Self-synchronization of coupled chaotic FitzHugh–Nagumo systems with unreliable communication links

The problem for self-synchronization of coupled chaotic FitzHugh–Nagumo (FHN) systems with unreliable communication links is investigated in this paper. Different from ordinary coupled chaotic systems, the links between two neurons are long-distance and unreliable. Some special network characteristics, such as nonuniform sampling, transmission-induced delays and data packet dropouts, are analyzed in detail. The sufficient condition in terms of linear matrix inequality (LMI) is obtained to guarantee the asymptotical self-synchronization of coupled chaotic FHN systems with unreliable communication links. Lastly, an illustrative example is provided to show the validity of the proposed sufficient condition.     

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.     

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.     

Hyperbolic Chaos in Systems Based on FitzHugh – Nagumo Model Neurons

In the present paper we consider and study numerically two systems based on model FitzHugh–Nagumo neurons, where in the presence of periodic modulation of parameters it is possible to implement chaotic dynamics on the attractor in the form of a Smale–Williams solenoid in the stroboscopic Poincaré map. In particular, hyperbolic chaos characterized by structural stability occurs in a single neuron supplemented by a time-delay feedback loop with a quadratic nonlinear element.     

Control of space-time chaos in a system of equations of the FitzHugh–Nagumo type

We perform an analytic and numerical study of a system of partial differential equations that describes the propagation of nerve impulses in the heart muscle. We show that, for fixed parameter values, the system has infinitely many distinct stable wave solutions running along the spatial axis at arbitrary velocities and infinitely many distinct modes of space-time chaos, where the bifurcation parameter is the velocity of running wave propagation along the spatial axis, which does not explicitly occur in the original system of equations. We suggest an algorithm for controlling the space-time chaos in the system, which permits one to stabilize any of its unstable periodic running waves.     

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 \(\varepsilon ^{2/3}\).     

Pattern selection in the 2D FitzHugh–Nagumo model

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

     

Periodic solutions in an array of coupled FitzHugh–Nagumo cells

We analyse the dynamics of an array of N²$N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh–Nagumo type and the total system is _ZN×_ZN${\mathbb{Z}}_N\times {\mathbb{Z}}_N$-symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.     

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.     

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.     

Pullback attractors for the non-autonomous FitzHugh–Nagumo system on unbounded domains

The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rⁿ${\mathbb{R}}^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.     

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.     

Convexity of the complete elliptic integrals of the first kind with respect to Hölder means

In this paper, we establish a necessary and sufficient condition for the convexity of the complete elliptic integrals of the first kind with respect to Hölder means.