Nagumo System Research Articles

Asymptotic behavior of non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by linear mixed white noise

This paper is concerned with the asymptotic behavior of the non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by the linear mixed white noise, which simultaneously contains linear additive noise and multiplicative noise. For the sake of the long-term behavior of the system we considered, we need to utilize a different Ornstein–Uhlenbeck transformation than the general one. First, the existence and uniqueness of pullback random attractors are demonstrated. Then, we prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

Local Exponential Stabilization of Rogers–McCulloch and FitzHugh–Nagumo Equations by the Method of Backstepping

In this article, we study the exponential stabilization of some one-dimensional nonlinear coupled parabolic-ODE systems, namely Rogers–McCulloch and FitzHugh–Nagumo systems, in the interval (0, 1) by boundary feedback. Our goal is to construct an explicit linear feedback control law acting only at the right end of the Dirichlet boundary to establish the local exponential stabilizability of these two different nonlinear systems with a decay e−ωt, where ω ∈ (0, δ] for the FitzHugh–Nagumo system and ω ∈ (0, δ) for the Rogers–McCulloch system and δ is the system parameter that presents in the ODE of both coupled systems. The feedback control law, derived by the backstepping method forces the exponential decay of solution of the closed-loop nonlinear system in both L2(0, 1) and H1(0, 1) norms, respectively, if the initial data is small enough. We also show that the linearized FitzHugh–Nagumo system is not stabilizable with exponential decay e−ωt, where ω > δ.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Symbiosis of an artificial neural network and models of biological neurons: Training and testing

In this paper, we show the possibility of creating and identifying the features of an artificial neural network (ANN), which consists of mathematical models of biological neurons. The FitzHugh–Nagumo (FHN) system is used as a paradigmatic model demonstrating basic neuron activities. First, in order to reveal how biological neurons can be embedded within an ANN, we train the ANN with nonlinear neurons to solve a basic image recognition problem with an MNIST database; next, we describe how FHN systems can be introduced into this trained ANN. After all, we show that an ANN with FHN systems inside can be successfully trained with improved accuracy comparing with first trained ANN and then with inserted FHN systems. This approach opens up great opportunities in terms of the direction of analog neural networks, in which artificial neurons can be replaced by more appropriate biological ones.

The Exponential Attractor for Partially Dissipative Lattice System of Infinite Dimension

With the wide application of fractional differential equations, the study of dynamic systems of fractional differential equations has become one of the hotspots in the past decade. The attractor theory is one of the research directions of infinite dimensional lattice dynamical system. In paper, Eden et al. Systematically introduced the exponential attractor of deterministic autonomous dynamical system, which is a compact positive invariant set with finite fractal dimension and attracting orbits at exponential rate. The exponential attractor has a finite fractal dimension. Because of the exponential attractiveness, the exponential attractor is more stable under perturbation than the global attractor. Fitzhugh Nagumo equation is a nonlinear partial equation that describes the periodic oscillation of neuronal action potential under constant current stimulation above the threshold. For Fitzhugh Nagumo system, there are many reports on attractors, but few reports on exponential attractors. In this paper, the infinite dimensional lattice Fitzhugh Nagumo system [4] is studied. The specific contents are as follows: In part 1, the background, development and research status of FitzHugh Nagumo lattice system and the main work of this paper are introduced. In part 2, the basic concepts, inequalities and lemmas required in the research of FitzHugh Nagumo lattice system are introduced. In part 3, the initial value problem of FitzHugh Nagumo lattice system is studied. For any initial value ϕ(0), there exists a unique solution ϕ(t) of system (9) and family of mapping S(t) can generate continuous operator semigroup {S(t)} t≥0 on ,. Through lemma 3.2,3.3,3.4, it can be proved that the continuous operator semigroup {S(t)} t≥0 satisfies uniform continuity and is asymptotically compact, thus proving the existence of the exponential attractor of the system.

Existence and Stability of Standing Spot Solutions in a Three-Component FitzHugh–Nagumo System in R2

Existence and Stability of Standing Spot Solutions in a Three-Component FitzHugh–Nagumo System in R²

Dynamics of stochastic FitzHugh–Nagumo system on unbounded domains with memory

In this paper, we consider the non-autonomous stochastic FitzHugh–Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all N-dimensional space.

Non-reciprocity induces resonances ina two-field Cahn-Hilliard model.

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive thecorresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Concentration Phenomena in Fitzhugh–Nagumo Equations: A Mesoscopic Approach

We consider a spatially extended mesoscopic FitzHugh–Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh–Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of a Dirac distribution concentrated to an averaged membrane potential.

Intermittent chaotic spiking in the van der Pol–FitzHugh–Nagumo system with inertia

The van der Pol–FitzHugh–Nagumo neuron model with inertia was shown to exhibit a chaotic mixed-mode dynamics composed of large-amplitude spikes separated by an irregular number of small-amplitude chaotic oscillations. In contrast to the standard 2D van der Pol–FitzHugh–Nagumo model driven by noise, the inter-spike intervals distribution displays a complex arrangement of sharp peaks related to the unstable periodic orbits of the chaotic attractor. For many ranges of parameters controlling the excitability of the system, we observe that chaotic mixed-mode states consist of lapses of nearly regular spiking interleaved by others of highly irregular one. We explore here the emergence of these structures and show their correspondence to the intermittent transitions to chaos. In fact, the average residence times in the nearly-periodic firing state, obey the same scaling law – as a function of the control parameter – than the one at the onset of type I intermittency for dynamical systems in the vicinity of a saddle node bifurcation. We hypothesize that this scenario is also present in a variety of slow-fast neuron models characterized by the coexistence of a two-dimensional fast manifold and a one-dimensional slow one.

Instability of small-amplitude periodic waves from fold-Hopf bifurcation

We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction–diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh–Nagumo system, caricature calcium models, and other models in real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149–1199 (2012)] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearization about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh–Nagumo system with an applied current.

Periodically kicked feedforward chains of simple excitable FitzHugh–Nagumo neurons

This article communicates results on regular depolarization cascades in periodically kicked feedforward chains of excitable two-dimensional FitzHugh–Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

The interplay between 1D traveling pulses with oscillatory tails (TPOs) and heterogeneities of the bump type is studied to formulate a generalized three-component FitzHugh–Nagumo system of equations. We present that stationary pulses with oscillatory tails (SPOs) form a “snaky” structure in homogeneous spaces and then TPO branches form a “figure-eight-like stack of isolas” located adjacent to the snaky structure of the SPO. Here, we adopt input resources such as the voltage-difference as a bifurcation parameter. A drift bifurcation from the SPO to TPO is observed by introducing another parameter at which these two solution sheets merge. In contrast to the monotone tail case, in the heterogeneous problem, a nonlocal interaction appears between the TPO and the heterogeneity, creating infinitely many saddle solutions and finitely many stable stationary solutions distributed across the line. The response of the TPO shows various dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interact with the TPO in a complex manner, creating a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. For the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOPs) reveals that HIOPs contain all the asymptotic states after collision to facilitate the prediction of the solution results without solving the PDEs. The reduction method of finite-dimensional system enables the clarification of the detailed mechanism of the transitions from PEN to pinning and pinning to REB based on a dynamical system perspective. Consequently, the basin boundary between two distinct outputs against the heterogeneities yields infinitely many successive reconnections of heteroclinic orbits among those saddles when the strength of heterogeneity increases. This causes the aforementioned subtle dependence of initial condition.

Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays

In this paper, a coupled FitzHugh–Nagumo system with multiple delays is considered. In a first step, the critical point at which a zero root of multiplicity two occurs in the characteristic equation is constructed. In a second step, in order to ensure that all the roots of the characteristic equation except for the double zero root have negative real parts, the zeros of a third and a fourth degree exponential polynomial are studied. Moreover, the critical values where the Bogdanov–Takens bifurcation occurs are derived. By using the normal form theory and reduction on the centre manifold, the truncated normal form is obtained, and throughout the bifurcation diagram, its dynamical behaviours are studied. Finally, a numerical example is given to demonstrate our results.

Well-posedness and dynamics of double time-delayed lattice FitzHugh–Nagumo systems

This paper is concerned with the well-posedness and dynamics of FitzHugh–Nagumo systems with variable delay defined on infinite lattices. We first prove the existence and uniqueness of solutions as well as the existence of an evolution process for this system. We then establish the existence of tempered pullback attractors and invariant measures. Finally, we study the upper semicontinuity of pullback attractors as the delay time tends to zero.

Mathematical Modeling of Autowave Processes in a Cell with Nanostructured Liquid

In this paper, a solution of the mathematical model of an autowave process in a thin nearelectrode layer of a magnetic fluid is found. A modification of the FitzHugh–Nagumo system taken as the basis of the mathematical model made it possible to obtain results similar to a full-scale experiment. As a criterion for the adequacy of the model, we use a visual correspondence to the experiment and the presence in the solution of typical phenomena observed in the autowave process, such as reverbs, pacemakers, bypassing of obstacles, etc.

Dynamical behaviors in the FitzHugh–Nagumo system with a memory trace

In this paper, the dynamical behaviors of the FitzHugh–Nagumo (FHN) system with a memory trace, which has time-fractional derivatives, are investigated. For the case of a classical order, the constant input current can change the stability of the equilibrium point in a single FHN unit, and the equilibrium is unstable in a certain range of the current. A decrease of the order of the time-fractional derivative may lead to a linear reduction of the range and the appearance of a solution of mixed-mode oscillations, which consist of subthreshold small-amplitude oscillation and suprathreshold large-amplitude oscillation. In the parameter space of the input current and the fractional order, the region of existing the mixed-mode oscillation is linearly widened when the fractional order moves toward its small value. If a suprathreshold perturbation is periodically applied, there exist some obvious bands, on which the excited period is locked to the perturbation period according to some rational ratios. As a result, the bands can be narrowed by decreasing the value of fractional order and their location has a slight drift toward the small value of the perturbation period. In addition, the properties of solitary traveling waves and wave train solutions are also studied in the one-dimensional space. It is illustrated that the traveling pulse is wider for a smaller value of fractional order, and its velocity is larger. Further, some relations of wave trains have a great change when the value of the fractional order is changed.

Wong–Zakai approximations of the non-autonomous stochastic FitzHugh–Nagumo system on RN in higher regular spaces

In this paper, we consider the Wong–Zakai approximations of a non-autonomous stochastic FitzHugh–Nagumo system driven by a multiplicative white noise with an arbitrary intensity. The convergence of solutions of the path-wise deterministic system to that of the corresponding stochastic system is established in higher regular spaces by means of a new iteration technique and an optimal multiplier at different stages. Furthermore, we prove that the random attractor of the path-wise deterministic system converges to that of the non-autonomous stochastic FitzHugh–Nagumo system in higher regular spaces when the size of approximation vanishes, with much looser conditions on the nonlinearity.

Asymptotic preserving schemes for the FitzHugh–Nagumo transport equation with strong local interactions

This paper is devoted to the numerical approximation of the spatially extended FitzHugh–Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude of the interactions becomes large, this method provides a consistent discretization of the macroscopic reaction-diffusion FitzHugh–Nagumo system. We carry out some theoretical proofs and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.