Long-time behavior of a nonlocal and non-monotone SPDE-ODE system arising in electrophysiology

Tobias Gebäck; Oleksandr Misiats; Ioanna Motschan Ulander; Irina Pettersson

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arXiv · 2604.11260

Long-time behavior of a nonlocal and non-monotone SPDE-ODE system arising in electrophysiology

Tobias Gebäck·Oleksandr Misiats·Ioanna Motschan Ulander·Irina Pettersson

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Abstract

This paper concerns a coupled semilinear SPDE-ODE system modelling the electropermeabilization phenomenon, which designates a transient increase in cell membrane permeability induced by short, high-voltage electric pulses. We present a stochastically perturbed electroporation model that couples electrostatic equations for the electric potential in the extra- and intracellular domains and a nonlinear evolution law for the transmembrane potential jump with an ordinary differential equation describing the porosity degree of the membrane. We prove the existence and uniqueness of a variational solution of the resulting coupled stochastic PDE-ODE system. Its long-time behavior is governed by the corresponding invariant measure for which we establish the regularity of its support. The ergodicity of this invariant measure is further established for a truncated nonlinear reaction term, corresponding to the case of a bounded electric potential. The main technical challenge arises from the nonlinear reaction term, which is neither Lipschitz continuous nor locally monotone. We also present a numerical example, computing the solution and its time averages for both additive and multiplicative noise, that provides an indication for the existence of an invariant measure.

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Tobias Gebäck, Oleksandr Misiats, Ioanna Motschan Ulander, Irina Pettersson. 2026-04-13. Long-time behavior of a nonlocal and non-monotone SPDE-ODE system arising in electrophysiology. https://arxiv.org/abs/2604.11260

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