Kirchhoff's laws:

• Total current is conserved
  
  $$\nabla\cdot (\sigma_i\nabla\phi_i + \sigma_e \nabla\phi_e) = 0$$
  
  

• transmembrane current is balanced
  
  

  
  $$\chi (C_m{\partial\phi \over \partial \tau} + I_m ) = \nabla\cdot (\sigma_i\nabla\phi_i)$$

  
  

Individual terms:

• $C_m{\partial\phi \over \partial \tau}$ capacitive current
• $I_m$ transmembrane ionic current
• $\chi$ surface to volume ratio
• $\nabla\cdot( \sigma_i\nabla\phi_i)$ outward intracellular current

Boundary conditions:

$${\rm n}\cdot\sigma_i \nabla\phi_i = 0,\hspace{0.5in} {\rm n}\cdot\sigma_e\nabla\phi_e = I(t,x)$$

for $x \in \partial\Omega$, the boundary of the cardiac domain, and n is the outward unit normal to the boundary, and zero net current

$$\int_{\partial\Omega} I(t,x) dS = 0.$$