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.     

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.     

Derivation of a bidomain model for bundles of myelinated axons

The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHugh–Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of monotone operators.     

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.     

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.     

A note on strong convergence rate in averaging principle for stochastic FitzHugh–Nagumo system with two time-scales

In this note, we establish a strong convergence rate in averaging principle for stochastic FitzHugh–Nagumo system with two time-scales under weaker conditions.     

Construction and analysis of some nonstandard finite difference methods for the FitzHugh–Nagumo equation

In this work, we construct four versions of nonstandard finite difference schemes in order to solve the FitzHugh–Nagumo equation with specified initial and boundary conditions under three different regimes giving rise to three cases. The properties of the methods such as positivity and boundedness are studied. The numerical experiment chosen is quite challenging due to shock-like profiles. The performance of the four methods is compared by computing L₁, L_∞ errors, rate of convergence with respect to time and central processing unit time at given time, T = 0.5. Error estimates have also been studied for the most efficient scheme.     

Analytical and numerical approaches to nerve impulse model of fractional-order

We consider a fractional-order nerve impulse model which is known as FitzHugh–Nagumo (F–N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional-order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave-type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers.     

Frequency sensitivity of sub-excitable systems coupled with different topology

We have investigated numerically the behaviours of identical FitzHugh－Nagumo systems, which are coupled into topologies of regular one-dimensional lattice, small-world network and scale-free network, and driven by white noise and an external signal. We found that when a number of uniform systems are coupled into a network, the system's signal-to-noise ratio remains at a high level in a wider frequency range than in the case of a single oscillator. Different architectures manifest different impact, with the scale-free network being the most remarkable. Results presented here suggest a possible approach to improve the sensitivity of a system to external signals and are helpful for designing communication equipments.     

Stochastic resonance in feedforward acupuncture networks

Effects of noises and some other network properties on the weak signal propagation are studied systematically in feedforward acupuncture networks (FFN) based on FitzHugh–Nagumo neuron model. It is found that noises with medium intensity can enhance signal propagation and this effect can be further increased by the feedforward network structure. Resonant properties in the noisy network can also be altered by several network parameters, such as heterogeneity, synapse features, and feedback connections. These results may also provide a novel potential explanation for the propagation of acupuncture signal.