of (1.1), where µ represents the wave speed and z = x + µt represents a moving coordinate. Then the traveling wave front and the wave speed µ satisfy the integral differential equation First of all, we will derive formal but explicit representation of the traveling wave front of the model equation. Then, we will prove the existence and uniqueness of the wave speed. Finally, we will prove the existence and uniqueness of the traveling wave front. We will use the auxiliary functions and the speed index function introduced earlier to achieve our main goal.

The Representation of the Traveling Wave Front
The main goal of this subsection is to derive the representation of the traveling wave front of (1.1).
Suppose that the wave speed µ satisfies the condition 0 < µ < c 0 . For the first integral in the last equation, making the change of variable and then we find that

Now the above integral differential equation becomes
The traveling wave front is translation invariant. Suppose that the traveling wave front satisfies the conditions U < θ on (−∞, 0), U (0) = θ, U ′ (0) > 0 and U > θ on (0, ∞). Then the traveling wave equation becomes where in the first integral of the right hand side, we let Rewriting this equation as a first order nonhomogeneous linear differential equation, we have dτ.

Now let us use the auxiliary functions introduced in Subsection 1.4 to rewrite this equation in a simple way. We get
Multiplying the differential equation by the integrating factor κ 7 (µ, z) and integrating the result with respect to z, we get Therefore, the traveling wave front may be represented as Note that, by using integration by parts, we get Therefore, we obtain the formal representation of the traveling wave front By using the limits and by using L'Hospital's rule, we find that

The Existence and Uniqueness of the Wave Speed
The main goal of this subsection is to prove the existence and uniqueness of the wave speed µ 0 of the traveling wave front of (1.1).
Letting z = 0 in the solution representation, we have The wave speed µ 0 should be the solution of the speed equation U (0) = θ, that is Now let us use intermediate value theorem to prove the existence of the wave speed µ 0 , which is a solution of this above equation. Recall that κ 1 (µ, z), κ 2 (µ, z) and κ 7 (µ, z) are defined by It is easy to see that ] . Recall that the probability density functions ξ ≥ 0 and η ≥ 0 on (0, ∞), and the synaptic couplings K ≥ 0 and W ≥ 0 on R. By mean value theorem, we find that for all z < 0, Hence, by using the assumptions on the monotonicity of f 1 , for all z < 0, we get Let us derive lower bound and upper bound for the functions κ 1 (µ, z), κ 2 (µ, z) on (0, c 0 )×(−∞, 0) and the integral ∫ 0 −∞ κ 1 (µ, z)dz on (0, c 0 ). By using the definitions, Now, by using these lower bounds and upper bounds for κ 1 (µ, z), κ 2 (µ, z) and ∫ 0 −∞ κ 1 (µ, z)dz, we get the following valuable lower bound As a function of κ 1 (µ, 0), . Furthermore, we have the following important lower bound on (0, c 0 ) which is closely related to the speed index function ϕ(µ) The last inequality is due to assumption (1.9). Thus, we obtain the desired estimates Therefore, by using intermediate value theorem, the existence of the wave speed µ 0 is easy to prove. Now we will use mean value theorem to prove the uniqueness of the wave speed. It is not difficult to use condition (1.2) to find that and then use condition (1.3) to estimate the derivative ∂ ∂µ Additionally, by using condition (1.2) and the monotonicity of f 2 , we have 0). Similarly, we have Additional to all assumptions made in Subsection 1.3, let us make the last assumption Therefore, we obtain for all 0 < µ < c 0 . By using intermediate value theorem and mean value theorem, we know that there exists a unique wave speed µ 0 , such that ϕ(µ 0 ) = n − θ.

The Existence and Uniqueness of the Traveling Wave Front
The main goal of this subsection is to prove the existence and uniqueness of the traveling wave front of the model equation (1.1). Suppose that the kernel functions are nonnegative synaptic couplings defined on R. By the assumption on the monotonicity of f 1 , we have ∂ ∂z 0). Therefore, if we let µ = µ 0 in the formal representation of the traveling wave front, then we find that In particular, we have U ′ (0) > 0. Thus, the traveling wave front u(x, t) = U (x+µ 0 t) satisfies the desired conditions U < θ on (−∞, 0), U (0) = θ, U ′ (0) > 0 and U > θ on (0, ∞). This means that the traveling wave front is a solution of the model equation (1.1). The proof of Theorem (1.1) is completely finished.

Some Remarks
In this paper, assuming that both kernel functions K and W are nonnegative on R, we have proved the existence and uniqueness of the wave speed and the existence and uniqueness of the traveling wave front of the model equation (1.1). Nevertheless, we did not prove the same or similar results when the kernel functions represent lateral inhibitions or lateral excitations in synaptically coupled neuronal networks. We hope that these results are also true. The stability of the traveling wave front is very important/interesting in mathematical neuroscience, see [9,18,[23][24][25][26][27][28]. This is worth of rigorous mathematical analysis.

Summary
We have coupled together intermediate value theorem, mean value theorem and many techniques in dynamical systems to prove the existence and uniqueness of the wave speed µ 0 and the existence and uniqueness of the traveling wave front u(x, t) = U (x+µ 0 t) of the model equation in synaptically coupled neuronal networks The wave speed µ 0 is the unique solution of the speed equation A very important step in establishing the uniqueness of the wave speed µ 0 is to show that In the above model equation, u = u(x, t) stands for the membrane potential of a neuron at position x and time t in synaptically coupled neuronal networks. The kernel functions K ≥ 0 and W ≥ 0 represent synaptic couplings between neurons. The Heaviside step function H represents the gain function and it is defined by H(u − θ) = 0 for all u < θ, H(0) = 1 2 and H(u − θ) = 1 for all u > θ. The functions ξ and η are probability density functions. The function f (u) = m(u − n) represents a sodium current, where m > 0 is a positive constant and n is a real constant. The constants a ≥ 0, b ≥ 0, α ≥ 0, β ≥ 0, θ > 0 represent biological mechanisms. This model equation is motivated by previous models in synaptically coupled neuronal networks, see [1,3,5,7,9,17,18,21,[23][24][25][26][27][28].
Overall, there exists a unique traveling wave front u(x, t) = U (x + µ 0 t) to this model equation, where µ 0 = µ 0 (a, b, m, n, α, β, ξ, η, K, W, θ) represents the wave speed and z = x + µ 0 t represent the moving coordinate. The traveling wave front u(x, t) = U (x + µ 0 t) and the wave speed µ 0 satisfy the traveling wave equation and some boundary conditions at z = ±∞.

Open Problems
Consider the nonlinear singularly perturbed system of integral differential equations in synaptically coupled neuronal networks where u = u(x, t) represents the membrane potential of a neuron at position x and time t, w = w(x, t) represents the leaking current in synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The gain function is given by 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 traveling pulse solutions (u(x, t), w(x, t)) = ( U (x + µ(ε)t), W (x + µ(ε)t) ) and their wave speeds µ = µ(ε) would satisfy if they exist, where z = x + µ(ε)t represents a moving coordinate. However, the existence and stability/instability of fast/slow traveling pulse solutions of the nonlinear singularly perturbed system of integral differential equations have not been proved yet. They are worth of rigorous mathematical investigation.