Existence of Nonoscillatory Solutions of the Discrete FitzHugh-Nagumo Systems

In this work, we are concerned with a system of two functional differential equations of mixed type (with delays and advances), known as the discrete Fitzhugh-Nagumo equations, which arises in the modelling of impulse propagation in a myelinated axon: C dv dt (t) = 1 R (v(t +τ)−2v(t) +v(t −τ)) + f(v(t))−w(t) dw dt = σv(t)−γw(t). (1) In the case γ = σ = 0, this system reduces to a single equation, which is well studied in the literature. In this case it is known that for each set of the equation parameters (within certain constraints), there exists a value of τ (delay) for which the considered equation has a monotone solution v satisfying certain conditions at infinity. The main goal of the present work is to show that for sufficiently small values of the coefficients in the second equation of the system (1), this system has a solution (v,w) whose first component satisfies certain boundary conditions and has similar properties to the ones of v, in the case of a single equation. With this purpose we linearize the original system as t → −∞ and t → ∞ and analyse the corresponding characteristic equations. We study the existence of nonoscillatory solutions, based on the number and nature of the roots of these equations.