FitzHugh-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.

Deterministic analysis of stochastic FHN systems based on Gaussian decoupling

This paper systematically explores the deterministic characteristics of FitzHugh-Nagumo (FHN) systems’ response to synaptic noise in statistical sense. With the help of Gauss decoupling approximation, two substitute systems of FHN systems’ response to synaptic noise are established by ignoring the cumulants higher than the second order, and the feasibilities of using the substitute systems for response analysis are demonstrated through error analysis. Then, the deterministic analyses of FHN systems with synaptic noise are carried out by means of the two substitute systems. Numerical results show that whether it is the neuronal system or the network system, the synaptic noise can effectively regulate its dynamics, and induce the mode transitions of discharge activity. Based on the class II excitability of FHN neurons, the system activity has a nonlinear dependence on the noise parameter and the other variables of interest. Particularly, the synaptic noise not only makes the neuronal system transition from low-level to high-level narrow-amplitude oscillation by competing with the input signal, but also contributes to the response or detection of this system to weak input signals. This study reveals the deterministic characteristics of FHN systems with synaptic noise, which can provide a reference for the large-scale analysis.

Approximation of translation invariant Koopman operators for coupled non-linear systems.

Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg-de Vries equation, and a coupled FitzHugh-Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction.

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Impact of pulse exposure on chimera state in ensemble of FitzHugh-Nagumo systems.

In this article, we consider the influence of a periodic sequence of Gaussian pulses on a chimera state in a ring of coupled FitzHugh-Nagumo systems. We found that on the way to complete spatial synchronization, one can observe a number of variations of chimera states that are not typical for the parameter range under consideration. For example, the following modes were found: breathing chimera, chimera with intermittency in the incoherent part, traveling chimera with strong intermittency, and others. For comparison, here we also consider the impact of a harmonic influence on the same chimera, and to preserve the generality of the conclusions, we compare the regimes caused by both a purely positive harmonic influence and a positive-negative one.

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.

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Bifurcations to bursting oscillations in memristor-based FitzHugh-Nagumo circuit

To characterize neuronal firing activities and elucidate its bifurcation mechanisms, a memristor-based FitzHugh-Nagumo (FHN) circuit is designed based on the FHN circuit architecture combined with a first-order memristive simulator, and its normalized system with periodic and quasi-periodic bursting oscillations is established. With the change of externally applied excitation, the memristor-based FHN system has the time-varying equilibrium point where the number, position and stability evolve slowly over time. In an evolution period of the time-domain waveform, different fold and/or Hopf bifurcations are triggered, resulting in periodic or quasi-periodic bursting oscillations. To explain the intrinsic bifurcation mechanisms, the fold and Hopf bifurcation sets are built and the transitions between the resting and spiking states are demonstrated, thus identifying the Hopf/fold and Hopf/Hopf bursting oscillations. Finally, based on the circuit simulation model, analog circuit simulations and hardware circuit measurements are developed for the memristor-based FHN circuit to confirm MATLAB numerical simulations. In addition, it is worth noting that the proposed circuit is a simple non-autonomous memristive neuron circuit that is particularly easy to physically implement.

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.

Invariant measures for stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions in weighted space

<abstract><p>The focus of this paper lies in exploring the limiting dynamics of stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions and nonlinear noise in weighted space. To begin, we established the well-posedness of solutions to these stochastic delay lattice systems and subsequently proved the existence and uniqueness of invariant measures.</p></abstract>

Spectral resonance in Fitzhugh-Nagumo neuron system: relation with stochastic resonance and its role in EMG signal characterization.

This paper examines the existence of spectral resonance in the Fitzhugh-Nagumo (FHN) system driven by periodical signal and unbounded noise having Gaussian distribution. It is newly revealed that if the inter-spike-interval (ISI) distribution is accumulated on a single cluster, there exists a dual relationship between stochastic resonance and spectral resonance determined by commonly used metric normalized standard deviation of ISI. Furthermore, the ISI distribution is also concentrated on more than one cluster depending on different driving signal frequency. Consequently, the apparent regular spiking behavior is observed to occur at specified driving signal frequencies which result in a local minimum in entropy function indicating spectral resonance. Therefore it is proposed that occurrence of spectral resonance strongly depends on the shape of ISI distribution tuned by the stochastic and deterministic driving signal parameters and conventional metrics may not indicate entire resonance behavior. Correspondingly, the entropy function is utilized in this paper as an alternative metric to enable the detection of the spectral resonance occurrence. The ISI distribution obtained from the FHN system is investigated to relate the real electromyography (EMG) measurements under different conditions such as myokymia and neuromyotonia. It is seen that ISI distribution observed from myokymic EMG exhibits notably close behavior as in the case of spectral resonance generated by FHN whereas a wider distribution is monitored in the case of neuromyotonia. It is contributed that the modeling and parameterization based on ISI distribution can be potentially used to identify different neural activities.

Turing Patterns in a $$p$$-Adic FitzHugh-Nagumo System on the Unit Ball

Turing Patterns in a $$p$$-Adic FitzHugh-Nagumo System on the Unit Ball

Energy dependence of synchronization mode transitions in the delay-coupled FitzHugh-Nagumo system driven by chaotic activity

Energy dependence of synchronization mode transitions in the delay-coupled FitzHugh-Nagumo system driven by chaotic activity

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.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

Large coupling in a FitzHugh-Nagumo neural network: Quantitative and strong convergence results

We consider a mesoscopic model for a spatially extended FitzHugh-Nagumo neural network and prove that in the regime where short-range interactions dominate, the probability density of the potential throughout the network concentrates into a Dirac distribution whose center of mass solves the classical non-local reaction-diffusion FitzHugh-Nagumo system. In order to refine our comprehension of this regime, we focus on the blow-up profile of this concentration phenomenon. Our main purpose here consists in deriving two quantitative and strong convergence estimates proving that the profile is Gaussian: the first one in a L1 functional framework and the second in a weighted L2 functional setting. We develop original relative entropy techniques to prove the first result whereas our second result relies on propagation of regularity.

Cluster solutions for the FitzHugh-Nagumo system with Neumann boundary conditions

We study cluster solution of the following FitzHugh-Nagumo system:{ε2Δu+f(u)−v=0in Ω,Δv−γv+βu=0in Ω,∂u∂n=∂v∂n=0on ∂Ω, where Ω is a smooth bounded domain in R2 and ε,γ,β are positive parameters. Under appropriate parameter regimes, given any integer k≥2, this system admits a cluster solution uε with k-boundary spikes that accumulate at Q0, a nondegenerate local maximum point of boundary curvature, whenever the term βεγ is sufficiently large or sufficiently small. Moreover, we show that these k-boundary spikes are separated from each other at a distance of O(log⁡(βεγ)γ) as βεγ→∞ or O((βε)12) as βεγ→0, respectively. The latter case proposes a new concentration feature of the activator-inhibitor type reaction-diffusion system which is barely observed in Gierer-Meinhardt system. The mechanism behind the scene is driven by two competing forces: a short range gravitational force with mass positioned at a local maximum point of the curvature and a long range repelling force between spikes whose major part comes from the Green's function of screened Poisson operator on the half space.

Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law

Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law

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.