Nagumo Model Research Articles

Stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems

In this study, we consider the stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems (TSFICSs). It should be noted that the linear growth condition is removed in the investigation of high-order nonlinear TSFICSs, which weakens the conditions compared with previous studies of impulsive systems. In addition, the existing research methods do not work for high-order nonlinear TSFICSs, so novel impulsive differential inequalities with high-order terms are established to investigate the stabilization of high-order nonlinear TSFICSs and develop the classic impulsive differential inequalities for high-order nonlinear situations. Next, sufficient conditions for the stability are obtained based on the new impulsive differential inequalities together with graph theory and the Lyapunov method. Finally, the theoretical results are applied to modified impulsive delayed coupled Fitzhugh–Nagumo models, and numerical simulations are provided to illustrate the practical value of the derived results.

Numerical analysis of coupled dynamical biological networks: Modeling electrical information exchange among nerve cells using finite volume method

An innovative approach to modeling the conduction of electrical impulses via intricate neuronal structures is introduced in this paper, which offers a theoretical and computational examination of parameter estimation in a coupled FitzHugh–Nagumo model. With this goal in mind, we present a finite volume approach to solving the FitzHugh–Nagumo model and check the numerical method’s accuracy against previous findings. To further assess and contrast the efficacy and precision of the model’s outputs, a finite difference formulation is incorporated. To clarify the basic qualitative properties of the inhibitor–activator mechanism intrinsic to the coupled FitzHugh–Nagumo model, the analysis uses dynamical system approaches and linear stability analysis. The results show that the suggested schemes are very accurate, with conditional stability, reaching fourth-order spatial and second-order temporal precision. The results are given in both tabular and graphical forms. According to numerical results, the suggested finite volume method outperforms the finite difference method in accurately and efficiently solving the nonlinear coupled FitzHugh–Nagumo model. Neuronal activity and electrical communication are complex biological systems with a lot of investigated nonlinear differential equations; this research helps us understand more about these topics.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

Dynamical analysis of the FitzHugh–Nagumo model with memristive synapse

Understanding the transition mechanism between different activity patterns in realistic neuron models brings forward fundamental challenges for the dynamical systems theory. The knowledge about the mechanisms can present precious predictions and intuitions for determining fundamental principles of neuron functioning. This paper aims to present a dynamic analysis of the FitzHugh–Nagumo neuron model with a memristive synapse. In this article, the effects of memristive synapse's conductance gain, applied current, and the speed of electrical signal propagation along axons and dendrites on the system's behavior and the neuron's membrane potential are investigated. Analyzes are performed by investigating bifurcation and Lyapunov exponent diagrams versus various model parameters. The physiological role of some parameters and the effect of parameter changes on the model behaviors are discussed. The results demonstrate that when no current is applied to the system, the maximum value of membrane potential is larger, and also, a lower memristive synapse conductance gain leads to a larger maximum value of membrane potential. So, it can be concluded that the neuron in an amplitude death state may be triggered and start firing by decreasing the conductance gain or not applying current.

Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions

Abstract The generalization of the classical FitzHugh–Nagumo model provides a more accurate description of the physical phenomena of neurons by incorporating both nonlinearity and fractional derivatives. In this article, we present a numerical method for solving the time-fractional FitzHugh–Nagumo equation (TFFNE) in the sense of the Atangana–Baleanu fractional derivative using B-spline functions. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while θ \theta -weighted scheme is used to discretize the space directions. The efficiency of the scheme is demonstrated through numerical results and rate of convergence. The convergence order and error norms are studied at different values of the noninteger parameter, temporal directions, and spatial directions. Finally, the effectiveness of the proposed methodology is examined through the analysis of three applications.

A New Two-Step Hybrid Block Method for the FitzHugh–Nagumo Model Equation

This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to evaluate the effectiveness of the proposed ETHBM, three numerical examples are presented and compared with the results obtained through existing methods. The results demonstrate that the proposed ETHBM produces more efficient results than some other numerical approaches in the literature.

A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection–diffusion–reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh–Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques.

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.

An efficient high-order compact approach for spiral wave dynamics by the FHN model

We simulate spiral waves through the famous Fitzhugh–Nagumo (FHN) model of excitable media by employing zero flux and periodic boundary conditions. This model has been solved numerically by modifying a recently developed implicit high-order compact (HOC) finite difference scheme. We also propose a novel approach for implementing the HOC scheme on boundaries of the domain for the periodic boundary conditions. Henceforth, we investigate the effects of the zero flux and periodic boundary conditions on the patterns in excitable media and the resulting dynamics. The introduction of periodic boundary conditions is seen to trigger the breakup of the spiral waves as opposed to the ones found by imposing no-flux boundary conditions. Our computed grid independent solutions are found to be extremely close to well-known numerical results. In the process, we also probe the unconditional stability of the HOC discretization for the FHN model.

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.

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.

Analytical wave solutions of an electronically and biologically important model via two efficient schemes

We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expa function and extended sinh-Gordon equation expansion (EShGEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.

A Multistable Discrete Memristor and Its Application to Discrete-Time FitzHugh–Nagumo Model

This paper presents a multistable discrete memristor that is based on the discretization of a continuous-time model. It has been observed that the discrete memristor model is capable of preserving the characteristics of the continuous memristor model. Furthermore, a three-dimensional memristor discrete-time FitzHugh–Nagumo model is constructed by integrating the discrete memristor into a two-dimensional FitzHugh–Nagumo (FN) neuron model. Subsequently, the dynamic behavior of the proposed neuron model is analyzed through Lyapunov exponents, phase portraits, and bifurcation diagrams. The results show multiple kinds of coexisting hidden attractor behaviors generated by this neuron model. The proposed approach is expected to have significant implications for the design of advanced neural networks and other computational systems, with potential applications in various fields, including robotics, control, and optimization.

Delay-induced self-oscillation excitation in the Fitzhugh–Nagumo model: Regular and chaotic dynamics

The stochastic FitzHugh–Nagumo model with time delayed-feedback is often studied in excitable regime to demonstrate the time-delayed control of coherence resonance. Here, we show that the impact of time-delayed feedback in the FitzHugh–Nagumo neuron is not limited by control of noise-induced oscillation regularity (coherence), but also results in excitation of the regular and chaotic self-oscillatory dynamics in the deterministic model. We demonstrate this numerically by means of simulations, linear stability analysis, the study of Lyapunov exponents and basins of attraction for both positive and negative delayed-feedback strengths. It has been established that one can implement a route to chaos in the explored model, where the intrinsic peculiarities of the Feigenbaum scenario are exhibited. For large time delay, we complement the study of temporal evolution by the interpretation of the dynamics as patterns in virtual space.

Determinants of collective failure in excitable networks.

We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.

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.

Dynamics and control of spiral waves under feedback derived from a moving measuring point

Feedback control of spiral waves by the signal derived from a moving measuring point is investigated in the two-dimensional FitzHugh–Nagumo model, where the measuring location is changed with the motion of the spiral tip according to two schemes. In the first scheme, the direction of the line joining the spiral tip and the moving measuring point at the same time remains unchanged during the feedback control, and the application of the feedback signal can make the spiral tip gradually drift to the non-flux boundary of the system, which eventually causes the disappearance of the whole spiral pattern. It is interesting that the drift unit can always travel in a straight line after a period of time, which is more effective for eliminating spiral waves. The direction angles of the straight drift and the initial drift depend on the parameters for determining the measuring location. The second scheme is designed to has a fixed included angle between the joining line and the tangent line of the tip path at the instantaneous tip position during the control. In response to the feedback control, the system can show a few types of dynamical behaviors, including the transition from the inward-petal to outward-petal tip paths, the shrinkage of the attractor region for outward-petal tip paths with a larger unexcited core, and the transition from meandering to rigidly rotating spirals.

Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer–van der Pol oscillator and Nagumo–Sato piecewise-linear discontinuous one-dimensional map

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map.

Hidden extreme multistability and synchronicity of memristor-coupled non-autonomous memristive Fitzhugh–Nagumo models

Hidden extreme multistability and synchronicity of memristor-coupled non-autonomous memristive Fitzhugh–Nagumo models