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Biology Articles » Biomathematics » A mathematical model for electrical stimulation of a monolayer of cardiac cells » Methods

The equation governing a two-dimensional sheet of tissue perfused by a three-dimensional bath is similar to the equation derived by Rattay for a one-dimensional nerve axon [13]. The tissue obeys the conservation of current,

where β is the ratio of membrane surface area to tissue volume, J_{m }is the membrane current density, and J_{ix }and J_{iy }represent x and y components of the intracellular current density. Ohm's Law gives

J_{m }= G_{m }V_{m}, (4)

where V_{i }is the intracellular potential, V_{m }is the transmembrane potential, G_{m }is membrane conductivity per unit area, and σ_{ix }and σ_{iy }are the intracellular conductivities parallel to (x) and perpendicular to (y) the fiber axis. Substituting Eqs. (2), (3), and (4) into Eq. (1), and letting V_{i }= V_{m }+ V_{e}, we find that

where λ_{x }and λ_{y }are defined as

In order to solve Eq. (5), we must first determine the extracellular potential, V_{e}. We assume that V_{e }is from a point electrode in an infinite, homogeneous bath

where , σ_{e }is the conductivity of the bath, I is the stimulus current, and d is the distance from the tissue sheet (z = 0) to the electrode.

We discretize Eq. (5) using a finite difference formula, and solve it using a relaxation method. The number of nodes in each direction is 100, the space step is 0.2 mm (implying a tissue size of 20 × 20 mm), σ_{e }is 1 S/m, I is 1 mA, and d is 1 mm.

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