Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations

Fitzhugh, Richard

doi:10.1085/jgp.43.5.867

Cited by 445 publications

(214 citation statements)

References 7 publications

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“…The two-current model is similar in spirit to the FitzHugh-Nagumo model (FitzHugh, 1960(FitzHugh, , 1961), but we believe that the two-current model is more closely related to the physiology of the heart for two reasons. A minor issue: the twocurrent model does not exhibit voltage overshoot, a phenomenon observed in neural but not cardiac tissue.…”

Section: The Modelmentioning

confidence: 87%

A two-current model for the dynamics of cardiac membrane

Mitchell

Schaeffer

2003

Bulletin of Mathematical Biology

298

318

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In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon-subcritical alternansthat does not occur for the commonly used exponential map.

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Section: The Modelmentioning

confidence: 87%

A two-current model for the dynamics of cardiac membrane

Mitchell

Schaeffer

2003

Bulletin of Mathematical Biology

298

318

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“…; a n j Þ 2 R n , (j = 1, 2, 3), are three distinct constant vectors, and ðU À a 1 ; U À a 2 Þ P n i¼1 ðU i À a i 1 ÞðU i À a i 2 Þ means the Euclidean scalar product of vectors (U À a 1 ) and (U À a 2 ) and $ 2 = D = o 2 /ox 2 + o 2 /oy 2 + o 2 /oz 2 is the Laplace operator, a, b are real constants. In the scalar case, when n = 1, a = 0, and in the one space dimension, for different choices of parameters a 1 , a 2 , a 3 the model reduces to the well-known nonlinear diffusion equations appearing in a different fields sometimes with different names: the Fitzhugh-Nagumo equation (a 1 = 0, a 2 = 1, a 3 = a) arising in population genetics [9] and models the transmission of nerve impulse [10], autocatalytic chemical reaction model introduced by Schlögl [11,12], generalized Fisher equation [2], Newell-Whitehead equation [13] or Kolmogorov-Petrovsky-Piscounov equation [14] (a 1 = 0, a 2 = 1, a 3 = À 1), Huxley equation (a 1 = a 2 = 0, a 3 = 1).…”

Section: Introductionmentioning

confidence: 99%

Solitary wave solution of nonlinear multi-dimensional wave equation by bilinear transformation method

Tanoğlu

2007

Communications in Nonlinear Science and Numerical Simulation

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The Hirota method is applied to construct exact analytical solitary wave solutions of the system of multi-dimensional nonlinear wave equation for n-component vector with modified background. The nonlinear part is the third-order polynomial, determined by three distinct constant vectors. These solutions have not previously been obtained by any analytic technique. The bilinear representation is derived by extracting one of the vector roots (unstable in general). This allows to reduce the cubic nonlinearity to a quadratic one. The transition between other two stable roots gives us a vector shock solitary wave solution. In our approach, the velocity of solitary wave is fixed by truncating the Hirota perturbation expansion and it is found in terms of all three roots. Simulations of solutions for the one component and one-dimensional case are also illustrated.

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“…A classical Fitzhugh-Nagumo PDE model [4,10] has been implemented here as a first attempt to perform numerical simulations on the computational meshes. The figures show what result can be obtained with a conventional computer equipment in a few hours.…”

Section: Introductionmentioning

confidence: 99%

Generation of computational meshes from MRI and CT-scan data

Sermesant

Frey

2005

ESAIM: Proc.

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Abstract. There are fields of engineering where accurate and personalised data are required; biomedical applications are one such example. We describe here a general purpose method to create computational meshes based on the analysis and segmentation of raw medical imaging data. The various ingredients are not new: a segmentation method based on deformable contours and a surface and volume mesh adaptation method based on discrete metric specifications; but the challenge that motivated this paper is to put them together in an attempt to design an automatic, easy to use and efficient 3D code.For non-engineering (like biomedical) applications, the user interface is often a key point. In this project, we put a great deal of emphasis on the automation of the whole procedure, making then possible to envisage large scale simulations with a minimal amount of user interaction. In particular, the user knowledge is required to help segmenting the image (the user is expected to have the knowhow of the body anatomy), all meshing steps rely on fully automatic algorithms depending on a few parameters (for the external surface approximation).One application example is presented and commented, for which the data preparation takes a few hours and results can be obtained overnight on a PC workstation.Résumé. Dans certains domaines du calcul scientifique il est nécessaire de disposer de données précises et personnalisées; les applications biomédicales en sont un bon exemple. Nous présentons ici une méthode générique permettant de générer des maillages de calculà partir de l'analyse et de la segmentation de données médicales discrètes. Si les divers composants ne sont pas nouveaux: une méthode de segmentation d'images basées sur des contours déformables et des techniques d'adaptation de maillage de surfaces et de volumes basées sur la notion de métrique, le challenge est ici d'assembler ces constituants de manièreà obtenir un code de simulation efficace et facileà utiliser.Pour des applications telles que biomédicales, l'interface utilisateur est un point clé. Dans ce projet, nous avons particulièrement soigné l'automatisation de la procédure, ce qui permet d'envisager des simulations de larges tailles pratiquement sans intervention de l'utilisateur. En particulier, les compétences de celui-ci (i.e., ses connaissances en anatomie) sont utilisées pour la segmentation des images, lesétapes de maillageétant entièrement automatisées et ne faisant intervenir qu'un ensemble réduit de paramètres (contrôle de l'approximation des surfaces notamment). * We are grateful to the CEA-LIST for funding this work

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Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations

Cited by 445 publications

References 7 publications

A two-current model for the dynamics of cardiac membrane

A two-current model for the dynamics of cardiac membrane

Solitary wave solution of nonlinear multi-dimensional wave equation by bilinear transformation method

Generation of computational meshes from MRI and CT-scan data

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