arXiv subjects

Irina Pettersson

Publications and source records attributed to Irina Pettersson.

10 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2026

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.

math.AP↗

arXiv2025

Homogenization of an indefinite spectral problem arising in population genetics

We study an indefinite spectral problem for a second-order self-adjoint elliptic operator in an asymptotically thin cylinder. The operator coefficients and the spectral density function are assumed to be locally periodic in the axial direction of the cylinder. The key assumption is that the spectral density function changes sign, which leads to infinitely many both positive and negative eigenvalues. The asymptotic behavior of the spectrum, as the thickness of the rod tends to zero, depends essentially on the sign of the average of the density function. We study the positive part of the spectrum in a specific case when the local average is negative. We derive a one-dimensional effective spectral problem that is a harmonic oscillator on the real line, and prove the convergence of spectrum. A key auxiliary result is the existence of a positive principal eigenvalue of an indefinite spectral problem with the Neumann boundary condition on a periodicity cell. This study is motivated by applications in population genetics where spectral problems with sign-changing weight naturally appear.

math.AP↗

arXiv2025

Nonlinear multidomain model for nerve bundles with random structure

We present a derivation of a multidomain model for the electric potential in bundles of randomly distributed axons with different radii. The FitzHugh-Nagumo dynamics is assumed on the axons' membrane, and the conductivity depends nonlinearly on the electric field. Under ergodicity conditions, we study the asymptotic behavior of the potential in the bundle when the number of the axons in the bundle is sufficiently large and derive a macroscopic multidomain model describing the electrical activity of the bundle. Due to the randomness of geometry, the effective intracellular potential is not deterministic but is shown to be a stationary function with realizations that are constant on axons' cross sections. The technique combines the stochastic two-scale convergence and the method of monotone operators.

math.AP↗

arXiv2024

Cell Electropermeabilization Modeling via Multiple Traces Formulation and Time Semi-Implicit Coupling

We simulate the electrical response of multiple disjoint biological 3D cells undergoing an electropermeabilization process. Instead of solving the boundary value problem in the unbounded volume, we reduce it to a system of boundary integrals equations--the local Multiple Traces Formulation--coupled with nonlinear dynamics on the cell membranes. Though in time the model is highly non-linear and poorly regular, the smooth geometry allows for boundary unknowns to be spatially approximated by spherical harmonics. This leads to spectral convergence rates in space. In time, we use a multistep semi-implicit scheme. To ensure stability, the time step needs to be bounded by the smallest characteristic time of the system. Numerical results are provided to validate our claims and future enhancements are pointed out.

cs.CE↗

arXiv2022

Derivation of a bidomain model for bundles of myelinated axons

The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHugh-Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of monotone operators.

math.AP↗

arXiv2021

A feedforward neural network for modelling of average pressure frequency response

The Helmholtz equation has been used for modelling the sound pressure field under a harmonic load. Computing harmonic sound pressure fields by means of solving Helmholtz equation can quickly become unfeasible if one wants to study many different geometries for ranges of frequencies. We propose a machine learning approach, namely a feedforward dense neural network, for computing the average sound pressure over a frequency range. The data is generated with finite elements, by numerically computing the response of the average sound pressure, by an eigenmode decomposition of the pressure. We analyze the accuracy of the approximation and determine how much training data is needed in order to reach a certain accuracy in the predictions of the average pressure response.

cs.LG↗

arXiv2018

Derivation of cable equation by multiscale analysis for a model of myelinated axons

The paper concerns the multiscale modeling of a myelinated axon. Taking into account the microstructure with alternating myelinated parts and nodes Ranvier, we derive a nonlinear cable equation describing the potential propagation along the axon. We assume that the myelin is not a perfect insulator, and assign a low (asymptotically vanishing) conductivity in the myelin. Compared with the case when myelin is assumed to have zero conductivity, an additional potential arises in the limit equation. The coefficient in front of the effective potential contains information about the geometry of the myelinated parts.

math.AP↗

arXiv2017

Stationary convection-diffusion equation in an infinite cylinder

We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

math.AP↗

arXiv2017

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

The aim of this paper is to adapt the notion of two-scale convergence in $L^p$ to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients in a thin cylinder with locally periodic rapidly varying thickness.

math.AP↗