FitzHugh-Nagumo System Research Articles

Desynchronization in Oscillatory Networks Based on Yakubovich Oscillatority

The desynchronization problems in oscillatory networks is considered. A new desynchronization notion is introduced and desynchronization conditions are provided. The desynchronization notion is formulated in terms of Yakubovich oscillatority of the auxiliary synchronization error system. As an example, the network of diffusively coupled FitzHugh-Nagumo systems with undirected graph is considered. The simple inequality guaranteeing network desynchronization is derived. The simulation results confirm the validity of the obtained analytical results.

Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process

In this paper, we study the long term behavior of non-autonomous fractional FitzHugh-Nagumo systems with random forcing given by an approximation of white noise, called Wong-Zakai approximation. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation fractional FitzHugh-Nagumo systems, and then establish the upper semicontinuity of attractors of system driven by a linear multiplicative Wong-Zakai approximations as random forcing approaches white noise in some sense.

Pullback Random Attractors for Non-Autonomous Stochastic Fractional FitzHugh-Nagumo System

This paper is concerned with the asymptotic behavior of solutions for a class of non-autonomous fractional FitzHugh-Nagumo equations deriven by additive white noise. We first provide some sufficient conditions for the existence and uniqueness of solutions, and then prove the existence and uniqueness of tempered pullback random attractors for the random dynamical system generated by the solutions of considered equations in an appropriate Hilbert space. The proof is based on the uniform estimates and the decomposition of dynamical system.

A myogenic motor pattern in mice lacking myenteric interstitial cells of Cajal explained by a second coupled oscillator network.

The interstitial cells of Cajal associated with the myenteric plexus (ICC-MP) are a network of coupled oscillators in the small intestine that generate rhythmic electrical phase waves leading to corresponding waves of contraction, yet rhythmic action potentials and intercellular calcium waves have been recorded from c-kit-mutant mice that lack the ICC-MP, suggesting that there may be a second pacemaker network. The gap junction blocker carbenoxolone induced a "pinstripe" motor pattern consisting of rhythmic "stripes" of contraction that appeared simultaneously across the intestine with a period of ~4 s. The infinite velocity of these stripes suggested they were generated by a coupled oscillator network, which we call X. In c-kit mutants rhythmic contraction waves with the period of X traveled the length of the intestine, before the induction of the pinstripe pattern by carbenoxolone. Thus X is not the ICC-MP and appears to operate under physiological conditions, a fact that could explain the viability of these mice. Individual stripes consisted of a complex pattern of bands of contraction and distension, and between stripes there could be slide waves and v waves of contraction. We hypothesized that these phenomena result from an interaction between X and the circular muscle that acts as a damped oscillator. A mathematical model of two chains of coupled Fitzhugh-Nagumo systems, representing X and circular muscle, supported this hypothesis. The presence of a second coupled oscillator network in the small intestine underlines the complexity of motor pattern generation in the gut.NEW & NOTEWORTHY Physiological experiments and a mathematical model indicate a coupled oscillator network in the small intestine in addition to the c-kit-expressing myenteric interstitial cells of Cajal. This network interacts with the circular muscle, which itself acts as a system of damped oscillators, to generate physiological contraction waves in c-kit (W) mutant mice.

Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh–Nagumo system

We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (−∂h∂y(x,y),∂h∂x(x,y),f(x,y))T. This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y):=(−∂h∂y(x,y),∂h∂x(x,y))T with a one-dimensional system.We analyze the quasi-homogeneous orbital normal forms for this kind of systems, by introducing a new splitting for quasi-homogeneous three-dimensional vector fields. The obtained results are applied to the nondegenerate Hopf-zero singularity that falls into this kind of systems. Beyond the Hopf-zero normal form, a parametric normal form is obtained, and the analytic expressions for the normal form coefficients are provided. Finally, the results are applied to a case of the three-dimensional Fitzhugh–Nagumo system.

α-stable noise-induced coherence on a spatially extended Fitzhugh–Nagumo system

In this paper, we study the noise-induced temporal and spatial coherence in a spatially extended Fitzhugh–Nagumo (FHN) neuronal system. The system is modeled by a randomly connected small-world network, in which an -stable noise is imposed on the first neuron. We focus on the impact of additive temporally and spatially uncorrelated non-Gaussian -stable noise that determines the interactions amongst coupled FHN neurons. Using numerical simulations, we find that the system presents a typical coherence characteristic on temporal scale, while a decoherence characteristic on spatial scale. Then, the temporal and spatial coherence under the influences of -stable noises are studied, respectively. We find that has a significant impact on the temporal coherence but little significance on the spatial coherence. It shows that in contrast to Gaussian noise, -stable noises lead to poor temporal coherence.

Dynamics of stochastic FitzHugh–Nagumo systems with additive noise on unbounded thin domains

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

Synchronization in ensembles of delay-coupled nonidentical neuronlike oscillators

We study both numerically and experimentally the synchronization in an ensemble of nonidentical neuronlike oscillators described by the FitzHugh–Nagumo equations. The cases of constant values of time-delayed couplings between the oscillators and adaptively controlled values of time-delayed couplings are considered. For the experimental study of the ensemble of neuronlike oscillators, we construct a radio engineering setup, in which the ability to specify both constant values and adaptively tuned values of couplings between the oscillators is implemented. Moreover, it is possible to specify an arbitrary architecture and type of dynamical couplings between oscillators in the setup. By the example of a system of two bidirectionally coupled nonidentical oscillators and a ring consisting of ten unidirectionally coupled nonidentical FitzHugh–Nagumo systems, it is shown that the using of an adaptively controlled time-delayed coupling allows one to achieve the in-phase synchronization of all oscillators in the ensemble even in the case of a large parameter mismatch. The results obtained in the physical experiment are in good agreement with the results of the numerical simulation.

Random attractors of FitzHugh–Nagumo systems driven by colored noise on unbounded domains

We investigate the pathwise asymptotic behavior of the FitzHugh–Nagumo systems defined on unbounded domains driven by nonlinear colored noise. We prove the existence and uniqueness of tempered pullback random attractors of the systems with polynomial diffusion terms. The pullback asymptotic compactness of solutions is obtained by the uniform estimates on the tails of solutions outside a bounded domain. We also examine the limiting behavior of the FitzHugh–Nagumo systems driven by linear colored noise as the correlation time of the colored noise approaches zero. In this respect, we prove that the solutions and the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise.

Effects of Lévy noise on the Fitzhugh–Nagumo model: A perspective on the maximal likely trajectories

This paper considers about the escape paths of the FitzHugh–Nagumo neural system driven by symmetric α-stable Lévy noise (non-Gaussian noise). The existing research has shown that noise can make this system produce a spike pulse, which corresponds to a state transition. To analyze the effects of Lévy noise on the state transition, a novel statistical quantity called maximal likely trajectory, which is obtained by recording the maximizer of the probability density function at every moment, is used to characterize the escape paths of the equilibrium and revel the relationship between state transition and noise intensity or Lévy motion index. The numerical experiments show that for fixed Lévy motion index, the larger noise intensity can promote this neural system to an excitatory state. In addition, the influence of Lévy motion index on the state transition depends on the selection of noise intensity in this neural system. Meanwhile, as a comparison, the case driven by Brownian motion (Gaussian noise) is also taken into account, which shows that in some situations Lévy noise makes the FitzHugh–Nagumo system excited in shorter time. In addition, the maximal likely trajectory provides us with a new perspective to show the existence of a separatrix in the stochastic setting of the FitzHugh–Nagumo model and also depict the rough shape of the middle part of this separatrix.

Reversible signal transmission in an active mechanical metamaterial.

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse-grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the time scale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.

Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise

Abstract In this paper, the dynamical behavior of simplified FitzHugh-Nagumo (FHN) neuron system under the co-excitation of Levy noise and Gaussian white noise are studied. Consideration from two aspects: the mean first-passage time (MFPT) and the probability density function (PDF) of the first-passage time (FPT). Using Janicki–Weron algorithm to generate Levy noise, and through the fourth-order Runge–Kutta algorithm to simulate the system response, the FPT of the 2 × 104 response tracks are calculated, and then the MFPT and the PDF are obtained. Finally, the effects of the multiplicative Gaussian noise and additive Levy noise on the MFPT and PDF of the FPT are discussed. In addition, it is found that the noise enhanced stability (NES) and resonance activation (RA) phenomena in the system.

Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor

In this paper, we consider a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. We construct continuous spatially heterogeneous steady states near, as well as far from, constant steady states and show that they are all unstable. In addition, we construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense defined by Weinberger.The results are quite different from those for classical reaction-diffusion systems where all species diffuse.

Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains

This article is concerned with the asymptotic behavior of solutions of a class of non-autonomous stochastic Fitzhugh-Nagumo systems driven by multiplicative white noise on unbounded thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the systems defined on (n+1)−dimensional unbounded narrow domains. We also establish the upper semicontinuity of these attractors when a family of (n+1)−dimensional unbounded thin domains collapses onto an n−dimensional unbounded domain. A cut-off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions.

Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.

We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical path-following methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.

Invariant algebraic surfaces of the FitzHugh-Nagumo system

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of the FitzHugh-Nagumo system by the called characteristic curves method which involves the theory of weight homogeneous polynomials. It has been ever widely used to give a complete classification of Darboux polynomials of some systems. However, it does not work for the FitzHugh-Nagumo system. Thus, we introduce an “assistant” system and classify its invariant algebraic surfaces finally.

Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol–Duffing oscillators and the neuron-like FitzHugh–Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.

Minimal lamellar structures in a periodic FitzHugh–Nagumo system

A singular limit of a FitzHugh–Nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions. We study the stationary lamellar set and give a criterion to select out the one with the lowest energy. Such an optimal structure is called a minimal lamella. While the empty set or the full torus is a global minimizer for appropriate parameter regimes, the minimal lamellae beat both in other circumstances. The concept of minimal lamella points out that a preferred 1D mesh size is universal.

Fronts in subdiffusive FitzHugh–Nagumo systems

Front solutions of subdiffusive FitzHugh–Nagumo equations are studied for a piece-wise linear nonlinearity. Multiple solutions of the problem and the dependence of the propagation velocity on the parameters are discussed.

Solitary states in multiplex networks owing to competing interactions.

Recent researches in network science demonstrate the coexistence of different types of interactions among the individuals within the same system. A wide range of situations appear in ecological and neuronal systems that incorporate positive and negative interactions. Also, there are numerous examples of systems that are best represented by the multiplex configuration. The present article investigates a possible scenario for the emergence of a newly observed remarkable phenomenon named as solitary state in coupled dynamical units in which one or a few units split off and behave differently from the other units. For this, we consider dynamical systems connected through a multiplex architecture in the presence of both positive and negative couplings. We explore our findings through analysis of the paradigmatic FitzHugh-Nagumo system in both equilibrium and periodic regimes on the top of a multiplex network having positive inter-layer and negative intra-layer interactions. We further substantiate our proposition using a periodic Lorenz system with the same scheme and show that an opposite scheme of competitive interactions may also work for the Lorenz system in the chaotic regime.