Meromorphic solutions in the FitzHugh–Nagumo model

The FitzHugh–Nagumo model is studied in the framework of analytic theory of differential equations. The Nevanlinna theory is used to find all meromorphic solutions of a second-order ordinary differential equation related to the FitzHugh–Nagumo model. As a consequence new exact solutions of the FitzHugh–Nagumo system are obtained in explicit form.     

Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations

This paper studies standing pulse solutions to the FitzHugh–Nagumo equations. Since the reaction terms are coupled in a skew-gradient structure, a standing pulse solution is a homoclinic orbit of a second order Hamiltonian system. In this work, an index theory for the Hamiltonian system is employed to study the stability of standing pulses for the FitzHugh–Nagumo equations. Related results for more general skew-gradient systems are also obtained.     

Resolvent Estimates for Boundary Value Problems on Large Intervals Via the Theory of Discrete Approximations

In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic–parabolic equations. Furthermore, we show that our results apply to the FitzHugh–Nagumo system.     

Two-frequency self-oscillations in a FitzHugh–Nagumo neural network

A new mathematical model of a one-dimensional array of FitzHugh–Nagumo neurons with resistive-inductive coupling between neighboring elements is proposed. The model relies on a chain of diffusively coupled three-dimensional systems of ordinary differential equations. It is shown that any finite number of coexisting stable invariant two-dimensional tori can be obtained in this chain by suitably increasing the number of its elements.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

Stability of Traveling Pulses with Oscillatory Tails in the FitzHugh–Nagumo System

The FitzHugh–Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter \(\varepsilon \) goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Recently, the existence of fast pulses with oscillatory tails was proved for the FitzHugh–Nagumo equations. In this paper, we prove that the fast pulses with oscillatory tails are also nonlinearly stable. Similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with \(\varepsilon \), while the relevant scaling in the oscillatory case is \(\varepsilon ^{2/3}\).     

Singleton sets random attractor for stochastic FitzHugh–Nagumo lattice equations driven by fractional Brownian motions

The paper is devoted to the study of the dynamical behavior of the solutions of stochastic FitzHugh–Nagumo lattice equations, driven by fractional Brownian motions, with Hurst parameter greater than 1/2. Under some usual dissipativity conditions, the system considered here features different dynamics from the same one perturbed by Brownian motion. In our case, the random dynamical system has a unique random equilibrium, which constitutes a singleton sets random attractor.     

Existence and Stability of Traveling Pulses in a Reaction–Diffusion-Mechanics System

In this article, we analyze traveling waves in a reaction–diffusion-mechanics (RDM) system. The system consists of a modified FitzHugh–Nagumo equation, also known as the Aliev–Panfilov model, coupled bidirectionally with an elasticity equation for a deformable medium. In one direction, contraction and expansion of the elastic medium decreases and increases, respectively, the ionic currents and also alters the diffusivity. In the other direction, the dynamics of the R–D components directly influence the deformation of the medium. We demonstrate the existence of traveling waves on the real line using geometric singular perturbation theory. We also establish the linear stability of these traveling waves using the theory of exponential dichotomies.     

Simulating the heart’s electric activity: Numericalmethods for inverse problems

For a two-dimensional modified FitzHugh–Nagumo mathematical model, the inverse problem is considered to find a coefficient of the system of partial differential equations, depending on spatial variables. Additional dynamic measuring of the potential is done throughout the inner boundary of the domain. A numerical way of solving the specified inverse problem is proposed and the results from numerical experiments are presented.     

Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh’ theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrödinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh–Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeros.     

Synchronization of FitzHugh–Nagumo neurons in external electrical stimulation via nonlinear control

Synchronization of FitzHugh–Nagumo neural system under external electrical stimulation via the nonlinear control is investigated in this paper. Firstly, the different dynamical behavior of the nonlinear cable model based on the FitzHugh–Nagumo model responding to various external electrical stimulations is studied. Next, using the result of the analysis, a nonlinear feedback linearization control scheme and an adaptive control strategy are designed to synchronization two neurons. Computer simulations are provided to verify the efficiency of the designed synchronization schemes.     

The global existence and asymptotic stability of solutions for a reaction–diffusion system

This paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples.     

Canonical reduction of self-dual Yang–Mills equations to Fitzhugh–Nagumo equation and exact solutions

The (constrained) canonical reduction of four-dimensional self-dual Yang–Mills theory to two-dimensional Fitzhugh–Nagumo and the real Newell–Whitehead equations are considered. On the other hand, other methods and transformations are developed to obtain exact solutions for the original two-dimensional Fitzhugh–Nagumo and Newell–Whitehead equations. The corresponding gauge potential A_μ and the gauge field strengths F_μν are also obtained. New explicit and exact traveling wave and solitary solutions (for Fitzhugh–Nagumo and Newell–Whitehead equations) are obtained by using an improved sine-cosine method and the Wu’s elimination method with the aid of Mathematica.     

Weak Solutions for Stochastic FitzHugh–Nagumo Equations

This article studies the existence of weak solutions for a stochastic version of the FitzHugh–Nagumo equations. The random elements are introduced through initial values and forcing terms of associated Cauchy problem, which may be white noise in the time. Moreover there is a dependence of a stochastic parameter.     

Complex behavior of polynomial FitzHugh–Nagumo cellular neural network model

In this paper, a cellular neural network (CNN) model of FitzHugh–Nagumo system with two diffusion coefficients is introduced. The dynamical behavior of the polynomial CNN model is investigated using harmonic balance method. Local activity domain and edge of chaos domain of the parameter space were found for the model. Numerical simulations of the CNN dynamics confirm the so-called edge of chaos phenomena and help in the better understanding of genesis and emergence of complexity in FitzHugh–Nagumo system.     

Threshold phenomena for the first initial boundary value problem of the FitzHugh–Nagumo equations

Under biologically reasonable assumptions the threshold phenomena for the first initial boundary value problem of FitzHugh–Nagumo equations is proved.     

Standing Waves Joining with Turing Patterns in FitzHugh–Nagumo Type Systems

An article by Kondo and Asai demonstrated that the pattern formation and change on the skin of tropical fishes can be predicted well by reaction-diffusion models of Turing type. As being observed, a common pattern structure is the rearrangement of stripe pattern, and defect like heteroclinic solution appeared between the patterns with different number of stripes. We consider FitzHugh–Nagumo type reaction-diffusion systems with anisotropic diffusion. Under a sufficient condition in diffusivity, we apply variational arguments to show the existence of standing waves joining with Turing patterns.     

On the bidomain problem with FitzHugh–Nagumo transport

The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.     

Analytical and numerical solutions of the Fitzhugh–Nagumo equation and their multistability behavior

In this paper, we propose an analytical method and a modification of explicit exponential finite difference method (EEFDM) for analytical and numerical solutions of the Fitzhugh–Nagumo (FN) and Newell–Whitehead (NW) equations. The method is improved computationally by using the Padé approximation technique. Furthermore, multistability behavior of traveling wave solutions of the FN and NW equations are examined in presence of external forcing. It is observed that there exist coexisting periodic and quasiperiodic orbits for the FN equation, where as only quasiperiodic orbits is observed in case of NW equation.