Homogenized continuum models in the form of homogeneous partial differential equations are employed frequently to simplify complex and multi-scale phenomena, such as biochemical ones. However, for some particular problems, for example, under weak coupling, such homogenization techniques and continuum models may fail to capture important features of the phenomena. Possible alternatives are detailed and multi-scale models, usually based on heterogeneous partial differential equations, or discrete models with effective or homogenized parameters. We discuss the limitations of these models and present a new mathematical model that overcomes these limitations. This new model is cast as a partial differential equation but with a parameter that depends not only on the heterogeneities of the phenomena, but also on the discretization mesh. We apply this model to describe different phenomena: the propagation of action potential in cardiac tissue, in myelinated axons of neurons, and the concentration waves of chemical reactions.
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