The nonlinear multidomain model: a new formal asymptotic analysis

Amato, Stefano; Bellettini, Giovanni; Paolini, Maurizio

doi:10.1007/978-88-7642-473-1_2

Cited by 11 publications

(5 citation statements)

References 16 publications

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“…Related speculation on the behavior of fronts for the bidomain Allen-Cahn equation is discussed in [3][4][5][6]9]. What distinguishes our conjecture from previous discussion is the presence of the planar front stability condition.…”

Section: Spreading Frontsmentioning

confidence: 75%

“…First, the anisotropic Allen-Cahn equation is wellposed only if the function A.k/ (D Q.k/=2) is convex, while no such restriction is needed for Q in (1.5) in the bidomain Allen-Cahn equation. The symbol Q can indeed be nonconvex for a large class of positive symmetric definite matrices A i and A e [3,5,6,9]. Another important difference concerns the maximum principle.…”

Section: Introductionmentioning

confidence: 98%

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Stability of Front Solutions of the Bidomain Equation

Mori

Matano

2016

Comm Pure Appl Math

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The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen-Cahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical AllenCahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under longwavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.

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Section: Spreading Frontsmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 98%

Stability of Front Solutions of the Bidomain Equation

Mori

Matano

2016

Comm Pure Appl Math

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“…We refer to [3,40] for works investigating a generalized multidomain system and eikonal models with non-convex indicatrix function˚.…”

Section: Well-posedness Results For Reduced Modelsmentioning

confidence: 99%

“…[104,125] These two choices allow us to also test the performance of our multilevel preconditioner in presence of severe domain deformations. The values of the conductivity coefficients used in the numerical tests are reported in Table 8.1 and the capacitance per unit volume is set to c m D 1 mF cm 3 . The values of the coefficients and parameters in the LR1 model are given in the original paper [308].…”

Section: Numerical Results For Multilevel Schwarz Preconditionersmentioning

confidence: 99%

“…The cardiac domain˝H considered in this study is a cartesian slab of dimensions 0:96 0:96 0:32 cm 3 , modeling a portion of the left ventricular wall. The endocardial surface is in contact with a smaller slab of dimensions 0:96 0:96 0:16 cm 3 , modeling the extracardiac bath, where b D 6e 3 1 cm 1 similar to the blood conductivity. Stimulation site.…”

Section: Myocardial Conductivity Tensors and Fiber Architecturementioning

confidence: 99%

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