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ContinuousMathematicsThe BidomainEquations are a coupled set of partial differential equations defined over a continuous space. The "bi-" refers to the conceptual separation of intracellular and extracellular space, although both are interpenetrating and occupy exactly the same space. The separation comes in through the cellular membrane - current which leaves one domain must go through the membrane before reaching the other domain. This membrane is where the non-linearities arise. The BidomainEquations are shown here:  The voltages Phi_i and Phi_e are for the intra- and extra-cellular domains, respectively. The transmembrane voltage, V_m, is the difference: V_m = Phi_i - Phi_e. The q variable is actually a group of state variables. The M(q,V_m) and I_ion terms use these state variables to produce the non-linear membrane effects. Beta is the surface-to-volume ratio for the tissue (ie. a constant), C_m is the membrane capacitance (also a constant), the S_i and S_e terms are optional stimulus currents, usually used to initially drive the system to activation. Finally, the sigma_i and sigma_e terms are spatially variant conductivity tensors which describe the flow of electrical current through the intra- and extra-cellular domains, respectively. We often make a simplification in the math to ease the computational burdens of the solution. In particular, we may assume that the extracellular space is highly conductive (ie. sigma_e is much larger than sigma_i) and that the extracellular domain is grounded, hence Phi_e = 0 and one of the above equations drops out. A similar reduction occurs if we assume that the sigma_i and sigma_e tensors are simple multiples of one another, it then turns out that both Phi_i and Phi_e are simple multiples of V_m and the bottom two equations can be merged. The results are called the MonodomainEquations and are shown here:  This reduces both the memory requirements of the simulation (only one voltage to store instead of two) and also reduces the amount of computation we need to perform. See also DiscreteMathematics |