FitzHugh-Nagumo Model Research Articles

A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model

A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.

A Comparative Analysis of Laplace Residual Power Series and a New Iteration Method for Fitzhugh-Nagumo Equation in the Caputo Operator Framework

A Comparative Analysis of Laplace Residual Power Series and a New Iteration Method for Fitzhugh-Nagumo Equation in the Caputo Operator Framework

Limit cycle of Fitzhugh–Nagumo equation

In the 1950s Hodgkin and Huxley created a model to describe the electrochemical transmission of neuronal signals along the cell membrane, it’s a very important model. However, in view of its complexity, this paper takes the next step to study its simplified model -- Fitzhugh–Nagumo equation. In these equations x is similar to the voltage and represents the excitability of the system, the variable y represents a combination of other forces that tend to return the system to rest. The parameter I is a stimulus parameter that leads to excitation of the system, I is like an applied current. Many studies have been made on the traveling wave solutions and solitary wave solutions of Fitzhugh–Nagumo equation under different parameters, and many people have also studied its numerical solutions. In this paper, the existence and uniqueness of periodic solutions of Fitzhugh-Nagumo equation with different parameters are studied by using geometric analysis and the results of Van der Pol equation.

Chirality and curvature determine the meandering of spirals in multilayer excitable media

We study the emergence of meandering spiral waves in a multilayer structure where two spirals, originating independently, evolve in space–time. The FitzHugh-Nagumo model, enhanced with electromagnetic induction effects, defines the nodal dynamics. The layers are chemically coupled, and the drift of spirals is influenced by interlayer flux coupling. An external magnetic flux force is also necessary to destabilize and unpin spiral rotors. The effects of unique characteristics of spiral patterns, like chirality and tip curvature, are assessed. We find that spirals often drift within a bounded meandering area; however, determining the drift directions and overall meandering regions is complex. The forced spiral generally drifts a greater distance, except for identical co-rotating spirals, which drift synchronously. Spirals with opposite chirality exhibit asynchronous drift and wavefront asymmetry, even with identical tip curvature. Regardless of chirality, non-identical spirals are geometrically aligned and amplified before drifting. Stimulating the more loosely curved spiral ensures both spirals drift.

Impact of time varying interaction: Formation and annihilation of extreme events in dynamical systems.

This study investigates the emergence of extreme events in two different coupled systems: the FitzHugh-Nagumo neuron model and the forced Liénard system, both based on time-varying interactions. The time-varying coupling function between the systems determines the duration and frequency of their interaction. Extreme events in the coupled system arise as a result of the influence of time-varying interactions within various parameter regions. We specifically focus on elucidating how the transition point between extreme events and regular events shifts in response to the duration of interaction time between the systems. By selecting the appropriate interaction time, we can effectively mitigate extreme events, which is highly advantageous for controlling undesired fluctuations in engineering applications. Furthermore, we extend our investigation to networks of oscillators, where the interactions among network elements are also time dependent. The proposed approach for coupled systems holds wide applicability to oscillator networks.

The spectrum and stability of travelling pulses in a coupled FitzHugh-Nagumo equation

The spectrum and stability of travelling pulses in a coupled FitzHugh-Nagumo equation

Dynamics of the FitzHugh-Nagumo equation with numerical methods

In this paper, our research focuses on investigating the behavior of the FitzHugh-Nagumo (FHN) equation dynamics, a mathematical model that describes the spiking behavior of neurons. We begin by presenting the mathematical formulation of the FHN equation as a system of two coupled ordinary differential equations. We then apply methods from the theory of dynamical systems to analyze the behavior of the system, including stability analysis and phase-plane analysis. We also discuss the bifurcations that occur in the system as its parameters are varied. In addition, we present numerical methods for solving the FHN equation, including explicit and implicit methods. We compare the accuracy and efficiency of these methods and discuss their suitability for different types of problems.Our results show that the FHN equation exhibits a rich range of dynamical behaviors, including periodic and chaotic solutions, bistability, and hysteresis.Additionally, FHN equation can be used to simulate the behavior of networks of excitable cells, such as neural circuits or cardiac tissue,and FHN models can also be used to study the effects of drugs or other interventions on the behavior of excitable cells. The numerical methods we present provide a powerful tool for studying the FHN equation and its variants, allowing us to explore the parameter space and predict the behavior of the system under different conditions.

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Scroll Waves and Filaments in Excitable Media of Higher Spatial Dimension.

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of codimension two in an N-dimensional space, and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have nonlocal connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.

Nonmonotonic enhancement of diversity-induced resonance in systems of mobile oscillators.

Diversity is omnipresent in natural and synthetic extended systems, the phenomenon of diversity-induced resonance (DIR), wherein a moderate degree of the diversity can provoke an optimal collective response, provides researchers a brand-new strategy to amplify and utilize the weak signal. As yet the relevant advances focus mostly on the ideal situations where the interactions among elements are uncorrelated with the physical proximity of agents. Such a consideration overlooks interactions mediated by the motion of agents in space. Here, we investigate the signal response of an ensemble of spatial mobile heterogeneous bistable oscillators with two canonical interacting modes: dynamic and preset. The oscillators are considered as mass points and perform random walks in a two-dimensional square plane. Under the dynamic scheme, the oscillators can only interact with other oscillators within a fixed vision radius. For the preset circumstance, the interaction among oscillators occurs only when all of them are in a predefined region at the same moment. We find that the DIR can be obtained in both situations. Additionally, the strength of resonance nonmonotonically rises with respect to the increase of moving speed, and the optimal resonance is acquired by an intermediate magnitude of speed. Finally, we propose reduced equations to guarantee the occurrence of such mobility-optimized DIR on the basis of the fast switching approximation theory and also examine the robustness of such phenomenon through the excitable FitzHugh-Nagumo model and a different spatial motion mechanism. Our results reveal for the first time that the DIR can be optimized by the spatial mobility and thus has promising potential application in the communication of mobile agents.

Application of Gegenbauer Wavelet Method on Initial Value Problem of Fractional Order Fitz Hugh-Nagumo Equation

This paper provides a detailed introduction to the fractional order differentials and the matrix operations of the Gegenbauer wavelet method for solving fractional order differential equations. By comparing the derived trigonometric fractional order differential form solution with the reconstructed Gegenbauer wavelet solution in exploring the initial value problem of the fractional order FitzHugh-Nagumo equation, this paper not only verifies the suitability of the defined fractional order differential form solution for numerical computations but also highlights the high accuracy of the Gegenbauer wavelet method in solving fractional order differential equations. Furthermore, by comparing the exact solution with the reconstructed solution using the Gegenbauer wavelet method, it is concluded that the synchronization rate between neurons in the fractional order FitzHugh-Nagumo model is faster than the corresponding integer-order synchronization rate.

Super-extreme events in a forced Bonhoeffer–van der Pol oscillator

Super-extreme events in a forced Bonhoeffer–van der Pol oscillator

Exponential synchronization of 2D cellular neural networks with boundary feedback

In this work we propose a new mathematical model of 2D cellular neural networks (CNN) in terms of the lattice FitzHugh-Nagumo equations with boundary feedback. The model features discrete Laplacian operators and periodic boundary feedback instead of the interior-clamped or mean-field feedback. We first prove the globally dissipative dynamics of the solutions through a priori uniform estimates. In the main result we show that the 2D cellular neural networks are exponentially synchronized if the computable threshold condition is satisfied by the synaptic gap signals of pairwise boundary cells of the grid and the system parameters with a linear feedback coupling coefficient tunable in applications.

Deep learning-based parameter estimation of stochastic differential equations driven by fractional Brownian motions with measurement noise

This study proposes a general parameter estimation neural network (PENN) to jointly identify the system parameters and the noise parameters of a stochastic differential equation driven by fractional Brownian motion (FBM) from a short sample trajectory. It separately extracts deep features from the trajectory and fuses the information of sampling frequency by a two-stage neural network architecture such that the sample trajectories with variable lengths and sampling times can be properly processed. In addition, by considering additive Gaussian measurement noise in the training stage and utilizing suitable loss functions, the PENN can quantitatively estimate the level of measurement noise and reduce its negative impacts on estimating the governing parameters. Experiments on Fitzhugh–Nagumo model, Duffing oscillator and genetic toggle switch model demonstrate that the PENN can accurately estimate the system parameters, the noise intensity and Hurst exponent of the process noise as well as the signal-to-noise ratio of the measurement noise with high speed.

Generalized FitzHugh–Nagumo model with tristable dynamics: Deterministic and stochastic bifurcations

We propose an extension of the Fitzhugh-Nagumo model, which possesses a regime of three coexisting stable states: resting equilibrium and two stable oscillatory states. Such a regime is absent in the original Fitzhugh-Nagumo model but it is known to exist in higher-dimensional conductance based neuronal models. Thus, the proposed system provides a simpler two-dimensional model with such a property. Using numerical bifurcation analysis as well as Lindsted’s method, we explore parameter regions and bifurcations leading to the tristability. Considering the effects of channel fluctuations as Gaussian white noise, phenomenological bifurcations of the corresponding stochastic system are analyzed using a Fokker–Planck approach. We investigate how the interplay between the system parameters and the noise intensity induces a switching of neural activities between silence, subthreshold, and spiking.

Excitable FitzHugh-Nagumo model with cross-diffusion: long-range activation instabilities

AbstractIn this paper, we shall study a spatially extended version of the FitzHugh-Nagumo model, where one describes the motion of the species through cross-diffusion. The motivation comes from modeling biological species where reciprocal interaction influences spatial movement. We shall focus our analysis on the excitable regime of the system. In this case, we shall see how cross-diffusion terms can destabilize uniform equilibrium, allowing for the formation of close-to-equilibrium patterns; the species are out-of-phase spatially distributed, namely high concentration areas of one species correspond to a low density of the other (cross-Turing patterns). Moreover, depending on the magnitude of the inhibitor’s cross-diffusion, the pattern’s development can proceed in either case of the inhibitor/activator diffusivity ratio being higher or smaller than unity. This allows for spatial segregation of the species in both cases of short-range activation/long-range inhibition or long-range activation/short-range inhibition.

Excitable FitzHugh-Nagumo model with cross-diffusion: close and far-from-equilibrium coherent structures

In this paper, we shall study the formation of stationary patterns for a reaction-diffusion system in which the FitzHugh-Nagumo (FHN) kinetics, in its excitable regime, is coupled to linear cross-diffusion terms. In (Gambino et al. in Excitable Fitzhugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023), we proved that the model supports the emergence of cross-Turing patterns, i.e., close-to-equilibrium structures occurring as an effect of cross-diffusion. Here, we shall construct the cross-Turing patterns close to equilibrium on 1-D and 2-D rectangular domains. Through this analysis, we shall show that the species are out-of-phase spatially distributed and derive the amplitude equations that govern the pattern dynamics close to criticality. Moreover, we shall classify the bifurcation in the parameter space, distinguishing between super-and sub-critical transitions. In the final part of the paper, we shall numerically investigate the impact of the cross-diffusion terms on large-amplitude pulse-like solutions existing outside the cross-Turing regime, showing their emergence also in the case of lateral activation and short-range inhibition.

Mixed-mode oscillations and extreme events in fractional-order Bonhoeffer-van der Pol oscillator.

In the present study, we investigate the dynamic behavior of the fractional-order Bonhoeffer-van der Pol (BVP) oscillator. Previous studies on the integer-order BVP have shown that it exhibits mixed-mode oscillations (MMOs) with respect to the frequency of external forcing. We explore the effect of fractional-order on these MMOs and observe interesting phenomena. For fractional-order q1, we find that as we vary the frequency of external forcing, the system exhibits increasingly small amplitude oscillations. Eventually, as q1 decreases, the MMOs disappear entirely, indicating that lower fractional orders eliminate the presence of MMOs in the BVP oscillator. On the other hand, for the fractional-order q2, we observe more complex MMOs compared to q1. However, we find that the elimination of MMOs occurs with less variation from the integer order 1. Intriguingly, as we change q2, the fractional-order BVP oscillator undergoes a phenomenon known as a crisis, where the attractor expands and extreme events occur. Overall, our study highlights the rich dynamics of the fractional-order BVP oscillator and its ability to display various modes of oscillations and crises as the order is changed.

The FitzHugh–Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results.