Derivation of nonlinear time-dependent macroscopic conductivity for an electropermeabilization model via homogenization
We study a phenomenological electropermeabilization model in a periodic medium representing biological tissue. Starting from a cell-level model describing the electric potential and the degree of porosity, we perform dimension analysis to identify a relevant scaling in terms of a small parameter $\ve$ - the ratio between the cell and the tissue size. The electric potential satisfies electrostatic equations in the extra- and intracellular domains, while its jump across the cell membrane evolves according to a nonlinear law coupled with an ordinary differential equation for the porosity degree. We prove the well-posedness of the microscopic problem, derive a priori estimates, obtain formal asymptotics, and rigorously justify the expansion combining two-scale convergence with monotonicity arguments. The resulting macroscopic model exhibits memory effects and a nonlinear, time-dependent effective current. It captures the nontrivial evolution of effective conductivity, including a characteristic drop reflecting the capacitive behavior of the lipid bilayer, in agreement with experimental data. Numerical computations of the effective conductivity confirm that, although microscopic conductivity is constant, tissue conductivity varies nonlinearly with electric field strength, showing a sigmoid trend. This suggests a rigorous mathematical explanation for experimentally observed conductivity dynamics.