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              <p>The bidomain model is the current most sophisticated model used in cardiac electrophysiology. The monodomain model is a simplification of the bidomain model that is less computationally intensive but only valid under equal conductivity ratio. We propose in this paper optimal monodomain approximations of the bidomain model. We first prove that the error between the bidomain and monodomain solutions is bounded by the error $\|B-A\|$ between the bidomain and monodomain conductivity operators. Optimal monodomain approximations are defined by minimizing the distance $\|B-A\|$, which reduces for solutions over all $\rr^d$ to minimize the $L^p$ norm of the difference between the operator symbols. Similarly, comparing the symbols pointwise amounts to compare the propagation of planar waves in the bidomain and monodomain models. We prove that any monodomain model properly propagates at least $d$ planar waves in $\rr^d$. We next consider and solve the optimal problem in the $L^\infty$ and $L2$ norms, the former providing minimal propagation error uniformly over all directions. The quality of these optimal monodomain approximations is compared among themselves and with other published approximations using two sets of test cases. The first one uses periodic boundary conditions to mimic propagation in $\rr^d$ while the second is based on a square domain with common Neumann boundary conditions. For the first test cases, we show that the error on the propagation speed is highly correlated with the error on the symbols. The second test cases show that domain boundaries control propagation directions, with only partial impact from the conductivity operator used.</p>
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              <p>Le modèle bidomaine est le plus complet pour représenter la propagation des potentiels d'action cardiaque. Le modèle monodomaine est une simplification de celui-ci, moins coûteuse numériquement, mais valable seulement sous une hypothèse d'égale anisotropie. Nous proposons des approximations optimales de type monodomaine d'un modèle bidomaine fixé. Nous montrons d'abord que l'erreur entre les solutions du bidomaine et du monodomaine est bornée par la différence $\|B-A\|$ entre les opérateurs correspondants. Un opérateur monodomaine optimal doit donc minimiser la distance $\|B-A\|$, ce qui est équivalent dans $\rr^d$ à minimiser l'erreur entre les symboles des opérateurs en norme $L^p$. On peut aussi comparer ces symboles dans une direction donnée, ce qui revient à comparer la propagation des ondes planes dans cette direction. Nous démontrons qu'un modèle monodomaine peut propager $d$ ondes planes d'un modèle bidomaine. Puis nous résolvons le problème de minimisation sur les symboles en norme $L^\infty$ et $L^2$. Dans le premier cas, on minimise donc l'erreur de propagation des ondes planes de manière uniforme sur les directions. Nous comparons ces approximations avec celles de la littérature sur deux séries de cas tests. D'abord avec des conditions aux limites périodiques pour retrouver les résultats démontrés dans $\rr^d$, puis sur un domaine borné avec conditions de Neumann homogènes. Dans le premier cas, l'erreur de propagation est corrélée à la distance entre les symboles, alors que dans le second, les conditions aux limites dirigent le front et perturbent les résultats théoriques.</p>
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