FitzHugh-Nagumo System Research Articles

Time-delayed feedback control of coherence resonance chimeras.

Using the model of a FitzHugh-Nagumo system in the excitable regime, we investigate the influence of time-delayed feedback on noise-induced chimera states in a network with nonlocal coupling, i.e., coherence resonance chimeras. It is shown that time-delayed feedback allows for the control of the range of parameter values where these chimera states occur. Moreover, for the feedback delay close to the intrinsic period of the system, we find a novel regime which we call period-two coherence resonance chimera.

Coherence resonance in a network of FitzHugh-Nagumo systems: Interplay of noise, time-delay, and topology.

We systematically investigate the phenomena of coherence resonance in time-delay coupled networks of FitzHugh-Nagumo elements in the excitable regime. Using numerical simulations, we examine the interplay of noise, time-delayed coupling, and network topology in the generation of coherence resonance. In the deterministic case, we show that the delay-induced dynamics is independent of the number of nearest neighbors and the system size. In the presence of noise, we demonstrate the possibility of controlling coherence resonance by varying the time-delay and the number of nearest neighbors. For a locally coupled ring, we show that the time-delay weakens coherence resonance. For nonlocal coupling with appropriate time-delays, both enhancement and weakening of coherence resonance are possible.

Fokas’s Unified Transform Method for linear systems

We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.

Analytical properties of the perturbed FitzHugh–Nagumo model

Analytical properties of the well-known FitzHugh–Nagumo model are studied. It is shown that the standard FitzHugh–Nagumo model does not pass the Painlevé test in the general case and does not have any meromorphic solutions because there are not any expansions of the general solution in the Laurent series. We demonstrate that the introduction of a nonlinear perturbation into the standard system of equations does not lead to the Painlevé property as well. However, in this case there are expansions of the general solution of the system of equations in the Laurent series for some values of parameters. This allows us to look for some exact solutions of the system of the perturbed FitzHugh–Nagumo model. We find some exact solutions of the perturbed FitzHugh–Nagumo system of equations in the form of kinks. These exact solutions can be used for testing numerical simulations of the system of equations corresponding to the FitzHugh–Nagumo model.

An autowave model of the bifurcation behavior of transformed cells in response to polysaccharide

Isolated Helianthus tuberosus L. polysaccharide (HTLP) with a molecular weight of 1–2 MDa has a wide range of biological activities, including the ability to switch transformed cells to an apoptotic pathway. A qualitative explanation for this effect has previously been proposed on the basis of the interactions among components of intracellular signaling pathways associated with the Dectin-1, TLR-6, and TNFR1 receptors and intracellular factors involved in regulating apoptosis. In this work, an autowave model is proposed to explain the switch between alternative modes of cell behavior in response to polysaccharide. The model is based on the modified FitzHugh–Nagumo system of equations. The parameters of the model were estimated numerically.

Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system.

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

On the propagation of a low frequency excitation in a perturbed FitzHugh–Nagumo system: Simulation and experiments

In this paper, we first report the response of an elementary FitzHugh–Nagumo electronic circuit excited by a low frequency sine wave and perturbed by an additive high frequency deterministic perturbation. This preliminary study constitutes a reference to analyze the collective behavior of a chain of 45 coupled elementary cells. In particular, we focus on the propagation of a low frequency sine wave which is only applied on the first fifteen cells of the lattice. It is shown that a high frequency sine wave which perturbs the whole network can enable or disable the propagation of this low frequency signal. By reducing the strength of the intercellular coupling below a critical value, we also establish that the propagation fails whatever the amplitude of the perturbation. Finally, by adding a stochastic perturbation to the high frequency deterministic perturbation, we numerically and experimentally investigate their combined effects on the propagation of the low frequency component. Numerical and experimental results reveal that, under certain conditions, noise can assist the propagation of the low frequency excitation.

Hopf Bifurcation in an Oscillatory-Excitable Reaction–Diffusion Model with Spatial Heterogeneity

We focus on the qualitative analysis of a reaction–diffusion model with spatial heterogeneity. The system is a generalization of the well-known FitzHugh–Nagumo system, in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of a Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.

A study on limit cycle and non-homoclinic orbits for FitzHugh-Nagumo System

A study on limit cycle and non-homoclinic orbits for FitzHugh-Nagumo System

Chimera patterns in two-dimensional networks of coupled neurons.

We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion.

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems

A game control problems of the Schlogl and FitzHugh-Nagumo equations are considered. The problems are investigated both from the viewpoint of the first player (the partner) and of the second player (the opponent). For both players, their own procedures for forming feedback controls are specified.

Pullback attractors of FitzHugh-Nagumo system on the time-varying domains

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

An Invariant Winding Number for the FitzHugh--Nagumo System with Applications to Cardiac Dynamics

The FitzHugh--Nagumo (FHN) system of PDEs is a generic model for excitable media, often used to build a qualitative understanding of electrophysiological phenomena. A well-characterized traveling pulse solution to FHN serves as a model for action potentials in cardiac tissue and other contexts. The stability of the traveling pulse has been well studied, but the more global problem of predicting when an arbitrary initial condition will converge to the uniform rest solution and when it will converge to the traveling pulse remains unsolved. Here we prove the existence of an invariant winding number in an asymptotic limit of the FHN system called the singular FHN system (SFHN) that provides a crucial step toward a global convergence result. This result is valid for a relatively general class of nonlinearities. In addition, we provide evidence that our SFHN results extend with limitations to FHN and outline conditions under which the SFHN approximation fails. The invariant winding number provides explanations for several observations of physiological relevance. For example, it explains the requirements on stimulus protocols that allow the formation and elimination of reentrant rhythms in cardiac tissue.

A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity

Chen, Kung and Morita [ 5 ] studied a variational problem corresponding to the FitzHugh-Nagumo type reaction-diffusion system (FHN type RD system), and they proved the existence of a heteroclinic solution to the system. Motivated by [ 5 ], we consider a variational problem corresponding to FHN type RD system which involves heterogeneity. We prove the existence of a heteroclinic solution to the problem under certain conditions on the heterogeneity. Moreover, we give some information about the location of the transitions.

REGULARITY OF PULLBACK ATTRACTORS AND RANDOM EQUILIBRIUM FOR NON-AUTONOMOUS STOCHASTIC FITZHUGH-NAGUMO SYSTEM ON UNBOUNDED DOMAINS

This paper is concerned with the stochastic Fitzhugh-Nagumo system with non-autonomous terms as well as Wiener type multiplicative noises. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, the existences and upper semi-continuity of pullback attractors are proved for the generated random cocycle in Ll(RN)×L2(RN) for any l ∈ (2,p]. The asymptotic compactness of the first component of the system in Lp(RN) is proved by a new asymptotic a priori estimate technique, by which the plus or minus sign of the nonlinearity at large values is not required. Moreover, the condition on the existence of the unique random fixed point is obtained, in which case the influence of physical parameters on the attractors is analysed.

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems in weighted spaces

In this paper, we first consider the existence of random attractors for the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise in a weighted space $l_{\rho }^{2} \times l_{\rho }^{2} $ ; then we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Autowave self-organization in heterogeneous natural–anthropogenic ecosystems

This article considers the spatio–temporal model of natural–anthropegenic ecosystems as a conjugated active media that takes the heterogeneity of anthropogenic and natural factors into account. The approach aims to identify the threshold values of the control parameters. The theoretical basis of the system analysis of the sustainability of the ecosystems is synergistic data on autowave self-organization in active media. The mathematical model is based on the modified FitzHugh–Nagumo system of equations.

Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.