A family of approximate inertial manifolds for a Van der Pol/FitzHugh–Nagumo perturbation problem

We investigate the stability of a family of approximate inertial manifolds (AIMs) obtained from an ODE containing a perturbation parameter. For two choices of the parameter, the dynamics associated with the equations are already well known: in one case, we have a Van der Pol equation, while in the other setting we obtain a FitzHugh–Nagumo equation. Recently, it has been shown that (a modified version of) each equation admits a sequence of AIMs which converges to the inertial manifold. We show that our model admits a family of AIMs depending on the perturbation parameter. We then investigate the stability of the family of AIMs as the perturbation parameter approaches two different vanishing coefficient limits. These results are intended to shed insight into the continuity properties of inertial manifolds.     

Nagumo型方程解的blow-up

本文研究Nagumo型方程的初边值问题解的blow-up问题,得到了一些在非负非恒零初始条件下,使问题的古典解blow-up的充分条件.     

Computational modeling of cardiac electrophysiology: A novel finite element approach

The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.     

Spatial heterogeneity enhances and modulates excitability in a mathematical model of the myometrium

The muscular layer of the uterus (myometrium) undergoes profound changes in global excitability prior to parturition. Here, a mathematical model of the myocyte network is developed to investigate the hypothesis that spatial heterogeneity is essential to the transition from local to global excitation which the myometrium undergoes just prior to birth. Each myometrial smooth muscle cell is represented by an element with FitzHugh–Nagumo dynamics. The cells are coupled through resistors that represent gap junctions. Spatial heterogeneity is introduced by means of stochastic variation in coupling strengths, with parameters derived from physiological data. Numerical simulations indicate that even modest increases in the heterogeneity of the system can amplify the ability of locally applied stimuli to elicit global excitation. Moreover, in networks driven by a pacemaker cell, global oscillations of excitation are impeded in fully connected and strongly coupled networks. The ability of a locally stimulated cell or pacemaker cell to excite the network is shown to be strongly dependent on the local spatial correlation structure of the couplings. In summary, spatial heterogeneity is a key factor in enhancing and modulating global excitability.     

An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.     

Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.     

Realistic model of compact VLSI FitzHugh–Nagumo oscillators

In this article, we present a compact analogue VLSI implementation of the FitzHugh–Nagumo neuron model, intended to model large-scale, biologically plausible, oscillator networks. As the model requires a series resistor and a parallel capacitor with the inductor, which is the most complex part of the design, it is possible to greatly simplify the active inductor implementation compared to other implementations of this device as typically found in filters by allowing appreciable, but well modelled, nonidealities. We model and obtain the parameters of the inductor nonideal model as an inductance in series with a parasitic resistor and a second order low-pass filter with a large cut-off frequency. Post-layout simulations for a CMOS 0.35 μm double-poly technology using the MOSFET Spice BSIM3v3 model confirm the proper behaviour of the design.     

Unconditionally positivity and boundedness preserving schemes for a FitzHugh–Nagumo equation

In this paper, a semi-explicit scheme is constructed for the space-independent FitzHugh–Nagumo equation. Qualitative stability analysis shows that the semi-explicit scheme is dynamically consistent with the space independent equation. Then, the semi-explicit scheme is extended to construct a new finite difference scheme for the full FitzHugh–Nagumo equation in one- and two-space dimensions, respectively. According to the theory of M-matrices, it is proved that these new schemes are able to preserve the positivity and boundedness of solutions of the corresponding equations for arbitrary step sizes. The consistency and numerical stability of these schemes is also analysed. Combined with the property of the strictly diagonally dominant matrix, the convergence of these schemes is established. Numerical experiments illustrate our results and display the advantages of our schemes in comparison to some other schemes.     

Delay-induced Hopf bifurcation in a noise-driven excitable neuron model

A FitzHugh–Nagumo neuron model with cubic nonlinearity and discrete delay is considered, in which the time delay is regarded as a parameter. The effect of time delay on the linear stability and Hopf bifurcation of the model is studied. The existence, stability and direction of the local and global Hopf bifurcation are derived. Some numerical simulations are employed to validate the main results of this work.