The bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium. It is composed of coupled parabolic and elliptic partial differential equations, as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes. The purpose of this paper is to propose and analyze several implicit, semi-implicit, and explicit time-stepping methods to solve that model, in particular to avoid the expensive resolution of a nonlinear system through the Newton-Raphson method. We identify necessary stability conditions on the time step Δt for the proposed methods through a theoretical analysis based on energy estimates. We next compare the methods for oneand two-dimensional test cases, in terms of both stability and accuracy of the numerical solutions. The theoretical stability conditions are seen to be consistent with those observed in practice. Our analysis allows us to recommend using either the Crank-Nicolson/Adams-Bashforth method or the second order semi-implicit backward differention method. These semi-implicit methods produce a good numerical solution; unlike the explicit methods, their stability does not depend on the spatial grid size; and unlike the implicit methods, they do not require the resolution of a system of nonlinear equations. Introduction.The bidomain model is used in electrophysiology to model the propagation of electrical potential waves in the myocardium. It is obtained by homogenization over the discrete cells of the myocardium, in particular using the regular arrangement of these in fibers [7,14]. At each point in the computational domain, two electrical potentials, namely, the intracellular potential u i and the extracellular potential u e , are recovered, representing the average of the electrical potential over the extracellular and the intracellular space, respectively, in the vicinity of that point.There are many ways to write the bidomain model's equations. Many of them are introduced in [11], where their respective merits are also discussed. We will use the following formulation:

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including a property of stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.

The computation of multidimensional persistent Betti numbers for a sublevel filtration on a suitable topological space equipped with a R n-valued continuous filtering function can be reduced to the problem of computing persistent Betti numbers for a parameterized family of one-dimensional filtering functions. A notion of continuity for points in persistence diagrams exists over this parameter space excluding a discrete number of so-called singular parameter values. We have identified instances of nontrivial monodromy over loops in nonsingular parameter space. In other words, following cornerpoints of the persistence diagrams along nontrivial loops can result in them switching places. This has an important incidence, e.g., in computer-assisted shape recognition, as we believe that new, improved distances between shape signatures can be defined by considering continuous families of matchings between cornerpoints along paths in nonsingular parameter space. Considering that nonhomotopic paths may yield different matchings will therefore be necessary. In this contribution we will discuss theoretical properties of the monodromy in question and give an example of a filtration in which it can be shown to be nontrivial.

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.

Abstract. Bidomain models are commonly used for studying and simulating electrophysiological waves in the cardiac tissue. Most of the time, the associated PDEs are solved using explicit finite difference methods on structured grids. We propose an implicit finite element method using unstructured grids for an anisotropic bidomain model. The impact and numerical requirements of unstructured grid methods is investigated using a test case with re-entrant waves.Mathematics Subject Classification. 35K57, 65M60, 92Cxx.