FitzHugh-Nagumo Model Research Articles

Optimal Control of the FitzHugh–Nagumo Stochastic Model with Nonlinear Diffusion

We consider the existence and first order conditions of optimality for a stochastic optimal control problem inspired by the celebrated FitzHugh–Nagumo model, with nonlinear diffusion term, perturbed by a linear multiplicative Brownian-type noise. The main novelty of the present paper relies on the application of the rescaling method which allows us to reduce the original problem to a random optimal one.

Codimension two bifurcations of discrete Bonhoeffer–van der Pol oscillator model

The non-degeneracy is one of the conditions to check for bifurcation analysis. Therefore, we need to compute the critical normal form coefficients to verify the non-degeneracy of the listed bifurcations. Using the critical normal form coefficients method to examine the bifurcation analysis makes it avoid calculating the central manifold and converting the linear part of the map into Jordan form. This is one of the most effective methods in the bifurcation analysis that has not received much attention so far. So in this article, we turn our attention to this method. In this study, the dynamic behaviors of the discrete Bonhoeffer–van der Pol (BVP) model are discussed. It is shown that the BVP model undergoes codimension one (codim-1) bifurcations such as pitchfork, fold, flip (period doubling) and Neimark–Sacker. Besides, codimension two (codim-2) bifurcations including resonance 1:2, 1:3, 1:4 and Chenciner have been achieved. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. Bifurcation curves of the fixed points are drawn with the aid of numerical continuation techniques. Besides, a numerical continuation not only confirms our analytical results but also reveals richer dynamics of the model especially in the higher iteration.

Solving the Magnetocardiography Forward Problem in a Realistic Three-Dimensional Heart-Torso Model

The forward problem in magnetocardiography (MCG) is important for understanding the relationship between the electric activity of the heart and the body surface magnetic field (BSM), and providing insight into the clinical application of MCG. In this paper, we proposed a computational framework based on the finite element method (FEM) to solve the MCG forward problem. For the subject-specific heart-torso geometry established from the medical image, the modified FitzHugh-Nagumo (FHN) equation was used to describe the volumetric myocardial dynamic transmembrane potential (TMP), then the quasi-static Maxwell equations was applied to simulate the propagation of cardiac magnetic field produced by TMP. The two parts were validated on the simplified one-dimensional FHN equation and the source model of the straight wire respectively, in which the analytical solutions exist. Further, under a realistic geometry heart-torso model, the distribution of the body surface magnetic vector field was presented, the component in the direction perpendicular to the body surface ( B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</sub> ) of which was in very good agreement with the actual observations from the same subject on a pulse-pumped Rb atomic magnetometer.

Analytical methods for non-linear fractional Kolmogorov-Petrovskii-Piskunov equation: Soliton solution and operator solution

Kolmogorov-Petrovskii-Piskunov equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry and biology. In this paper, two fractional ex-tended versions of the non-linear Kolmogorov-Petrovskii-Piskunov equation are solved by analytical methods. Firstly, a new and more general fractional derivative is defined and some properties of it are given. Secondly, a solution in the form of operator representation of the non-linear Kolmogorov-Petrovskii-Piskunov equation with the defined fractional derivative is obtained. Finally, some exact solutions including kink-soliton solution and other solutions of the non-linear Kolmogorov-Petrovskii-Piskunov equation with Khalil et al.?s fractional derivative and variable coefficients are obtained. It is shown that the fractional-order affects the propagation velocity of the obtained kink-soliton solution.

TURING INSTABILITY AND PATTERNS OF THE FITZHUGH-NAGUMO MODEL IN SQUARE DOMAIN

<abstract> In this paper, critical conditions of Turing instability for Fitzhugh-Nagumo (FHN) model with diffusion under Neumann boundary conditions are derived. Moreover, different from previous works about the FHN model, we obtain simple bifurcation, double bifurcation, and four-fold bifurcation with stripe pattern, rectangular pattern, spot pattern, square pattern, and highly developed square pattern, respectively. Meanwhile, the theoretical results are applied to two coupled FHN model with diffusion, and the process of the coupling strengths affecting the stability of the model is presented by numerical simulations. </abstract>

Feedback synchronization of FHN cellular neural networks

&lt;p style='text-indent:20px;'&gt;In this work we study the synchronization of ring-structured cellular neural networks modeled by the lattice FitzHugh-Nagumo equations with boundary feedback. Through the uniform estimates of solutions and the analysis of dissipative dynamics, the synchronization of this type neural networks is proved under the condition that the boundary gap signal exceeds the adjustable threshold.&lt;/p&gt;

An effective heterogeneous whole-heart mathematical model of cardiac induction system with heart rate variability.

The main objective of this research work is to develop an effective mathematical model of cardiac conduction system using a heterogeneous whole-heart model. The model is in the form of a system of modified Van der Pol and FitzHugh-Nagumo differential equations capable of describing the heart dynamics. The proposed model extends the range of normal and pathological electrocardiogram (ECG) waveforms that can be generated by the model. The effects of the respiratory sinus arrhythmia (RSA) and the Mayer waves (MW) are both incorporated to modulate the intrinsic frequency of the main oscillator that represents the sinoatrial node. Also, three pathological conditions are incorporated into the model. The heart rate variability (HRV) phenomenon is incorporated into the synthetic ECGs produced which yields valuable information about the cardiovascular health and the performance of the autonomic nervous system. The spectral analysis of the generated RR tachogram delivers power spectrums that resemble those obtained from real recordings. Also, the proposed model generates synthetic ECGs that characteristic the three considered pathological conditions, namely, the tall T wave, the ECG with U wave, and the Wolff-Parkinson-White syndrome. In general, the significance of this research work is in developing a mathematical model that represents the interactions between different pacemakers and allows analysis of cardiac rhythms. To show the effectiveness and the accuracy of the presented model, the results are compared to published results. The proposed model can be a useful tool to study the influences of different physiological conditions on the profile of the ECG. The synthetic ECG signals produced can be used as signal sources for the assessment of diagnostic ECG signal processing devices.

Turing instability in the reaction-diffusion network.

It is an established fact that a positive wave number plays an essential role in Turing instability. However, the impact of a negative wave number on Turing instability remains unclear. Here, we investigate the effect of the weights and nodes on Turing instability in the FitzHugh-Nagumo model, and theoretical results reveal genesis of Turing instability due to a negative wave number through the stability analysis and mean-field method. We obtain the Turing instability region in the continuous media system and provide the relationship between degree and eigenvalue of the network matrix by the Gershgorin circle theorem. Furthermore, the Turing instability condition about nodes and the weights is provided in the network-organized system. Additionally, we found chaotic behavior because of interactions between I Turing instability and II Turing instability. Besides, we apply this above analysis to explaining the mechanism of the signal conduction of the inhibitory neuron. We find a moderate coupling strength and corresponding number of links are necessary to the signal conduction.

Dynamics for stochastic Fitzhugh–Nagumo systems with general multiplicative noise on thin domains

This paper is devoted to studying the dynamics for stochastic Fitzhugh–Nagumo equations with general multiplicative noise on thin domains, where the noise is described by a general stochastic process instead of the usual Wiener process. The measurability of the solution operator is proved, which yields to the measurability of the random attractor. We then show the existence and upper semi‐continuity of these attractors when a family of (n + 1)‐dimensional thin domains degenerates onto an n‐dimensional domain under the topology of p‐times Lebesgue space.

Ultradiscrete bifurcations for one dimensional dynamical systems

Bifurcations of one dimensional dynamical systems are discussed based on some ultradiscrete equations. The ultradiscrete equations are derived from normal forms of one-dimensional nonlinear differential equations, each of which has saddle-node, transcritical, or supercritical pitchfork bifurcations. An additional bifurcation, which is similar to the flip bifurcation, is found in ultradiscrete equations for supercritical pitchfork bifurcations. Dynamical properties of these ultradiscrete bifurcations can be characterized with graphical analysis. As an example of application of our treatment, we focus on an ultradiscrete equation of the FitzHugh–Nagumo model and discuss its dynamical properties.

Radial Basis Function Pseudospectral Method for Solving Standard Fitzhugh-Nagumo Equation

In this article, a pseudospectral approach based on radial basis functions is considered for the solution of the standard Fitzhugh-Nagumo equation. The proposed radial basis function pseudospectral approach is truly mesh free. The standard Fitzhugh-Nagumo equation is approximated into ordinary differential equations with the help of radial kernels. An ODE solver is applied to solve the resultant ODEs. Shape parameter which decides the shape of the radial basis function plays a significant role in the solution. A cross-validation technique which is the extension of the statistical approach leave-one-out-cross-validation is used to find the shape parameter value. The presented method is demonstrated with the help of numerical results which shows a good understanding with the exact solution. The stability of the proposed method is demonstrated with the help of the eigenvalues method numerically.

Spontaneous Activity Induced by Gaussian Noise in the Network-Organized FitzHugh-Nagumo Model.

In this paper, we show some dynamical and biological mechanisms of the short-term memory (the fixed point attractor) through the toggle switch in the FitzHugh-Nagumo model (FN). Firstly, we obtain the bistable conditions, show the effect of Gaussian noise on the toggle switch, and explain the short-term memory's switch mechanism by mean first passage time (MFPT). Then, we obtain a Fokker-Planck equation and illustrate the meaning of the monostable and bistable state in the short-term memory. Furthermore, we study the toggle switch under the interaction of network and noise. Meanwhile, we show that network structure and noise play a vital role in the toggle switch based on network mean first passage time (NMFPT). And we illustrate that the modest clustering coefficient and noise are necessary to maintain memories. Finally, the numerical simulation shows that the analytical results agree with it.

On new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients

AbstractIn this article, Heir‐equations method has been used to investigate nonclassical symmetries and new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients. Different types of variable coefficients with corresponding exact solutions have been considered.

Index theory for traveling waves in reaction diffusion systems with skew gradient structure

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation.

Optimal perturbation iteration method for solving fractional FitzHugh-Nagumo equation

Abstract In this study, a modified fractional form of FitzHugh-Nagumo equation is investigated via a newly developed semi-analytical method. The classical equation has been modified with a new fractional operator and the optimal perturbation iteration algorithms have been adapted accordingly for solving the fractional model. An illustration has been deeply analyzed for different values of physical parameters. Figures and tables are given to show the errors of different order approximations. Obtained results prove the accuracy and effectiveness of the proposed technique.

Preconditioned iterative methods for the convective FitzHugh–Nagumo equations

We consider preconditioned iterative methods for numerical solution of the convective FitzHugh–Nagumo equations. By eliminating the symmetric indefinite matrix derived from the single nonlinear term of the convective FitzHugh–Nagumo equations, a preconditioner that is robust with respect to the involved model parameters is proposed to solve the arising discretized linear system. Spectral properties of the corresponding preconditioned system are analyzed. Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner.

New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation

This work presents the new exact solutions of nonlinear partial differential equations (PDEs). The solutions are acquired by using an effectual approach, the first integral method (FIM). The suggested technique is implemented to obtain the solutions of space-time Kolmogorov Petrovskii Piskunov (KPP) equation and its derived equations, namely, Fitzhugh Nagumo (FHN) equation and Newell-Whitehead (NW) equation. The considered models are significant in biology. The KPP equation describes genetic model for spread of dominant gene through population. The FHN equation is imperative in the study of intercellular trigger waves. Similarly, the NW equation is applied for chemical reactions, Faraday instability, and Rayleigh-Benard convection. The proposed technique FIM can be applied to find the exact solutions of PDEs.

Structural anomalies in brain networks induce dynamical pacemaker effects.

Dynamical effects on healthy brains and brains affected by tumor are investigated via numerical simulations. The brains are modeled as multilayer networks consisting of neuronal oscillators whose connectivities are extracted from Magnetic Resonance Imaging (MRI) data. The numerical results demonstrate that the healthy brain presents chimera-like states where regions with high white matter concentrations in the direction connecting the two hemispheres act as the coherent domain, while the rest of the brain presents incoherent oscillations. To the contrary, in brains with destructed structures, traveling waves are produced initiated at the region where the tumor is located. These areas act as the pacemaker of the waves sweeping across the brain. The numerical simulations are performed using two neuronal models: (a) the FitzHugh-Nagumo model and (b) the leaky integrate-and-fire model. Both models give consistent results regarding the chimera-like oscillations in healthy brains and the pacemaker effect in the tumorous brains. These results are considered a starting point for further investigation in the detection of tumors with small sizes before becoming discernible on MRI recordings as well as in tumor development and evolution.

Kinky periodic pulse and interaction of bell wave with kink pulse wave propagation in nerve fibers and wall motion in liquid crystals

In this article, we apply a direct influential approach namely enhanced modified simple equation (EMSE) method to integrate the Burgers–Huxley (BH) and FitzHugh–Nagumo (FHN) equations which explain nerve pulse propagation in nerve fibers, circuit theory as well as wall motion in liquid crystals. The EMSE technique usually used to obtain pulse wave with time variety constants. In fact, we present a set of explicit traveling wave solutions for BH model and three sets of such solutions for FHN model using the technique. Choosing various particular values of the free parameters, we determine kinky periodic pulse, single solitonic pulse, bell type pulse in the kink waves. In particular, we obtain both bright and dark type kink and bell pulse waves from the solutions and interactions between time varying periodic waves with the solitons. We incorporate some 2D and 3D plots to illustrate the wave characteristics (pulses, directions and amplitudes) of the obtained solutions. The solutions of BH and FHN equations can explain nerve pulse propagation in nerve fibers in the population genetics, circuit theory as well as wall motion in liquid crystals.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations.

A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.