Abstract
The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.
Highlights
Among these is the bidomain model [54], which is regarded as an apt spatial model of the electrical properties of cardiac tissue [13,52]
The idiom “bidomain” reflects that the intra- and extracellular tissues are viewed as two superimposed anisotropic continuous media, with different longitudinal and transversal conductivities. If these conductivities are equal, we have the so-called monodomain model
The purpose of the present paper is to introduce and analyze a bidomain model that accounts for random effects, by way of a few well-placed stochastic terms
Summary
Hodgkin and Huxley [29] introduced the first mathematical model for the propagation of electrical signals along nerve fibers. Conduction of electrical signals in cardiac tissue rely on the flow of ions through so-called ion channels in the cell membrane. This similarity has led to a number of cardiac models based on the Hodgkin–Huxley formalism [11,13,32,42,45,52]. The idiom “bidomain” reflects that the intra- and extracellular tissues are viewed as two superimposed anisotropic continuous media, with different longitudinal and transversal conductivities If these conductivities are equal, we have the so-called monodomain model (elliptic PDE reduces to an algebraic equation). For work devoted to cardiac cells, see [19,36,42]
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