Fast and Slow Waves in the FitzHugh–Nagumo Equation

Krupa, Martin; Sandstede, Björn; Szmolyan, Peter

doi:10.1006/jdeq.1996.3198

Cited by 115 publications

(118 citation statements)

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“…In Figure 7.2 we plot (a, c) curves for one-pulse solutions (these are actually (a, c) curves obtained when there exists ξ 1 > 0 such that g(ξ 1 ) = 0) for various values of the parameters d, γ, b, r. The plot illustrates the difference between the behavior with continuous and discrete diffusion. In the case of continuous diffusion it is known that the fast waves, i.e., those above the tip on the (a, c) curve, are stable (see [43,30,25,29] Figure 7.8 is obtained by superimposing two identical one-pulse solutions, using the superimposed one-pulse solutions as an initial guess and then applying Newton's method. The onepulse solution is obtained from (d, γ, c, b, r) = (1, 0, 1, 1, 0) and the pulses are put at a distance (in ξ) of 40 units apart.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Notable work on the existence and stability on monotone traveling fronts for the Nagumo PDE include that of Aronson and Weinberger [2] and Fife and McLeod [21] and the original work of Nagumo, Arimoto, and Yoshizawa [34]. Existence and stability of fronts and pulses for the FitzHugh-Nagumo PDE begins from the work of FitzHugh [22] (see also [34]) and includes the work on stability of Jones [25], Maginu [30], and Yanagida [43] (see also [29]), the work on existence of Deng [10] and existence and stability results of Evans [15,16,17,18], Feroe [20], Wang [41,42], Rinzel and Keller [35], and McKean [33] for the piecewise linear nonlinearity considered here. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Spatially Discrete FitzHugh--Nagumo Equations

Elmer

Vleck²

2005

SIAM J. Appl. Math.

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Abstract. We consider pulse and front solutions to a spatially discrete FitzHugh-Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discrete diffusion terms.Key words. traveling fronts and pulses, discrete diffusion AMS subject classifications. 35K57, 74N99 DOI. 10.1137/S003613990343687X1. Introduction. By considering an electrical circuit model with a complicated nonlinear resistor, Hodgkin and Huxley (along with Katz) modeled the ionic conductances that generate the action potential of nerve fibers. To develop their model they performed voltage and space clamping experiments on the giant axon of squids, axons which were relatively easy to work with because of their size. The Hodgkin-Huxley equations are a four-variable model which may be reduced to a two-variable model (the FitzHugh-Nagumo (FH-N) ODE model) which preserves much of the dynamics of the Hodgkin-Huxley system by considering fast and slow variables and slaving the other variables. When considering a chain of electrical circuits, diffusion is added as a means of propagation in the spatial variable, thus obtaining the FH-N PDE model. Since the seminal work of Hodgkin, Huxley, and Katz, similar experiments have been performed on nerve axons of vertebrates and its been discovered that, electrically, nerve fibers behave as spatially discrete periodic structures in vertebrates. This is due to the periodically spaced active channels (nodes of Ranvier) in the myelin insulation (in the coating by Schwann cells or oligodendrocytes). Thus it is not only appropriate but correct to model motor nerves in vertebrates with equations which also have a spatially discrete periodic structure, to model with nonlinear differentialdifference equations (DDEs), in particular an FH-N DDE model. Our contribution in this paper is to consider front and pulse solutions for a FitzHugh-Nagumo system with both continuous and discrete diffusion, thus allowing one to compare and contrast the dynamics generated by spatially continuous and spatially discrete models of action potential propagation. By employing a piecewise linear bistable nonlinearity we reduce the problem to a linear inhomogeneous equation, for which candidate solutions can be derived using transform methods. The candidate solutions are then shown to be consistent with our ansatz of a front or pulse solution, a necessary condition for existence. We focus on one-front and one-pulse solutions and prove their existence using two approaches: (1) we show that consistency of the

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spatially Discrete FitzHugh--Nagumo Equations

Elmer

Vleck²

2005

SIAM J. Appl. Math.

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“…Results on inclination-flips in Z 2 -equivariant ODEs, in which Lorenz-like strange attractors appear, can be found in Theorems 5.61 and 5.81 below. Applications in which inclination-flips appear include travelling waves in FitzHugh-Nagumo equation [240], 1 : 2 spatial resonances in systems with broken O(2) symmetry [313], and models for instabilities in thermal convection [293].…”

Section: Inclination Flipsmentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

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164

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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“…In fact, careful path-following techniques for homoclinic orbits [12] reveal that solitary pulse solutions lie on a C-shaped curve to the left of the U in the (p, s) parameter plane, which is also depicted in Figure 1. For more on computations of such traveling wave solutions in various different forms of the FitzHugh-Nagumo model, see [11,36] and references therein. The C-shape of the pulse curve implies that more than one solitary pulse exists for a range of values of p, with two different wave speeds.…”

mentioning

confidence: 99%

When Shil'nikov Meets Hopf in Excitable Systems

Champneys¹,

Kirk²,

Knobloch³

et al. 2007

SIAM J. Appl. Dyn. Syst.

111

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Abstract. This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh-Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a Ushaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these "CU systems." A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov-Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov-Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

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Fast and Slow Waves in the FitzHugh–Nagumo Equation

Cited by 115 publications

References 24 publications

Spatially Discrete FitzHugh--Nagumo Equations

Spatially Discrete FitzHugh--Nagumo Equations

Homoclinic and Heteroclinic Bifurcations in Vector Fields

When Shil'nikov Meets Hopf in Excitable Systems

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